Analysis and optimal control for chemotaxis-consumption models
Abstract
En esta tesis investigamos el siguiente modelo de quimiotaxis-consumo en dominios acotados de RN (N = 1, 2, 3): ∂tu − Δu = −∇ · (u∇v), ∂tv − Δv = −usv, donde s ≥ 1, dotado de condiciones de contorno aisladas y condiciones iniciales para (u, v), con u y v representando la densidad de células y la concentración de la señal química, respectivamente. Bajo hipótesis poco exigentes sobre la regularidad del dominio y a través de la convergencia de las soluciones de un modelo truncado adecuado, se establecen dos resultados principales: existencia de soluciones débiles uniformes en el tiempo en dominios 3D, y unicidad y regularidad en dominios 2D (o 1D). Utilizando la teoría desarrollada en este análisis teórico, proponemos y estudiamos un esquema discreto en tiempo implícito tipo Backward Euler para dicho modelo combinado con el uso de una variable auxiliar, probando existencia de solución, estimaciones a priori uniformes en el tiempo y convergencia hacia una solución débil (u, v) del modelo quimiotaxis-consumo. A continuación abordamos problemas de control óptimo sujetos al siguiente modelo de quimiotaxis-consumo controlado de forma bilineal en un dominio acotado Ω ⊂ R3 durante un intervalo de tiempo (0, T): ∂tu − Δu = −∇ · (u∇v), ∂tv − Δv = −usv + fv1Ωc , siendo f el control que actúa en un subdominio Ωc ⊂ Ω. En primer lugar, abordamos un problema de control óptimo relacionado con las soluciones débiles del modelo de quimiotaxis-consumo controlado. Demostramos la existencia de soluciones débiles que satisfacen una desigualdad de energía, la existencia de control óptimo sujeto a controles acotados y discutimos la relación entre el problema de control considerado y otros dos relacionados que pueden ser de interés. A continuación estudiamos un problema de control óptimo sujeto a soluciones fuertes del citado modelo de quimiotaxis-consumo controlado. Demostramos un criterio de regularidad que nos permite obtener existencia y unicidad de soluciones fuertes globales en el tiempo, mostramos la existencia de una solución óptima global y, utilizando un teorema de multiplicadores de Lagrange, establecemos condiciones de optimalidad de primer orden para cualquier solución óptima local, probando existencia, unicidad y regularidad de los multiplicadores de Lagrange asociados. Finalmente, en el capítulo de conclusiones, discutimos una serie de posibles trabajos futuros relacionados con los resultados presentados en esta tesis.
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University of Seville Doctorate Program “Mathematics” Doctoral Thesis ANALYSIS AND OPTIMAL CONTROL FOR CHEMOTAXIS-CONSUMPTION MODELS Author: André Luiz Corrêa Vianna Filho Supervisor: Francisco Manuel Guillén-González SEVILLA 2023
ANDRÉ LUIZ CORRÊA VIANNA FILHO ANALYSIS AND OPTIMAL CONTROL FOR CHEMOTAXIS-CONSUMPTION MODELS Doctoral thesis submitted in partial fulfillment of the requirements for the degree of Doctor in Mathematics at the University of Seville. Supervisor: Francisco Manuel Guillén- González SEVILLA 2023
To my wife, Jessica.
Resumen En esta tesis investigamos el siguiente modelo de quimiotaxis-consumo en dominios acotados de RN(N= 1,2,3): ∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv, donde s≥1, dotado de condiciones de contorno aisladas y condiciones iniciales para (u, v), con uyvrepresentando la densidad de células y la concentración de la señal química, respectivamente. Bajo hipótesis poco exigentes sobre la regularidad del dominio y a través de la convergencia de las soluciones de un modelo truncado adecuado, se establecen dos resultados principales: existencia de soluciones débiles uniformes en el tiempo en dominios 3D, y unicidad y regularidad en dominios 2D (o 1D). Utilizando la teoría desarrollada en este análisis teórico, proponemos y estudiamos un esquema discreto en tiempo implícito tipo Backward Euler para dicho modelo combinado con el uso de una variable auxiliar, probando existencia de solución, estimaciones a priori uniformes en el tiempo y convergencia hacia una solución débil (u, v)del modelo quimiotaxis-consumo. A continuación abordamos problemas de control óptimo sujetos al siguiente modelo de quimiotaxis-consumo controlado de forma bilineal en un dominio acotado Ω⊂R3durante un intervalo de tiempo (0, T): ∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv+fv1Ωc, siendo fel control que actúa en un subdominio Ωc⊂Ω. En primer lugar, abordamos un problema de control óptimo relacionado con las soluciones débiles del modelo de quimiotaxis-consumo controlado. Demostramos la existencia de soluciones débiles que satisfacen una desigualdad de energía, la existencia de control óptimo sujeto a controles acotados y discutimos la relación entre el problema de control considerado y otros dos relacionados que pueden ser de interés. A continuación estudiamos un problema de control óptimo sujeto a soluciones fuertes del citado modelo de quimiotaxis-consumo controlado. Demostramos un criterio de regularidad que nos permite obtener existencia y unicidad de soluciones fuertes globales en el tiempo, mostramos la existencia de una solución óptima global y, utilizando un teorema de multiplicadores de Lagrange, establecemos condiciones de optimalidad de primer orden para cualquier solución óptima local, probando existencia, unicidad y regularidad de los multiplicadores de Lagrange asociados. Finalmente, en el capítulo de conclusiones, discutimos una serie de posibles trabajos futuros relacionados con los resultados presentados en esta tesis.
Abstract In this thesis we investigate the following chemotaxis-consumption model in bounded domains of RN(N= 1,2,3): ∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv, where s≥1, endowed with isolated boundary conditions and initial conditions for (u, v), with uand vrepresenting the cell density and chemical signal concentration, respectively. Under mild regularity assumptions on the domain and through the convergence of solutions of an adequate truncated model, two main results are established: existence of uniform in time weak solutions in 3D domains, and uniqueness and regularity in 2D (or 1D) domains. Using the theory developed in this theoretical analysis, we propose and study a Backward Euler implicit time discrete scheme combined with the use of an auxiliary variable for the aforementioned model, proving existence of solution, uniform in time a priori estimates and convergence towards a weak solution (u, v)of the chemotaxis-consumption model. In the sequel we approach optimal control problems subject to the following bilinear controlled chemotaxis-consumption model in a bounded domain Ω⊂R3during a time interval (0, T): ∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv+fv1Ωc, with fbeing the control acting in a subdomain Ωc⊂Ω. First, we approach an optimal control problem related to weak solutions of the controlled chemotaxis-consumption model. We prove the existence of weak solutions satisfying an energy inequality, the existence of optimal control subject to bounded controls and discuss the relation between the considered control problem and two other related ones that might be of interest. Next we study an optimal control problem subject to strong solutions of the aforementioned controlled chemotaxis-consumption model. We prove a regularity criterion that allows us to get existence and uniqueness of global-in-time strong solutions, we show the existence of a global optimal solution and, using a Lagrange multipliers theorem, we establish first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers. Finally, in the conclusions chapter, we discuss a series of possible future works related to the results presented in this thesis.
Acknowledgements I thank my supervisor, Professor Francisco, for his guidance. His knowledge and experience were indispensable for the realization of this work. I also thank him for his friendship and encouragement throughout the realization of this thesis. I thank the professors of the Mathematics Department of the University of Seville for their attention and good will in every interaction I had with them. I also acknowledge that this work has been supported by Grant PGC2018-098308- B-I00 (MCI/AEI/FEDER, UE). I thank professors Diego and Silvia for their friendship, motivation and moments of fun. I thank my friend Cori who, together with Francisco, contributed a lot to make my stay in Seville a very happy one. The truth is that all these wonderful people I had the privilege of meeting made me feel at home, even being in a foreign country. I thank my family for all the support I have always received. I thank my wife, Jessica, for all the encouragement and support I always had from her, which were essential for me to be able to reach the end of this stage.
CONTENTS INTRODUCTION 9 Background................................. 10 MainContributions............................. 12 1 PRELIMINARY RESULTS 16 2 UNIFORM IN TIME SOLUTIONS 25 2.1 MainResults ................................ 25 2.2 The Regularized Problem . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Existence and uniqueness of problem (2.1) . . . . . . . . . . . . 28 2.2.2 Regularity up to infinity time of problem (2.1) . . . . . . . . . 29 2.3 ProofofTheorem2.2............................ 35 2.3.1 An energy law appears: formal computations . . . . . . . . . . 35 2.3.2 Rigorous justification of the energy inequalities . . . . . . . . . 36 2.3.3 m-independent estimates and passage to the limit as m→ ∞ 46 m-independent estimates for ∇vm................ 46 m-independent estimates for (um, vm)and passage to the limit for s∈[1,2) ....................... 47 m-independent estimates for (um, vm)and passage to the limit for s≥2......................... 52 2.4 Regularity and Uniqueness in 2D..................... 55 2.4.1 Uniqueness in 2D.......................... 56 2.4.2 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . 58 3 CONVERGENCE OF A TIME DISCRETE SCHEME 62 3.1 MainResults ................................ 62 3.2 Study of the (u, z)-Scheme(3.3) ..................... 64 3.2.1 Existence of solution of (3.3) . . . . . . . . . . . . . . . . . . . 64 3.2.2 First uniform in time estimates . . . . . . . . . . . . . . . . . . 68 3.2.3 Energy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Energy Estimates and Passage to the Limit as (m, k)→(∞,0) . . . 71 3.3.1 Estimates for ∇zk,r m........................ 72 3.3.2 Estimates for (uk,r m, zk,r m)and passage to the limit for s∈[1,2) 73 3.3.3 Estimates for (uk,r m, zk,r m)and passage to the limit for s≥2. 81 3.4 Convergence of vk,r m............................ 83 3.4.1 vk,r mgivenby(3.4)......................... 83
3.4.2 vk,r mgivenby(3.5)......................... 83 4 OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS 85 4.1 MainResults ................................ 85 4.2 Existence for the Bilinear Controlled Model . . . . . . . . . . . . . . . 89 4.2.1 A L∞function that bounds vmfrom above . . . . . . . . . . . 90 4.2.2 Existence for the controlled truncated problem and the first uniformestimates.......................... 92 4.2.3 Energy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2.4 m-independent estimates and passage to the limit as m→ ∞ 96 m-independent estimates for ∇vm............. 96 Case s∈[1,2) .......................... 97 Case s≥2............................. 103 4.3 Existence of an Optimal Control . . . . . . . . . . . . . . . . . . . . . 106 4.3.1 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.2 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . . 109 5 OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS 111 5.1 MainResults ................................ 111 5.2 Regularity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2.1 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Existence of Global Optimal Solution . . . . . . . . . . . . . . . . . . . 122 5.4 First Order Necessary Conditions for a Local Optimal Solution . . . . 123 5.4.1 Abstract setting and a Lagrange multipliers theorem . . . . . . 124 5.4.2 Local optimal solutions are regular points . . . . . . . . . . . . 125 5.4.3 Proof of Theorem 5.6 . . . . . . . . . . . . . . . . . . . . . . . . 128 5.4.4 Proof of Theorem 5.9 . . . . . . . . . . . . . . . . . . . . . . . . 130 CONCLUSION 131 Futureworks ................................ 133 A Hypothesis (H1) 137 B Proof of Lemma 2.10 139 C Existence of solution for a generic linear system 142 BIBLIOGRAPHY 148
9 INTRODUCTION In microbiology, chemotaxis is understood as the directed migration of cells in response to a concentration gradient of a certain chemical substance, either toward attractant chemicals or away from repelents [53]. Chemotaxis plays an essential role in many biological processes such as wound healing, the immune cells migration, the migration of bacteria, among others. It is also an important factor in some undesired events such as tumor growth, cancer metastasis and inflamatory diseases, for example [46, 59]. In unicellular organisms, chemotaxis is frequently related to the search for nutrients and there are studies on its applications to the degradation of polluting substances [47, 48]. The introduction of one of the first mathematical models for chemotaxis is attributed Keller and Segel in two works from 1970 and 1971 [38, 39] which are also regarded by some authors as a development of the work of Patlak [50]. Since then, the research on this topic gave rise to different related models such as models with chemoattraction or chemorepulsion, combined with production or consumption of the chemical substance, with the presence of a logistic growth of the population of cells, models for angiogenesis, haptotaxis and so on, covering a wide variety of applications of very practical interest. From the mathematical point of view, the aforementioned models possess interesting and challenging features that attracted the attention of many authors along the years and make these models still relevant nowadays [4, 33, 34]. Among the various models for chemotaxis that were mentioned, in this thesis, we focus on a model that describes a situation where, inside a bounded and connected region of the d-dimensional space, with d= 1,2,3, the cells are attracted by the concentration gradient of a chemical substance that, in its turn, diffuses and is consumed by the cells. Let Ωbe a bounded domain of Rdand Γbe its boundary. Let u=u(t, x)and v=v(t, x)be the density of cell population and the concentration of chemical substance, respectively, on x∈Ωand t > 0. This model is governed by the initial-boundary PDE problem (∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv, ∂nu|Γ=∂nv|Γ= 0, u(0) = u0, v(0) = v0,(1) where ∇·(u∇v)is the chemotaxis term and usvis the consumption term, with s≥1. ∂nudenotes the normal derivative of uon the boundary. We assume that the initial conditions u0and v0are nonnegative functions. In the present thesis, we aim for contributing for three distinct areas: (i) the analysis, (ii) the numerical approximation and (iii) the optimal control theory related to the
16 Chapter 1 PRELIMINARY RESULTS In this section we present some technical tools which will be used in the rest of the thesis. For p∈[1,∞], we denote by Lp(Ω), the usual Banach spaces of p-integrable Lebesgue-mensurable functions, with the norm ∥·∥Lp(Ω). We denote by p′the conjugate exponent of p. We recall that L2(Ω) is a Hilbert space with the inner product (f, g) = ZΩ f(x)g(x)dx. We also denote by Wl,p(Ω), with l∈N, the usual Sobolev space, equipped with the usual norm ∥·∥Wl,p(Ω); for p= 2, we denote Wl,2(Ω) by Hl(Ω), with norm ∥·∥Hl(Ω). If X(Ω) is a Banach space, then Lp(0, T;X(Ω)) is the Bochner space with the norm ∥v∥Lp(0,T;X(Ω)) =ZT 0∥v(t)∥p X(Ω) dt1/p ,∥v∥L∞(0,T;X(Ω)) = ess sup t∈(0,T)∥v(t)∥X(Ω), C([0, T]; X(Ω)) is the Bochner space of functions defined in [0, T]and continuous with values in X(Ω) and Cw([0, T]; X(Ω)) is the Bochner space of functions defined in [0, T]and weakly continuous with values in X(Ω). To simplify the notation, from now on, we denote the spaces Lp(0, T, X(Ω)) by Lp(X),C([0, T]; X(Ω)) by C(X)and Cw([0, T]; X(Ω)) by Cw(X), suppressing both the time interval (0, T )and the domain Ω. If p= 2 and Xis a Hilbert space then L2(X)is a Hilbert space with the inner product (u, v)L2(X)=ZT 0 (u(t), v(t))Xdt, ∀u, v ∈L2(X), where (·,·)Xdenotes the inner product of X. Next we present some interpolation inequalities and other results which will be of frequent use in the article. Unless otherwise stated, we consider Ω⊂RN(N= 2,3) to be an open, bounded and locally Lipschitz domain, whose boundary we denote by Γ. Lemma 1.1. ([6, 58]) We have the following interpolation inequalities: 1. Let 1≤
Chapter 1. PRELIMINARY RESULTS 17 p < q ≤ ∞,θ∈(0,1) and r∈(p, q), with 1 r=θ p+1−θ q. If f∈Lp(Ω) ∩Lq(Ω) then f∈Lr(Ω) and ∥f∥Lr(Ω) ≤ ∥f∥θ Lp(Ω)∥f∥1−θ Lq(Ω). 2. There exist (different) constants β > 0such that (i) if N= 2 then ∥v∥L4(Ω) ≤β∥v∥1/2 L2(Ω)∥v∥1/2 H1(Ω),∀v∈H1(Ω); (1.1) (ii) if N= 3 then ∥v∥L4(Ω) ≤β∥v∥1/4 L2(Ω)∥v∥3/4 H1(Ω),∀v∈H1(Ω). ∥v∥L3(Ω) ≤β∥v∥1/2 L2(Ω)∥v∥1/2 H1(Ω),∀v∈H1(Ω). ∥v∥L10/3(Ω) ≤C∥v∥2/5 L2(Ω)∥v∥3/5 H1(Ω),∀v∈H1(Ω),(1.2) ∥v∥L10(Ω) ≤C∥v∥4/5 H1(Ω)∥v∥1/5 H2(Ω),∀v∈H2(Ω).(1.3) Lemma 1.2. ([16]) Let Ω⊂RNbe a bounded Lipschitz domain. Then the interpolation inequality ∥w∥Wα,r(Ω) ≤C∥w∥λ Wβ,˜p(Ω)∥w∥1−λ Wγ,˜q(Ω) (1.4) holds for 0≤α, β, γ, λ ≤1and 1<˜p, ˜q, r < ∞such that α=λβ + (1 −λ)γand 1 r=λ ˜p+(1 −λ) ˜q. Lemma 1.3 (Poincare’s Inequality, [14]). There is a constant C > 0such that ∥v−1 |Ω|ZΩ v(x)dx∥W1,p(Ω) ≤C∥∇v∥Lp(Ω),∀v∈W1,p(Ω). Lemma 1.4. ([30]) Let Ω⊂RNand p1, q, p2,˜p, ˜q≥1be such that 1 ˜q=(1 −θ) q+θ1 p1−r N,and 1 ˜p=θ p2 ,with θ∈[0,1] and r > 0, then L∞(Lq)∩Lp2(Wr,p1),→L˜p(L˜q). Lemma 1.5. ([58]) Let Xand Ybe two Banach spaces such that X⊂Ywith a continuous injection. If ϕ∈L∞(0, T;X)and ϕ∈C([0, T]; Y), then ϕ∈Cw([0, T]; X).
18 Chapter 1. PRELIMINARY RESULTS Due to the nature of our problem, it is convenient to have information on the Poisson-Neumann problem (−∆z+z=fin Ω, ∂nz|Γ= 0.(1.5) Then we give the following definition. Definition 1.6. Let z∈H1(Ω) is a weak solution of (1.5) with f∈Lp(Ω). If this implies that z∈W2,p(Ω) with ∥z∥W2,p(Ω) ≤C∥−∆z+z∥Lp(Ω), then we say that the Poisson-Neumann problem (1.5) has the W2,p-regularity. In the hilbertian case p= 2 we say H2-regularity. □ Remark 1.7. According to Grisvard [20], if f∈Lp(Ω),p∈[1,∞], and the boundary Γis at least C1,1, then the Neumann problem (1.5) has the W2,p-regularity for all p∈ [1,∞]. The aforementioned result is also true if Ωis a polygon, that is, a polyhedron in R2, or if Ωis convex and p= 2. For more regular domains, if f∈Wl,p(Ω) and the boundary Γis at least Cl+1,1, then the solution of the Neumann problem (1.5) belongs to Wl+2,p(Ω).□ Lemma 1.8. Let Ωbe a Lipschitz domain such that the Poisson-Neumann problem (1.5) has the W2,p-regularity. There is a constant C > 0such that ∥∇z∥W1,p(Ω) ≤C∥∆z∥Lp(Ω),∀z∈W2,p(Ω) such that ∂nz|Γ= 0.(1.6) Proof. Suppose that the result is false, that is, for each n∈Nthere is zn∈W2,p(Ω) with ∂nzn|Γ= 0 such that ∥∇zn∥W1,p(Ω) > n∥∆zn∥Lp(Ω).(1.7) Without loss of generality, we can take znsuch that ZΩ zndx = 0 and ∥∇zn∥W1,p(Ω) = 1 (1.8) Accounting for (1.7), (1.8) and Lemma 1.3 we have (zn),(∇zn)bounded in W1,p(Ω) and ∆zn−→ ∆z= 0 strongly in Lp(Ω).(1.9) Using the W2,p-regularity of the Poisson-Neumann problem (1.5) we have ∥zn∥W2,p(Ω) ≤C(∥∆zn∥Lp(Ω) +∥zn∥Lp(Ω))
Chapter 1. PRELIMINARY RESULTS 19 and thus (zn)is bounded in W2,p(Ω). This allows us to conclude, using compactness results in Sobolev spaces, that there is z∈W2,p(Ω) such that, up to a subsequence, zn−→ zweakly in W2,p(Ω) and strongly in W1,p(Ω),(1.10) Using again the W2,p-regularity of the Poisson-Neumann problem (1.5) we have ∥zi−zj∥W2,p(Ω) ≤C∥∆zi−∆zj∥Lp(Ω) +C∥zi−zj∥Lp(Ω),∀i, j ∈N, and accounting for (1.10) and (1.9) we conclude that zn−→ zstrongly in W2,p(Ω).(1.11) Now, considering the properties of each znand the convergences (1.9) and (1.11) we have ∆z= 0,with ∂nz|Γ= 0,ZΩ zndx = 0 and ∥∇z∥W1,p(Ω) = 1. But this is not possible because if zsatisfies ∆z= 0,∂nz|Γ= 0 and RΩz dx = 0, then we have z≡0and hence ∥∇z∥W1,p(Ω) = 0. Therefore we must have (1.6). ■ Lemma 1.9. ([15]) Let zn, zn−1∈L∞(Ω) and let f:R→Rbe a C2function. Then ZΩ δtznf′(zn) = δtZΩ f(zn)dx +1 2kZΩ f′′(cn(x))(zn(x)−zn−1(x))2dx, where cn(x)is an intermediate point between zn(x)and zn−1(x). In particular, if f is convex then we have ZΩ δtznf′(zn)dx ≥δtZΩ f(zn)dx. Lemma 1.10. Let w1and w2be nonnegative real numbers. For each s≥1we have |ws 2−ws 1| ≤ s|max {w1, w2}|s−1|w2−w1|. Proof. Indeed, we have |ws 2−ws 1|=s|Zw2 w1 rs−1dr| ≤ s|max {w1, w2}|s−1|w2−w1|. ■ Using Lemma 1.10, we arrive at the following. Lemma 1.11. Let p∈(1,∞)and let {wn}be a sequence of nonnegative functions in Lp(0, T;Lp(Ω)) such that wn→win Lp(0, T;Lp(Ω)) as n→ ∞. Then, for every r∈(1, p),wr n→wrin Lp/r(0, T ;Lp/r(Ω)) as n→ ∞.
20 Chapter 1. PRELIMINARY RESULTS Let Xand Ybe Banach spaces, we say that Xis continuously injected in Yand denote it by X ,→Yif X⊂Yand, moreover, there is a constant C > 0such that ∥φ∥Y≤C∥φ∥X,∀φ∈X. Lemma 1.12. ([6]) Let B be a Banach space, let {wn}be a sequence in Band w∈B. Either if wn→wweakly* or weakly in Bthen {wn}is bounded in Band ∥w∥B≤lim inf ∥wn∥B. Denote Q= (0, T)×Ωand let Xpbe the Banach space Xp={v∈C([0, T]; Wp−2/p)∩Lp(0, T ;W2,p)|∂tv∈Lp(Q)} endowed with the norm ∥v∥Xp=∥v∥L∞(0,T;Wp−2/p)+∥v∥Lp(0,T;W2,p)+∥∂tv∥Lp(Q). Lemma 1.13. ([16]) Let Ωbe a bounded domain of RNsuch that Γis of class C2. Let p∈(1,3),w0∈W2−2/p,p(Ω) and h∈Lp(Q). Then the problem ∂tw−∆w=hin Q, ∂nw|Γ= 0 on (0, T)×Γ, w(0, x) = w0(x)in Ω, has a unique solution wsuch that w∈Xp. Moreover, there is a positive constant C=C(p, T, Ω) such that ∥w∥Xp=∥w∥C([0,T];W2−2/p,p(Ω)) +∥w∥Lp(0,T;W2,p(Ω)) +∥∂tw∥Lp(Q) ≤C(∥h∥Lp(Q)+∥w0∥W2−2/p,p(Ω)).(1.12) Lemma 1.14. (Compactness in Bochner spaces, [52]) Let X, B and Ybe Banach spaces, let F=nf∈L1(0, T;Y)∂tf∈L1(0, T;Y)o. Suppose that X⊂B⊂Y, with compact embedding X⊂Band continuous embedding B⊂Y. We have: 1. if the set Fis bounded in Lq(0, T;B)∩L1(0, T;X), for 1< q ≤ ∞, and n∂tf, ∀f∈Fois bounded in L1(0, T;Y), then Fis relatively compact in Lp(0, T;B), for 1≤p<q; 2. if Fis bounded in L∞(0, T;X)and n∂tf, ∀f∈Fois bounded in Lr(0, T ;Y), for some r > 1. Then Fis relatively compact in C([0, T]; B).
Chapter 1. PRELIMINARY RESULTS 21 To define the regularized problems in Chapter 2, we use the following truncation of the identity function, from above and from below: am(u) = −1,if u≤ −2, C2extension,if u∈(−2,0), u, if u∈[0, m], C2extension,if u∈(m, m + 2), m+ 1,if u≥m+ 2. (1.13) Note that amis globally Lipschitz. Along the study of problem (1), for s≥2, we will need the following result. Lemma 1.15. Let p∈(1,∞). Let {wm}be a sequence of nonnegative functions which is uniformly bounded in L∞(0,∞;Lp(Ω)) with respect to mand defined for every t∈(0,∞). If there is α > 0such that ZΩ wm(t, x)dx ≥α, ∀t∈(0,∞),∀m∈N, then there exist β > 0and m0∈Nlarge enough such that ZΩ am(wm(t, x)) dx ≥β, ∀t∈(0,∞),∀m≥m0. Proof. For every t∈(0,∞), let Sm(t) = nx∈Ωwm(t, x)> mo. Then, for all t∈(0,∞), we have ZSm(t) mpdx +ZΩ\Sm(t) am(wm(t, x))pdx ≤ZΩ am(wm(t, x))pdx ≤C1(p). This implies |Sm(t)|=ZSm(t) dx ≤C1(p)1 mp .(1.14) We have ZΩ am(wm(t, x)) dx =ZΩ\Sm(t) wm(t, x)dx +ZSm(t) am(wm(t, x)) dx =ZΩ wm(t, x)dx −ZSm(t)wm(t, x)−am(wm(t, x))dx ≥α−ZSm(t)wm(t, x)−am(wm(t, x))dx. To finish the proof, we show that lim m→∞ ZSm(t)wm(t, x)−am(wm(t, x))dx = 0,
22 Chapter 1. PRELIMINARY RESULTS uniformly with respect to t∈(0,∞). In fact, using Hölder’s inequality and (1.14), we obtain ZSm(t)wm(t, x)−am(wm(t, x))dx ≤ ZSm(t)|wm(t, x)−am(wm(t, x))|pdx!1 p ZSm(t) dx!p−1 p ≤C2(p) ZSm(t) dx!p−1 p ≤C(p)1 mp−1 . for all t∈(0,∞). Therefore we have ZΩ am(wm(t, x)) dx ≥α−C(p)1 mp−1 and then we can choose m0large enough such that ZΩ am(wm(t, x)) dx ≥β=α 2, completing the proof. ■ Lemma 1.16 (Gronwall’s inequality). Let f,gand hbe nonnegative functions such that f∈W1,1(0, T)and g, h ∈L1(0, T), for some T > 0. Let G(t) = Zt 0 g(r)dr and H(t) = Zt 0 h(r)dr. If fis such that d dtf(t)≤g(t) + h(t)f(t), a.e. t ∈(0, T ), then f(t)≤G(t) + f(0)eH(t), a.e. t ∈(0, T). Lemma 1.17. Let Xand Ybe Banach spaces. Let S:X−→ Ybe a continuous linear map. If f∈L1((0,∞); X)then Sf ∈L1((0,∞); Y)and Z∞ 0 Sf dt =SZ∞ 0 f dt. Proof. See the section about the Bochner’s integral in the book of Yosida [64]. ■ We will apply this lemma for S:L2(Ω) −→ Rgiven by Sf =ZΩ f dx.
Chapter 1. PRELIMINARY RESULTS 23 Lemma 1.18. If w, wt∈L1 loc((0,∞); L1(Ω)) then d dt ZΩ w(·, x)dx∈L1 loc(0,∞) and d dt ZΩ w(·, x)dx=ZΩ wt(·, x)dx. Proof. We look at the integral Z∞ 0ZΩ wt(t, x)dx ψ(t)dt, for every ψ∈C∞ c((0,∞)). Since the integral over Ωis a linear map and ψ(t)is a real number, for each t∈(0,∞), we have ZΩ wt(t, x)dx ψ(t) = ZΩ wt(t, x)ψ(t)dx, for a.e. fixed t. Then we use lemma 1.17 with X=L1(Ω),Y=Rand f=wtψ∈ L1((0,∞); L1(Ω)) to write Z∞ 0ZΩ wt(t, x)dx ψ(t)dt =Z∞ 0ZΩ wt(t, x)ψ(t)dx dt =ZΩZ∞ 0 wt(t, x)ψ(t)dt dx =−ZΩZ∞ 0 w(t, x)ψt(t)dt dx =−Z∞ 0ZΩ w(t, x)dx ψt(t)dt, for all ψ∈C∞ c((0,∞)).■ Let C∞ c(0,∞;L2(Ω)) denote the space of the infinitely differentiable functions defined in [0,∞)with range in L2(Ω) and with compact support in (0,∞). Consider the space H1(0,∞;L2(Ω)) = nw∈L2(0,∞;L2(Ω)) wt∈L2(0,∞;L2(Ω))o, which is a Hilbert space with the norm ∥w∥H1(0,∞;L2(Ω)) =∥w∥2 L2(0,∞;L2(Ω)) +∥wt∥2 L2(0,∞;L2(Ω))1/2. Lemma 1.19. ([41]) C∞ c(0,∞;L2(Ω)) is dense in H1(0,∞;L2(Ω)). We end this section recalling the concept of positive and negative parts of a function. For w∈Lp(Ω),1≤p≤ ∞, the positive and negative parts of ware given by w+(x) = max {0, w(x)}and w−(x) = min {0, w(x)}, respectively. Then w=w++w−and |w|=w+−w−; besides, if w∈H1(Ω) then w+, w−∈H1(Ω) with ∇w+(x) = (∇w, if w(x)>0, 0,if w(x)<0,and ∇w−(x) = (∇w, if w(x)<0, 0,if w(x)>0.
24 Chapter 1. PRELIMINARY RESULTS For more details on truncations applied to H1(Ω) functions we suggest Gilbarg and Trudinger [19]. Definition 1.20. (Weak solution of (2))Let s≥1,q > 5/2. Let f∈Lq(Q)and let u0, v0∈W2−2/q,q(Ω) be non-negative functions. A pair (u, v)is called a weak solution of (2) if u(t, x), v(t, x)≥0a.e. (t, x)∈Q, satisfying, for s∈[1,2), u∈L5s/(3+s)(W1,5s/(3+s)), ∂tu∈L5s/(3+s)((W1,5s/(4s−3))′), for s≥2, u∈L2(H1), ∂tu∈L2((H1)′), for s≥1, u∈L∞(Ls)∩L5s/3(Q), v∈L∞(Q)∩L∞(H1)∩L4(W1,4)∩L2(H2), ∂tv∈L5/3(Q), and satisfying the initial conditions for (u, v), the u-equation of (2) and the boundary condition of uin the variational sense (∂tu, φ)+(∇u, ∇φ)=(u∇v, ∇φ),(1.15) for all φ∈L5s/(4s−3)(W1,5s/(4s−3)), the v-equation pointwisely (in fact, the v-equation is satisfied in L5/3(Q)) and, since ∆v∈L2(Q), the boundary condition of vin the sense of H−1/2(Γ).□ Definition 1.21. (Strong solution of (2))Let s≥1,q > 5/2. Let f∈Lq(Q)and let u0, v0∈W2−2/q,q(Ω) be non-negative functions. A pair (u, v)is called a strong solution of (2) if u(t, x), v(t, x)≥0a.e. (t, x)∈Q, with regularity (u, v)∈Xq×Xq and satisfying the initial and boundary conditions of (2), the u-equation and the v-equation of (2) pointwisely. □ Remark 1.22. Since q > 5/2, if (u, v)is a strong solution of (2) then, in particular, u, v ∈L∞(Q). Then, through a comparison argument we can prove that, for each fixed f∈Lq(Q), the strong solution of (2) is unique. We refer the reader to the proof of uniqueness in 2D domains from Chapter 2 that, in view of the regularity of the strong solution, can be adapted to 3D domains. □
25 Chapter 2 UNIFORM IN TIME SOLUTIONS FOR CHEMOTAXIS WITH POTENTIAL CONSUMPTION MODELS 2.1 Main Results To present the main results of this chapter, we introduce the following regularized problems, which depend on a truncation parameter m∈N, (∂tum−∆um=−∇·(am(um)∇vm), ∂tvm−∆vm=−am(um)svm, ∂num|Γ=∂nvm|Γ= 0, um(0) = u0 m, vm(0) = v0 m,(2.1) where u0 m≥0and v0 m≥0are suitable regular approximations of u0and v0, respectively, and am(·)is the truncation of the identity function (bounded from above and from below) defined in (1.13). With the objective of enlarging the class of considered domains, we state and demonstrate our results in terms of the regularity of the Poisson-Neumann (1.5) (see definition 1.6 in page 18), and, when necessary, in terms of the following technical hypothesis: Hypothesis (H1). For each z∈H2(Ω) such that ∂ηz|Γ= 0 there is a sequence {ρn} ⊂ C2(Ω) such that ∂ηρn|Γ= 0 and ρn→zin H2(Ω). Remark 2.1. In order to show that the Hypothesis (H1) is not too restrictive, we prove in Lemma A.1, in the Appendix A, that Hypothesis (H1) is satisfied if the Poisson-Neumann problem has the W3,p-regularity (see definition 1.6 in page 18), for p>N. This is true, in particular, if Γis C2,1(see [20]). □
32 Chapter 2. UNIFORM IN TIME SOLUTIONS 2umand using Lemma 2.8.2, we arrive at ∥um(t)∥2 L2(Ω) +Zt 0∥∇um(s)∥2 L2(Ω) ds ≤ ∥u0 m∥2 L2(Ω) +m2Zt 0∥∇vm(s)∥2 L2(Ω) ds ≤ ∥u0 m∥2 L2(Ω) +m2 2∥v0 m∥2 L2(Ω). The latter allows us to conclude that um∈L∞(0,∞;L2(Ω)) and ∇um∈L2(0,∞;L2(Ω)). Hence, using the Poincaré’s type inequality of Lemma 1.3 we can prove that um−u∗∈L∞(0,∞;L2(Ω)) ∩L2(0,∞;H1(Ω)).(2.15) Next, we use this fact and take vmas a test function in the following reformulation of the vm-equation of (2.1) (vm)t−∆vm+am(u∗)svm=−(am(um)s−am(u∗)s)vm, we obtain 1 2 d dt∥vm(t)∥2 L2(Ω) +∥∇vm(t)∥2 L2(Ω) +am(u∗)s∥vm(t)∥2 L2(Ω) =−ZΩ (am(um(t, x))s−am(u∗)s)vm(t, x)2dx. Using Lemma 1.10, the right hand side can be estimated by |ZΩ (am(um)s−am(u∗)s)v2dx| ≤ ∥v0 m∥L∞(Ω) ZΩ|am(um)s−am(u∗)s|vmdx ≤ ∥v0∥L∞(Ω) ZΩ|am(um) + am(u∗)|s−1|am(um)−am(u∗)|v dx ≤2s−1ms−1∥v0∥L∞(Ω)∥um−u∗∥L2(Ω)∥vm∥L2(Ω) ≤C(u∗, v0)ms−1∥um−u∗∥2 L2(Ω) +am(u∗)s 2∥vm∥2 L2(Ω), where C(u∗, v0)>0is a constant (independent of tand x). Now, considering also the terms of the left hand side, one has 1 2 d dt∥vm(t)∥2 L2(Ω) +∥∇vm(t)∥2 L2(Ω) +am(u∗)∥vm(t)∥2 L2(Ω) ≤C(u∗, v0)ms−1∥um(t)−u∗∥2 L2(Ω). Note that am(u∗)is a fixed positive real number if u0= 0 and ∥um(t)−u∗∥2 L2(Ω) ∈ L1(0,∞), because of (2.15). Hence we can conclude that vm∈L2(0,∞;L2(Ω)) and, together with (2.14), we finally conclude that vm∈L2(0,∞;H1(Ω)).■ The regularity given in Lemma 2.9 allows us to obtain the regularity (2.12). In
2.2. The Regularized Problem 33 fact, first we test the vm-equation of (2.1) by −∆vm∈L2 loc([0,∞); H1(Ω)). After some computations, we arrive at 1 2 d dt∥∇vm∥2 L2(Ω) +∥∆vm∥2 L2(Ω) +∥am(um)s/2∇vm(s)∥2 L2(Ω) ≤C(m, ∥v0∥L∞(Ω))(∥∇um∥2 L2(Ω) +∥∇vm∥2 L2(Ω)), and this allows us to conclude that vm∈L∞(0,∞;H1(Ω)) ∩L2(0,∞;H2(Ω)) (2.16) because, according to Lemmas 2.8 and 2.9 we have ∥∇um∥2 L2(Ω),∥∇vm∥2 L2(Ω) ∈ L1(0,∞). Next we take the gradient of the vm-equation of (2.1) and test the resulting equation by −∇∆vm∈L2 loc([0,∞); L2(Ω)), obtaining 1 2 d dt∥∆vm∥2 L2(Ω) +∥∇∆vm∥2 L2(Ω) =sZΩ a′ m(um)am(um)s−1vm∇um·∇∆vmdx +ZΩ am(um)s∇vm·∇∆vmdx ≤C(s, m, ∥v0∥L∞(Ω))(∥∇um∥2 L2(Ω) +∥∇vm∥2 L2(Ω)) + 1 2∥∇∆vm∥2 L2(Ω), which gives us vm∈L∞(0,∞;H2(Ω)) and ∆vm∈L2(0,∞;H1(Ω)).(2.17) From regularities (2.16) and (2.17), we can go back to the vm-equation of (2.1) and conclude that (vm)t∈L2(0,∞;H1(Ω)). Now, we can test the um-equation of (2.1) by −∆um∈L2 loc([0,∞); L2(Ω)). Considering the bounds of the truncation am(·)given in (2.3), the interpolation inequality of Lemma 1.1-2 and inequality (1.6), we have d dt∥∇um∥2 L2(Ω) +∥∆um∥2 L2(Ω) =ZΩ a′ m(um)∇um·∇vm∆umdx +ZΩ am(um)∆vm∆umdx ≤C∥∇um∥L3(Ω)∥∇vm∥L6(Ω)∥∆um∥L2(Ω) +m∥∆vm∥L2(Ω)∥∆um∥L2(Ω) ≤C∥∇um∥1/2 L2(Ω)∥∇vm∥L6(Ω)∥∆um∥3/2 L2(Ω) +m∥∆v∥L2(Ω)∥∆um∥L2(Ω) ≤C∥∇vm∥4 L6(Ω)∥∇um∥2 L2(Ω) +C(m)∥∆vm∥2 L2(Ω) +1 2∥∆um∥2 L2(Ω). After absorbing the term 1 2∥∆um∥2 L2(Ω), the other terms in the right hand side of the inequality belong to L1(0,∞). Hence, integrating the last inequality with respect to t, ∇um∈L∞(0,∞;L2(Ω)) and ∆um∈L2(0; ∞;L2(Ω)),
34 Chapter 2. UNIFORM IN TIME SOLUTIONS finishing the proof of the regularity (2.12). Finally, we consider the case in which we assume that the Poisson-Neumann problem (1.5) has the W2,3-regularity. In this case, as observed in the end of Subsection 2.2.1, the regularity (2.13) already holds if we consider finite intervals (0, T), for finite T > 0, instead of (0,∞). Then we can take the gradient of the u-equation of (2.1) and test the resulting equation by −∇∆um∈L2 loc([0,∞); L2(Ω)), obtaining 1 2 d dt∥∆um∥2 L2(Ω) +∥∇∆um∥2 L2(Ω) =ZΩ am(um)∇∆vm·∇∆umdx +ZΩ a′ m(um)∆vm∇um·∇∆umdx +ZΩ a′ m(um)∇um·D2vm∇∆umdx +ZΩ a′ m(um)∇vm·D2um∇∆umdx +ZΩ a′′ m(um)∇um·∇vm∇um·∇∆umdx. We recall that, from (2.3), we have |am(um)| ≤ mand |a′ m(um)|,|a′′ m(um)| ≤ C, for some C > 0. Then, using Hölder’s inequality we obtain 1 2 d dt∥∆um∥2 L2(Ω) +∥∇∆um∥2 L2(Ω) ≤m∥∇∆vm∥L2(Ω) +C∥∆vm∥L3(Ω)∥∇um∥L6(Ω) +C∥D2vm∥L3(Ω)∥∇um∥L6(Ω) +C∥∇vm∥L6(Ω)∥D2um∥L3(Ω) +C∥∇um∥2 L6(Ω)∥∇vm∥L6(Ω)∥∇∆um∥L2(Ω) Using the continuous embedding L6(Ω) ⊂H1(Ω), the W2,3-regularity, inequality (1.6), the interpolation inequality for the the L3-norm (Lemma 1.1) in 3D domains and Young’s inequality, we arrive at d dt∥∆um∥2 L2(Ω) +1 2∥∇∆um∥2 L2(Ω) ≤Cm∥∇∆vm∥2 L2(Ω) +Ch∥∆vm∥2 L3(Ω) +∥∆vm∥4 L2(Ω) +∥∆um∥2 L2(Ω)∥∆vm∥2 L2(Ω)i∥∆um∥2 L2(Ω). Now notice that the first term of the right hand side of the last inequality belongs to L1(0,∞)and, because of the regularity obtained so far, we have Ch∥∆vm∥2 L3(Ω) + ∥∆vm∥2 L2(Ω) +∥∆um∥4 L2(Ω)∥∆vm∥2 L2(Ω)i∈L1(0,∞). Therefore, using Grownwall’s inequality (Lemma 1.16) we can conclude that ∆um∈L∞(0,∞;L2(Ω)) ∩L2(0,∞;H1(Ω)) and hence um∈L∞(0,∞;H2(Ω)). Next we can go back to the um-equation of (2.1) and conclude that (um)t∈L2(0,∞;H1(Ω)), finishing the proof of the regularity (2.13).
2.3. Proof of Theorem 2.2 35 2.3 Proof of Theorem 2.2 In this section we will obtain m-independent estimates to (um, vm), the solution of problem (2.1), in order to pass to the limit as m→ ∞ and prove the existence of solution to the original problem (1). 2.3.1 An energy law appears: formal computations The basic idea to obtain additional a priori m-independent estimates is that the effects of the consumption and chemotaxis terms cancel. First of all, we present some formal calculations to illustrate how it works. Suppose (u, v)is a regular enough solution to the original problem (1) with u, v > 0. Consider the change o variable z=√v, then (1) can be rewritten as ∂tu−∆u=−∇·(u∇(z)2) ∂tz−∆z−|∇z|2 z=−usz 2 ∂nu|Γ=∂nz|Γ= 0 u(0) = u0, z(0) = √v0, (2.18) We are going to obtain estimates for uand zand then extract estimates for vfrom the estimates of z. For this, we consider a function g(u)such that g′′(u) = us−2. Formally, assuming u, z > 0we can use g′(u) = us−1 (s−1) ,if s > 1, ln(u),if s= 1. (2.19) as a test function in the u-equation of (2.18), obtaining d dt ZΩ g(u)dx +ZΩ g′′(u)|∇u|2dx =ZΩ ug′′(u)∇(z2)·∇u and, since ug′′(u) = us−1, we have d dt ZΩ g(u)dx +ZΩ us−2|∇u|2dx =ZΩ us−1∇(z2)·∇u dx =1 sZΩ∇(z2)·∇(us)dx. (2.20) On the other hand, we can test the z-equation of (2.18) by −∆z. Then we obtain 1 2 d dtZΩ|∇z|2dx +ZΩ|∆z|2dx +ZΩ |∇z|2 z∆z dx +1 2ZΩ us|∇z|2dx =−1 4ZΩ∇(us)·∇(z2)dx. (2.21)
36 Chapter 2. UNIFORM IN TIME SOLUTIONS Hence, if we add (2.21) to s/4times (2.20), then the two terms on the right hand side cancel each other and we obtain the time differential equation d dt s 4ZΩ g(u)dx +1 2ZΩ|∇z|2dx+s 4ZΩ us−2|∇u|2dx +1 2ZΩ us|∇z|2dx +ZΩ|∆z|2dx +ZΩ |∇z|2 z∆z dx = 0. (2.22) The main idea now is to estimate from below the term ZΩ|∆z|2dx +ZΩ |∇z|2 z∆z dx (2.23) (see Lemma 2.10 below). 2.3.2 Rigorous justification of the energy inequalities In the sequel, we consider the regularized problem (2.1) and its solution (um, vm) instead of the original problem (1) and (u, v). In this case, we have to deal with the truncation am(·)in the chemotaxis and consumption terms and with the fact that um is nonnegative, but not necessarily strictly positive (Lemma 2.7). In order to obtain time independent estimates, we will also need to separate the auxiliary variable z from zero, that’s why we are going to consider the change of variables z=√v+α, for α > 0to be chosen later, instead of √v. With this modification, we will obtain the corresponding version of (2.22). We will separate the cases s= 1,s∈(1,2) and s≥2. Note that if s= 1 then g′(u), given by (2.19), and g′′(u)have a singularity at u= 0. If s∈(1,2), then only g′′(u)is singular at u= 0 and, if s≥2, then neither g′(u)nor g′′(u)are singular. Let us consider the variable zm(t, x) = pvm(t, x) + α. Taking into account that the pair (um, vm)is a strong solution of (2.1), on has by straightforward calculations that (um, zm)satisfies the following equivalent problem: ∂tum−∆um=−∇·(am(um)∇(zm)2) ∂tzm−∆zm−|∇zm|2 zm =−1 2am(um)szm+α 2 am(um)s zm ∂num|Γ=∂nzm|Γ= 0 um(0) = u0, zm(0) = √v0+α. (2.24) In the present subsection, we drop the m-subscript and write (u, z)for (um, zm) to simplify the notation along the proofs of the forthcoming Lemmas. We remark that all the constants obtained in these Lemmas are independent of the parameter m∈Nand this is why the energy inequalities proved in this section allow us to obtain m-independent bounds in the rest of the chapter. We use the following lemma in order to estimate (2.23), whose proof can be found in the Appendix B.
2.3. Proof of Theorem 2.2 37 Lemma 2.10. Suppose that the Poisson-Neumann problem (1.5) has the H2-regularity and assume that Hypothesis (H1) holds. Then there exist positive constants C1, C2>0 such that ZΩ|∆z|2dx +ZΩ |∇z|2 z∆z dx ≥C1ZΩ|D2z|2dx +ZΩ |∇z|4 z2dx−C2ZΩ|∇z|2dx, for all z∈H2(Ω) such that ∂nz|Γ= 0 and z≥αin Ω, for some α > 0. Now we prove the following. Lemma 2.11. The solution (u, z)of (2.24), satisfies the inequality 1 2 d dt∥∇z∥2 L2(Ω) +C1ZΩ|D2z|2dx +ZΩ |∇z|4 z2dx +1 2ZΩ am(u)s|∇z|2dx ≤s 4ZΩ am(u)s−1∇(z2)·∇am(u)dx +s 2√αZΩ am(u)s−1|∇z||∇am(u)|dx +C2ZΩ|∇z|2dx. Proof. We begin by testing the z-equation of (2.24) by −∆z. This gives us 1 2 d dt∥∇z∥2 L2(Ω) +∥∆z∥2 L2(Ω) +ZΩ |∇z|2 z∆z dx +1 2ZΩ am(u)s|∇z|2dx ≤s 4ZΩ am(u)s−1∇(z2)·∇am(u)dx +s 2√αZΩ am(u)s−1|∇z||∇am(u)|dx. Then, applying Lemma 2.10, we obtain the desired inequality. ■ Define, for σ≥0, the functions gmand gm,j, adequate regularizations of the function gthat appears in the formal inequality (2.22), by gm(σ) = Zσ 0 g′ m(r)dr and gm,j(σ) = Zσ 0 g′ m,j(r)dr, (2.25) where g′ mand g′ m,j are defined for r≥0and given by g′ m(r) = ln(am(r) + 1),if s= 1, am(r)s−1 s−1,if s > 1,(2.26) and g′ m,j(r) = (am(r)+1/j)s−1 s−1,for s∈(1,2) and j∈N.(2.27) In Lemmas 2.14, 2.15 and 2.16 below we will be interested in the terms ZΩ ut(t, x)g′ m(u(t, x)) dx and ZΩ ut(t, x)g′ m,j(u(t, x)) dx,
38 Chapter 2. UNIFORM IN TIME SOLUTIONS recalling that u(t, x)denotes the function um(t, x)of the solution (um, zm)of (2.24). We have the following results on these terms. Theorem 2.12. The weak time derivatives of gm,j(u)and gm(u)belong to L2(0,∞;L2(Ω)) and are given by ∂tgm(u) = g′ m(u)ut(2.28) and ∂tgm,j(u) = g′ m,j(u)ut.(2.29) Proof. We are going to prove (2.29). The proof of (2.28) is analogous to the proof of (2.29). Because of Lemma 1.19, we know that there is a sequence (un)n∈N⊂ C∞ c(0,∞;L2(Ω)) that converges to uin the norm of H1((0,∞); L2(Ω)). For these functions un, which are very regular in the time variable, we can write −Z∞ 0 gm,j(un)φ′(t)dt =Z∞ 0 g′ m,j(un)un tφ(t)dt =Z∞ 0 (am(un)+1/j)s−1 s−1un tφ(t)dt, for all φ∈C∞ c(0,∞). Then we note that the convergence in the norm of H1((0,∞); L2(Ω)) is enough to pass to the limit as n→ ∞ in the first and in the last terms of the above equality, yielding −Z∞ 0 gm,j(u)φ′(t)dt =Z∞ 0 (am(u)+1/j)s−1 s−1utφ(t)dt, ∀φ∈C∞ c(0,∞). By the definition of weak derivative, this is precisely (2.29), as we wanted to prove. ■ Corollary 2.13. The weak time derivatives d dt ZΩ gm(u(t, x)) dx and d dt ZΩ gm,j(u(t, x)) dx belong to L2(0,∞)and are given by the expressions d dt ZΩ gm(u(t, x)) dx =ZΩ ut(t, x)g′ m(u(t, x)) dx (2.30) and d dt ZΩ gm,j(u(t, x)) dx =ZΩ ut(t, x)g′ m,j(u(t, x)) dx. (2.31) Proof. We are going to prove (2.31). The proof of (2.30) is analogous to the proof of (2.31). Accounting for Lemma 1.18 and (2.29), we obtain d dt ZΩ gm,j(u(t, x)) dx =ZΩ d dtgm,j(u(t, x)) dx =ZΩ ut(t, x)(am(u(t, x)) + 1/j)s−1 s−1dx. Since (am(u(t,x))+1/j)s−1 s−1is pointwisely bounded and ut(t, x)∈L2(0,∞;L2(Ω)) then d dt ZΩ gm,j(u(t, x)) dx ∈L2(0,∞). ■
2.3. Proof of Theorem 2.2 39 Now, we are in position to prove an energy inequality associated to the formal inequality (2.22). Lemma 2.14 (Energy inequality for s= 1). The solution (u, z)of the problem (2.24) satisfies, for sufficiently small α > 0, d dt 1 4ZΩ gm(u)dx +1 2ZΩ|∇z|2dx+CZΩ|∇[am(u) + 1]1/2|2dx +1 4ZΩ am(u)|∇z|2dx +C1ZΩ|D2z|2dx +ZΩ |∇z|4 z2dx≤CZΩ|∇z|2dx, (2.32) where gm(u)is given by (2.25). Proof. In order to prove (2.32) we will use the cancellation effect mentioned in Subsection 2.3.1. Since now we are dealing with the regularized problem (instead of the original problem), we must pay attention to two technical difficulties that arise: the presence of the truncation am(·)in the chemotaxis and consumption terms, the fact that uis nonnegative, but not strictly positive, and now we consider z=√v+α. For s= 1, this means that instead of using g′(u) = ln(u)as a test function in the um-equation of (2.1), we must use g′(am(u) +1) = ln(am(u)+ 1), in order to preserve the cancellation effect, avoid divisions by zero and invalid values for the argument of ln(·). We begin by using φ=ln(am(u) + 1) in the u-equation of problem (2.1) to obtain d dt ZΩ gm(u)dx +ZΩ a′ m(u) am(u)+1|∇u|2dx =am(u) am(u)+1∇(z2),∇am(u), where gm(r) = Zr 0 ln(am(θ) + 1) dθ is a primitive of ln(am(r) + 1). Due to the regularity of u,utand the functions gm(r) and g′ m(r), for r≥0, we can conclude that the weak derivative d dt RΩgm(u)dx = RΩg′ m(u)utdx and belongs to L2(0, T). Since 0≤a′ m(u)≤C, we have (a′ m(u))2≤Ca′ m(u), and we can write ZΩ a′ m(u) am(u)+1|∇u|2dx ≥CZΩ (a′ m(u))2 am(u)+1|∇u|2dx =CZΩ |∇am(u)|2 am(u)+1 dx ≥CZΩ|∇[am(u) + 1]1/2|2dx.
40 Chapter 2. UNIFORM IN TIME SOLUTIONS Hence, using that 1 (am(u) + 1) ≤1 pam(u)+1, we have d dt ZΩ gm(u)dx +CZΩ|∇[am(u) + 1]1/2|2dx = 2 am(u)+1−1 am(u)+1 z∇z, ∇am(u) =∇(z2),∇am(u)−2z∇z, ∇am(u) am(u)+1 ≤∇(z2),∇am(u)+ 2q∥v0∥L∞(Ω) +α∥∇z∥L2(Ω)∥∇[am(u) + 1]1/2∥L2(Ω), Then we obtain d dt ZΩ gm(u)dx +CZΩ|∇[am(u) + 1]1/2|2dx ≤∇(z2),∇am(u)+C∥∇z∥2 L2(Ω) (2.33) Now, using Lemma 2.11 for s= 1 we obtain 1 2 d dt∥∇z∥2 L2(Ω) +C1ZΩ|D2z|2dx +ZΩ |∇z|4 z2dx+1 2ZΩ am(u)|∇z|2dx ≤1 4ZΩ∇(z2)·∇am(u)dx +√α 2ZΩ|∇z||∇am(u)|dx +C2∥∇z∥2 L2(Ω). If we add the above inequality to 1/4times (2.33), then the terms ZΩ∇am(u)·∇(z2)dx cancel and we obtain d dth1 4ZΩ gm(u)dx +1 2∥∇z∥2 L2(Ω)i+CZΩ|∇[am(u) + 1]1/2|2dx +1 2ZΩ am(u)|∇z|2dx +C1ZΩ|D2z|2dx +ZΩ |∇z|4 z2dx ≤√α 2ZΩ|∇z||∇am(u)|dx +C2∥∇z∥2 L2(Ω) ≤ZΩ √α|∇[am(u) + 1]1/2||pam(u)+1||∇z|dx +C2∥∇z∥2 L2(Ω). (2.34) We can deal with the first term in the right hand side of the inequality using Hölder’s and Young’s inequality, ZΩ √α|∇[am(u) + 1]1/2||pam(u)+1||∇z|dx ≤α C(δ)ZΩ am(u)|∇z|2dx +δ∥∇[am(u) + 1]1/2∥2 L2(Ω) +α C(δ)ZΩ|∇z|2dx. Therefore, we can first choose δ > 0and then α > 0sufficiently small in order to use the terms on the left hand side of inequality (2.34) to absorb the first two terms on the right hand side of the above inequality and finally obtain the desired inequality (2.32). ■
2.3. Proof of Theorem 2.2 41 Lemma 2.15 (Energy inequality for s∈(1,2)). The solution (u, z)of the problem (2.24) satisfies, for sufficiently small α > 0, d dt s 4ZΩ gm(u)dx +1 2∥∇z∥2 L2(Ω)+1 4ZΩ am(u)s|∇z|2dx +C1ZΩ|D2z|2dx +ZΩ |∇z|4 z2dx≤CZΩ|∇z|2dx, (2.35) where gm(u)is given by (2.25). Proof. Analogously to the case s= 1 (Lemma 2.14) in order to preserve the cancellation effect and avoid divisions by zero, for s∈(1,2), instead of using g′(u)as a test function in the um-equation of (2.1), we should consider the sequence {1/j}j∈N and use g′ m,j(u)given by (2.27). Due to the complexity of the procedures that are involved, we divide the proof in three main steps: 1. Obtain an inequality from the um-equation of problem (2.24); 2. Use this inequality and Lemma 2.11 to obtain the corresponding version of (2.22); 3. Pass to the limit as j→ ∞ to obtain (2.35). STEP 1: By testing the um-equation of (2.24) by g′ m,j(u) = (am(u)+1/j)s−1/(s− 1) d dt ZΩ gm,j(u)dx+ZΩ a′ m(u) (am(u)+1/j)2−s|∇u|2dx =am(u) (am(u)+1/j)2−s∇(z2),∇am(u), where gm,j(r) = Zr 0 (am(θ)+1/j)s−1 (s−1) dθ is a primitive of (am(r)+1/j)s−1/(s−1). Since 0≤a′ m(u)≤C, we have (a′ m(u))2≤ Ca′ m(u), we can write ZΩ a′ m(u) (am(u)+1/j)2−s|∇u|2dx ≥CZΩ (a′ m(u))2 (am(u)+1/j)2−s|∇um|2dx ≥CZΩ|∇[am(u) + 1]s/2|2dx. and hence we obtain d dt ZΩ gm,j(u)dx +CZΩ|∇[am(u)+1/j]s/2|2dx =am(u)+1/j −1/j (am(u)+1/j)2−s∇(z2),∇am(u) =(am(u)+1/j)s−1∇(z2),∇am(u)−2 z∇z, 1/j (am(u)+1/j)1−s/2(1/j)s/2∇am(u) (am(u)+1/j)1−s/2! ≤(am(u)+1/j)s−1∇(z2),∇am(u)+4 sq∥v0∥L∞(Ω) +α(1/j)s/2∥∇z∥L2(Ω)∥∇[am(u)+1/j]s/2∥L2(Ω),
48 Chapter 2. UNIFORM IN TIME SOLUTIONS and thus we have d dt ZΩ g(um+ 1) dx +2 s2ZΩ|∇[um+ 1]s/2|2dx ≤1 4ZΩ am(um)s|∇vm|2dx. Integrating with respect to tfrom 0to T, for any fixed T∈(0,∞), we obtain ZΩ g(um(T) + 1) dx +2 s2ZT 0ZΩ|∇[um+ 1]s/2|2dx dt ≤1 4ZT 0ZΩ am(um)s|∇vm|2dx dt +ZΩ g(u0 m+ 1) dx. Then, because of Lemma 2.8.1, (2.4) and the definition of gand (2.45) we conclude that (um+ 1)s/2is bounded in L∞(0,∞;L2(Ω)),(2.47) in particular, umis bounded in L∞(0,∞;Ls(Ω)),(2.48) and ∇[um+ 1]s/2is bounded in L2(0,∞;L2(Ω)).(2.49) Consider the relation ∇um=∇(um+ 1) = ∇(um+ 1)s/22/s =2 s(um+ 1)1−s/2∇(um+ 1)s/2.(2.50) Taking into account that we are considering s∈[1,2), we can use (2.47) to obtain (um+ 1)1−s/2is bounded in L∞(0,∞;L2s/(2−s)(Ω)) and then (2.49) and (2.50) to conclude that ∇umis bounded in L2(0,∞;Ls(Ω)).(2.51) In conclusion, using (2.48), (2.51) and the Poincaré inequality for zero mean functions (Lemma 1.3), um−u∗is bounded in L∞(0,∞;Ls(Ω)) ∩L2(0,∞;W1,s(Ω)).(2.52) Considering the chemotaxis term of the um-equation of (2.1), we can write am(um)∇vm as am(um)∇vm=am(um)1−s/2am(um)s/2∇vm. Then, we have am(um)1−s/2bounded in L∞(0,∞;L2s/(2−s)(Ω)), because of (2.47), and am(um)s/2∇vmbounded in L2(0,∞;L2(Ω)), because of (2.45), and hence we can conclude that am(um)∇vmis bounded in L2(0,∞;Ls(Ω)).(2.53)
2.3. Proof of Theorem 2.2 49 Then, if we consider the um-equation of (2.1), from (2.52) and (2.53) we conclude that ∂tumis bounded in L20,∞;W1,s/(s−1)(Ω)′. Now we turn to the vm-equation, rewritten as ∂tvm−∆vm+am(u∗)svm=−(am(um)s−am(u∗)s)vm.(2.54) Analyzing the term on the right hand side of (2.54), we have am(um)s−am(u∗)sis bounded in L2(0,∞;L3/2(Ω)).(2.55) In fact, using Lemma 1.10, we obtain |am(um)s−am(u∗)s| ≤ s|am(um) + am(u∗)|s−1|um−u∗|. Then, considering the m-uniform bound (2.52) and the Sobolev embedding L3s/(3−s)(Ω)) ⊂ W1,s(Ω) we obtain (2.55). With this information, now we can test (2.54) by vm, obtaining 1 2 d dt∥vm∥2 L2(Ω) +∥∇vm∥2 L2(Ω) +am(u∗)s 2∥vm∥2 L2(Ω) ≤CZΩ|am(um)s−am(u∗)s|v2 mdx ≤C∥v0 m∥L∞(Ω)∥am(um)s−am(u∗)s∥L3/2(Ω)∥vm∥L3(Ω) ≤C(δ)∥am(um)s−am(u∗)s∥2 L3/2(Ω) +δ∥vm∥2 L2(Ω) +δ∥∇vm∥2 L2(Ω). Note that if u0≡ 0then have am(u∗) = u∗>0all m≥u∗. Hence, choosing δ > 0 small enough, we can conclude that, for m≥u∗, there is β > 0such that 1 2 d dt∥vm∥2 L2(Ω) +β∥∇vm∥2 L2(Ω) +β∥vm∥2 L2(Ω) ≤C∥am(um)s−am(u∗)s∥2 L3/2(Ω). Therefore, integrating with respect to tand using (2.55) we obtain vmis bounded in L2(0,∞;L2(Ω)).(2.56) Hence, in view of (2.44), (2.45) and (2.56) we have vmis bounded in L2(0,∞;H2(Ω)). With the m-uniform bounds obtained so far we can obtain a m-uniform bound for the function ∂tvmin L2(0,∞;L3/2(Ω)). In fact, going back to (2.54), reminding that vmis uniformly bounded in L∞(0,∞;L∞(Ω)) with respect to mand considering (2.56) and (2.55), we conclude that ∂tvmis bounded in L2(0,∞;L3/2(Ω)). Now we are going to obtain compactness for {um}which are necessary in order to
50 Chapter 2. UNIFORM IN TIME SOLUTIONS pass to the limit as m→ ∞ in the nonlinear terms of the equations of (2.1). Because of (2.47) and (2.49), we have that (um+ 1)s/2is bounded in L∞(0,∞;L2(Ω)) ∩L2(0, T;H1(Ω)),for every finite T > 0. Using the Sobolev inequality H1(Ω) ⊂L6(Ω) and interpolation inequalities we obtain (um)s/2is bounded in L10/3(0, T;L10/3(Ω)), which is equivalent to umis bounded in L5s/3(0, T;L5s/3(Ω)).(2.57) By using (2.47) and (2.57) in (2.50) (remind that s∈[1,2)), we also have umis bounded in L5s/(3+s)(0, T;W1,5s/(3+s)(Ω)). We observe that W1,5s/(3+s)(Ω) ⊂Lq(Ω), with continuous embedding for q= 15s/(9 −2s)and compact embedding for q∈[1,15s/(9 −2s)). Then, since s∈[1,2), we have 5s/3<15s/(9 −2s)and therefore the embedding W1,5s/(3+s)(Ω) ⊂L5s/3(Ω) is compact. Note also that q= 5s/3≥5/3>1. Now we can use Lemma 1.14 with X=W1,5s/(3+s)(Ω), B =L5s/3(Ω), Y =H3(Ω)′ and q= 5s/3, to conclude that there is a subsequence of {um}(still denoted by {um}) and a limit function usuch that um−→ uweakly in L5s/(3+s)(0, T;W1,5s/(3+s)(Ω)),∀T > 0, and um−→ ustrongly in Lp(0, T;Lp(Ω)),∀p∈[1,5s/3),∀T > 0.(2.58) Using the Dominated Convergence Theorem we can conclude from (2.58) that am(um)→ustrongly in Lp(0, T;Lp(Ω)),∀p∈(1,5s/3),∀T > 0.(2.59) It stems from the convergence (2.59) and Lemma 1.11 that (am(um))s→usstrongly in Lq(0, T;Lq(Ω)),∀q∈(1,5/3),∀T > 0.(2.60)
2.3. Proof of Theorem 2.2 51 The convergence of vmis better. There is a subsequence of {vm}(still denoted by {vm}) and a limit function vsuch that vm→vweakly* in L∞((0,∞)×Ω) ∩L∞(0,∞;H1(Ω)), vm→vweakly in L2(0,∞;H2(Ω)), ∇vm→ ∇vweakly in L4(0,∞;L4(Ω)), and ∂tvm→∂tvweakly in L2(0,∞;L3/2(Ω)). (2.61) Now we are going to use the weak and strong convergences obtained so far to pass to the limit as m→ ∞ in the equations of problem (2.1). We are going to identify the limits of the nonlinear terms related to chemotaxis and consumption, am(um)∇vmand am(um)svm, respectively, with u∇vand usv. In fact, considering the chemotaxis term, because of (2.59), (2.44) and (2.61), we can conclude that am(um)∇vm−→ u∇vweakly in L20s/(5s+12)(0, T;L20s/(5s+12)(Ω)),∀T > 0. Considering now the consumption term, considering (2.60) and (2.61) we conclude that am(um)svm−→ usvweakly in L5/3(0, T;L5/3(Ω)),∀T > 0. With these identifications and all previous convergences, it is possible to pass to the limit as m→ ∞ in each term of the equations of (2.1). In order to finish the proof of Theorem 2.2 we must obtain the regularity (up to infinite time) which is claimed for u. From (2.47) and (2.49) there exists a subsequence of {(um+ 1)s/2}, still denoted by {(um+ 1)s/2}, and a limit function φsuch that (um+ 1)s/2−→ φweakly* in L∞(0,∞;L2(Ω)) ∇(um+ 1)s/2−→ ∇φweakly in L2(0,∞;L2(Ω)). Then, using the strong convergence (2.58), the continuity of the function um7→ f(um) = (um+ 1)s/2and the Dominated Convergence Theorem, we prove that φ= (u+ 1)s/2. Analogously, because of (2.45) we can conclude that, up to a subsequence, there is a limit function ϕsuch that am(um)s/2∇vm−→ ϕweakly in L2(0,∞;L2(Ω)).
52 Chapter 2. UNIFORM IN TIME SOLUTIONS And using the convergences (2.60) and (2.61) we can conclude that ϕ=us/2∇v. Therefore we have proved the global in time regularity (u+ 1)s/2∈L∞(0,∞;L2(Ω)),∇(u+ 1)s/2∈L2(0,∞;L2(Ω)), us/2∇v∈L2(0,∞;L2(Ω)).(2.62) Considering (2.62) and proceeding as in the obtaining of (2.52) and (2.53) we conclude the global in time regularity u∈L∞(0,∞;Ls(Ω)),∇u, u∇v∈L2(0,∞;Ls(Ω)), finishing the proof of Theorem 2.2 in the case s∈[1,2). m-independent estimates for (um, vm)and passage to the limit for s≥2 The procedure for the case s≥2is slightly different. First we note that, integrating the energy inequality (2.39) from Lemma 2.16 with respect to t, we have ∇am(um)s/2is bounded in L2(0,∞;L2(Ω)).(2.63) We also remind that we defined g′ m(r) = am(r)s−1/(s−1), for s≥2. Then we have am(r)s=sZr 0 a′ m(θ)am(θ)s−1dθ ≤Cs Zr 0 am(θ)s−1dθ =Cs(s−1)gm(r). Therefore it also stems from integrating the energy inequality (2.39) with respect to tthat am(um)s/2is bounded in L∞(0,∞;L2(Ω)).(2.64) From (2.64) and (2.63) we can conclude that am(um)s/2is bounded in L10/3(0, T;L10/3(Ω)), that is, am(um)is bounded in L5s/3(0, T;L5s/3(Ω)).(2.65) For each fixed m∈N, consider the zero measure set N ⊂ (0,∞)such that um(t∗,·), vm(t∗,·)∈H1(Ω),∀t∗∈(0,∞)\N. Then, for each fixed t∗∈(0,∞)\N, let us consider the sets {0≤um≤1}=nx∈Ω0≤um(t∗, x)≤1o
2.3. Proof of Theorem 2.2 53 and {um≥1}=nx∈Ωum(t∗, x)≥1o. Now note that, since s≥2, we have ZΩ am(um(t∗, x)2|∇vm(t∗, x)|2dx ≤Z{0≤um≤1}|∇vm(t∗, x)|2dx +Z{um≥1} am(um(t∗, x))s|∇vm(t∗, x)|2dx ≤ZΩ|∇vm(t∗, x)|2dx +ZΩ am(um(t∗, x))s|∇vm(t∗, x)|2dx. The last inequality is valid for all t∗∈(0,∞)\N, then if we integrate in the variable twe obtain Z∞ 0ZΩ am(um(t, x))2|∇vm(t, x)|2dx dt ≤Z∞ 0ZΩ|∇vm(t, x)|2dx dt +Z∞ 0ZΩ am(um(t, x))s|∇vm(t, x)|2dx dt. Therefore by (2.14) and (2.45) we can conclude that am(um)∇vmis bounded in L2(0,∞;L2(Ω)).(2.66) Now we test the um-equation of problem (2.1) by um. This gives us 1 2 d dt∥um∥2 L2(Ω) +∥∇um∥2 L2(Ω) =ZΩ am(um)∇vm·∇umdx ≤1 2ZΩ am(um)2|∇vm|2dx +1 2∥∇um∥2 L2(Ω), hence we have d dt∥um∥2 L2(Ω) +∥∇um∥2 L2(Ω) ≤ZΩ am(um)2|∇vm|2dx. Integrating with respect to t, we conclude from (2.66) that umis bounded in L∞(0,∞;L2(Ω)) (2.67) and ∇umis bounded in L2(0,∞;L2(Ω)).(2.68) Then, if we consider the um-equation of (2.1), by applying (2.68) and (2.66) we conclude that ∂tumis bounded in L2(0,∞; (H1(Ω))′).(2.69) Let (am(um)s)∗=1 |Ω|ZΩ am(um)sdx, from (2.64) and (2.63), we can also conclude that ∇am(um)sis bounded in L2(0,∞;L1(Ω)).
54 Chapter 2. UNIFORM IN TIME SOLUTIONS In view of Lemma 1.3, the latter implies am(um)s−(am(um)s)∗is bounded in L2(0,∞;W1,1(Ω)) and, in particular, by the Sobolev embedding, we have am(um)s−(am(um)s)∗is bounded in L2(0,∞;L3/2(Ω)).(2.70) Now we consider the vm-equation of (2.1) written as ∂tvm−∆vm+ (am(um)s)∗vm=−(am(um)s−(am(um)s)∗)vm.(2.71) Testing (2.71) by vmand using Hölder’s inequality we can obtain 1 2 d dt∥vm∥2 L2(Ω) +∥∇vm∥2 L2(Ω) + (am(um)s)∗∥vm∥2 L2(Ω) ≤C∥v0 m∥L∞(Ω)∥am(um)s−(am(um)s)∗∥L3/2(Ω)∥vm∥L3(Ω) ≤C∥am(um)s−(am(um)s)∗∥2 L3/2(Ω) +δ∥vm∥2 L2(Ω) +δ∥∇vm∥2 L2(Ω). In order to bound (am(um)s)∗from below, we will apply Lemma 1.15. Indeed, (am(um)s)∗≥CZΩ am(um)dxs and applying Lemma 1.15 (with wm=um,p= 2, and using (2.67)) we conclude that there exist β > 0and m0large enough such that (am(um)s)∗≥β > 0,a.e. t ∈(,∞), for all m≥m0. Therefore 1 2 d dt∥vm∥2 L2(Ω) + (1 −δ)∥∇vm∥2 L2(Ω) + (β−δ)∥vm∥2 L2(Ω) ≤C∥am(um)s−(am(um)s)∗∥2 L3/2(Ω). Now, choosing δsmall enough, integrating the last inequality with respect to tand using (2.70) we obtain vmis bounded in L2(0,∞;L2(Ω)).(2.72) With the m-independent a priori bounds obtained so far we can also give an mindependent a priori bound for ∂tvm. In fact, if we consider again the equation (2.71), then the m-independent estimate in the L∞-norm for vmgiven by Lemma 2.7-2 and the m-independent a priori bounds (2.72), (2.70) and (2.45) allow us to conclude that ∂tvmis bounded in L2(0,∞;L3/2(Ω)).
2.4. Regularity and Uniqueness in 2D55 Now, using (2.67), (2.68) and (2.69) we can conclude that there is a subsequence of {um}, still denoted by {um}, and a limit function usuch that um−→ uweakly* in L∞(0,∞;L2(Ω)), ∇um−→ ∇uweakly in L2(0,∞;L2(Ω)), ∂tum−→ uweakly in L20,∞;H1(Ω)′. By applying the compactness result Lemma 1.14, one has um−→ ustrongly in L2(0, T;L2(Ω)),∀T > 0. Using the Dominated Convergence Theorem and (2.65) we can also prove that am(um)−→ ustrongly in Lp(0, T;Lp(Ω)),∀p∈(1,5s/3), and using Lemma 1.11, am(um)s−→ usstrongly in Lp(0, T;Lp(Ω)),∀p∈(1,5/3). From the global in time estimate (2.64) we can conclude that, up to a subsequence, am(um)→uweakly* in L∞(0,∞;Ls(Ω)), hence, in particular, u∈L∞(0,∞;Ls(Ω)). For s≥2, if we consider the functions vm, we have the same m-independent estimates that we had for s∈[1,2). Then we have the same convergences given in (2.61). Following the ideas of Subsection 2.3.3, we can identify the limits of am(um)∇vm and am(um)svmwith u∇vand usv, respectively. This finishes the proof of existence of solution to the original problem (1) as a limit of solutions of the regularized problems (2.1) for s≥2. 2.4 Regularity and Uniqueness in 2D In this section, we show that, for two dimensional domains, we can improve the results on the uniqueness and regularity of the solution of (1). The key point is the inequality (1.1), which allows us to improve the a priori estimates of umand then of vm, where (um, vm)is the solution of (2.1).
56 Chapter 2. UNIFORM IN TIME SOLUTIONS 2.4.1 Uniqueness in 2D Theorem 2.19. In the two dimensional case, we have uniqueness of solution in the class of functions (u, v)such that u∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)) ∩L4s−4(0, T;L4s−4(Ω)),(2.73) u∈L4(0, T;L4+ϵ(Ω)) if s= 2 (2.74) and v∈L∞(0, T;L∞(Ω)) ∩L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)).(2.75) Remark 2.20. The regularities (2.73) and (2.75) imply in particular that ut∈L2(0, T; (H1(Ω))′) and therefore, the solution ucan be taken as test function in the u-equation of (1). The regularity u∈L4s−4(0, T;L4s−4(Ω)) is an additional hypothesis only if s > 2, because in 2D domains, the regularity u∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)) implies u∈L4(0, T;L4(Ω)).□ Proof of Theorem 2.19. Suppose (u1, v1)and (u2, v2)are two solutions of the original problem (1) with the regularity given in (2.73)-(2.75). Define (u, v) = (u2−u1, v2− v1). Then (u, v)satisfies (ut(t), φ)+(∇u(t),∇φ)=(u(t)∇v2(t),∇φ)+(u1(t)∇v, ∇φ),∀φ∈H1(Ω), a.e t ∈(0, T), (2.76) and vt(t)−∆v(t) = −[(u2(t))s−(u1(t))s]v2(t)−(u1(t))sv(t), a.e t ∈(0, T),(2.77) with (u(0), v(0)) = (0,0). Note that we can conclude from (2.76) that uis a zero mean function. Now we test (2.76) by u. We obtain, first using the interpolation inequality from Lemma 1.1-2, for 2D domains, and the Poincaré inequality for zero mean functions from Lemma 1.3, and after Young’s inequality 1 2 d dt∥u∥2 L2(Ω) +∥∇u∥2 L2(Ω) ≤ ∥u∥L4(Ω)∥∇v2∥L4(Ω)∥∇u∥L2(Ω) +∥u1∥L4(Ω)∥∇v∥L4(Ω)∥∇u∥L2(Ω) ≤C∥u∥1/2 L2(Ω)∥∇v2∥L4(Ω)∥∇u∥3/2 L2(Ω) +C∥u1∥L4(Ω)∥∇v∥1/2 L2(Ω)∥∇v∥1/2 H1(Ω)∥∇u∥L2(Ω) ≤C(δ)∥∇v2∥4 L4(Ω)∥u∥2 L2(Ω) +C(δ)∥u1∥4 L4(Ω)∥∇v∥2 L2(Ω) +δ∥∇u∥2 L2(Ω) +δ∥∇v∥2 H1(Ω),
2.4. Regularity and Uniqueness in 2D57 for each δ > 0. Then, accounting for (1.6), we get 1 2 d dt∥u∥2 L2(Ω) +∥∇u∥2 L2(Ω) ≤C(δ)∥∇v2∥4 L4(Ω)∥u∥2 L2(Ω) +C(δ)∥u1∥4 L4(Ω)∥∇v∥2 L2(Ω) +δ∥∇u∥2 L2(Ω) +δC∥∆v∥2 L2(Ω). (2.78) Next we test (2.77) by v−∆v. Taking into account that v2∈L∞(Ω), we obtain 1 2 d dt∥v∥2 H1(Ω) +∥∆v∥2 L2(Ω) +∥∇v∥2 L2(Ω) +ZΩ us 1v2dx ≤ZΩ|(u2)s−(u1)s||v2||v−∆v|dx +ZΩ|us/2 1v||∆v|dx ≤C(δ)∥((u2)s−(u1)s)∥2 L2(Ω) +C(δ)∥(u1)sv∥2 L2(Ω) +δ∥v∥2 L2(Ω) + 2δ∥∆v∥2 L2(Ω), for each δ > 0. We must estimate the terms ∥((u2)s−(u1)s)∥2 L2(Ω) and ∥(u1)sv∥2 L2(Ω). For the first of these two terms we use |us 2−us 1| ≤ s|max {u1, u2}|s−1|u2−u1| ≤ s|u1+u2|s−1|u2−u1|, and, considering Lemma 1.3 applied to zero mean function u, we find ∥[(u2)s−(u1)s]∥2 L2(Ω) =ZΩ|(u2)s−(u1)s|2dx ≤s2ZΩ|u2+u1|2s−2|u2−u1|2dx ≤s2∥u2+u1∥2s−2 L4s−4(Ω)∥u∥2 L4(Ω) ≤ ∥u2+u1∥2s−2 L4s−4(Ω)∥u∥L2(Ω)∥∇u∥L2(Ω) ≤ ∥u2+u1∥4s−4 L4s−4(Ω)∥u∥2 L2(Ω) +δ∥∇u∥2 L2(Ω). For the second term we have, for any ε > 0, ∥(u1)sv∥2 L2(Ω) =ZΩ (u1)2sv2dx ≤ ∥u1∥2s L2s+ε(Ω)∥v∥2 L(2s+ε)/ε(Ω) ≤C(ε)∥u1∥2s L2s+ε(Ω)∥v∥2 H1(Ω). Using the estimates of these two terms we obtain 1 2 d dt∥v∥2 H1(Ω) +∥∆v∥2 L2(Ω) ≤C(δ)∥u2+u1∥4s−4 L4s−4(Ω)∥u∥2 L2(Ω) +δ∥∇u∥2 L2(Ω) +C(δ, ε)∥u1∥2s L2s+ε(Ω)∥v∥2 H1(Ω) +δ∥v∥2 L2(Ω) +δ∥∆v∥2 L2(Ω). (2.79) If we sum up (2.78) and (2.79) and choose δ > 0small enough so that the terms that are multiplied by δon the right hand side can be absorbed by the corresponding nonnegative terms on the left hand side, we obtain 1 2 d dt∥u∥2 L2(Ω) +∥v∥2 H1(Ω)≤C∥∇v2∥4 L4(Ω)∥u∥2 L2(Ω) +C∥u1∥4 L4(Ω)∥∇v∥2 L2(Ω) +C∥u2+u1∥4s−4 L4s−4(Ω)∥u∥2 L2(Ω) +C(ε)∥u1∥2s L2s+ε(Ω)∥v∥2 H1(Ω) +C∥v∥2 L2(Ω)
64 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME In particular, we also prove the result on existence of weak solutions to (1) in 3D domains given in Theorem 2.2 of Chapter 2 but, this time, as a consequence of the convergence of the time discrete scheme introduced in this chapter. In 2D domains, there exists a unique strong solution of (1), see Chapter 2. The proof is achieved through the obtaining of stronger m-independent estimates for the solution of an adequate truncated problem. Unfortunately, it is not clear how we could adapt these strong estimates for the time discrete scheme. Consequently, in 2D, the convergence of the whole sequence of solutions of the time discrete scheme towards the unique strong solution of (1) as (m, k)→(0,∞)remains as an open problem. In order to prove Theorem 3.2, the chapter is organized as follows. In Section 3.2 we establish the existence of solution (un m, zn m)of the (u, z)-scheme (3.3), some pointwise estimates independent of (m, k, n)and an energy inequality for (un m, zn m). In Section 3.3, starting from this energy inequality, we deduce additional a priori estimates for (un m, zn m), independent of (m, k, n), that allow us to pass to the limit as (m, k)→(∞,0), obtaining convergence of (3.3) towards the (u, z)-problem (3.1). Finally, in Section 3.4 we prove the convergence of (un m, vn m), with vn mdefined by (3.4) or (3.5), towards the (u, v)-problem (1). 3.2 Study of the (u, z)-Scheme (3.3) 3.2.1 Existence of solution of (3.3) For simplicity, in the present subsection and in the following one we drop the m subscript and denote the solutions of (3.3) by (un, zn). In Subsection 3.3 we go back to the notation (un m, zn m). Theorem 3.4. (Existence of solution to (3.3))Suppose (un−1, zn−1)∈L2(Ω) × L∞(Ω) with un−1(x)≥0and zn−1(x)≥α a.e. x ∈Ω. Then there is a solution (un, zn)of (3.3) which satisfies un m(x)≥0and ∥zn−1 m∥L∞(Ω) ≥zn m(x)≥α a.e. x ∈Ω. Proof. In order to avoid divisions by zero in some terms of (3.3) and obtain un(x)≥0 a.e. x ∈Ω, we define the lower truncation for z Tα(z) = (α, if z≤α, z, if z≥α, and the lower-upper truncation for u Tm 0(u) = 0,if u≤0, u, if u∈[0, m], m, if u≥m,
3.2. Study of the (u, z)-Scheme (3.3) 65 Then, we consider the auxiliary problem δtun−∆un=∇·Tm 0(un)∇(zn)2, δtzn−|∇zn|2 Tα(zn)−∆zn=−1 2Tm 0(un)szn−α2 Tα(zn),(3.6) with the same boundary and initial conditions of (3.3). We prove the existence of a solution (un, zn)to (3.6) via Leray-Schauder fixed point theorem [19]. Along this proof, we also have that any solution (un, zn)of (3.6) satisfies un(x)≥0and ∥zn−1∥L∞(Ω) ≥zn(x)≥α a.e. x ∈Ω, which implies that Tα(zn) = zn, Tm 0(un) = Tm(un)and therefore we conclude that (un, zn)is also a solution of (3.3). Now we proceed with the proof of existence for (3.6) which is divided in three steps. Step 1 (Definition of the compact mapping S): For all (u, z)∈W1,4(Ω)2, we define (u, z) = S(u, z)∈H2(Ω)2as the solution of u k−∆u= 2∇·Tm 0(u)z∇z+un−1 k,(3.7) z k−∆z+1 2Tm 0(u)sz−α2 Tα(z)=|∇z|2 Tα(z)+zn−1 k.(3.8) We can use standard results on linear elliptic problems to conclude that (u, z) = S(u, z)is well defined. In fact, given (u, z), we begin by solving the z-equation (3.8). Since 0<1 k+1 2Tm 0(u)s≤1 k+ms 2, we first prove the existence of a weak solution z∈H1(Ω) by means of the Lax-Milgram Theorem and then we use the H2-regularity of the Poisson-Neumann problem (1.5) to prove that z∈H2(Ω). Once proved the existence of z∈H2(Ω), we have ∇·Tm 0(u)z∇z∈L2(Ω) and therefore we are able to solve the u-equation (3.7) and, using again the H2-regularity of the Poisson-Neumann problem (1.5), we obtain u∈H2(Ω). Hence, S(u, z)∈H2(Ω)2and therefore Sis a compact mapping defined in W1,4(Ω)2. Step 2 (Pointwise bounds for uand z/λ for any (u, z) = λS(u, z)): Let λ∈[0,1]. We will study the pairs (u, z)such that (u, z) = λS(u, z). If we consider λ= 0 then S(u, z)∈H2(Ω)2is well defined and (u, z) = (0,0). Once the case where λ= 0 is treated, we consider λ∈(0,1], therefore we can write S(u, z) = (1/λ)(u, z) and we have (u, z)satisfying u k−∆u=∇·Tm 0(u)∇(z)2+λun−1 k, 1 k z λ−|∇z|2 Tα(z)−∆z λ=−1 2Tm 0(u)sz λ−α2 Tα(z)+1 kzn−1. (3.9) If we test the u-equation of (3.9) by the negative part of udefined, as u−(x) = min {0, u(x)}, we conclude that u≥0. Now let c=∥zn−1∥L∞(Ω). We rewrite the
66 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME z-equation of (3.9) as 1 k(z λ−c)−λ2|∇(z λ−c)|2 Tα(z)−∆(z λ−c) =−Tm 0(u)s 2(z λ−c) + Tm 0(u)s 2(α2 Tα(z)−c) + (zn−1−c) k. (3.10) Now we test (3.10) by the positive part of (z λ−c), defined as (z λ−c)+(x) = max {0,(z λ−c)(x)} ≥ 0. Since c=∥zn−1∥L∞(Ω), we have zn−1−c≤0. Moreover, note that (z λ−c)+≥0 and (z λ−c)+= 0 ⇐⇒ z > λc. Then, an analysis taking account of the possible cases α < λc and α≥λc leads us to (z λ−c)+ Tα(z)≤1 λ. Hence, reminding that λ∈(0,1] and Tα(z)≥α, if we test (3.10) by (z/λ −c)+we obtain 1 k∥(z λ−c)+∥2 L2(Ω) +∥∇(z λ−c)+∥2 L2(Ω) ≤λ∥∇(z λ−c)+∥2 L2(Ω) +1 2Tm 0(u)s(α−c)(z λ−c)+. By hypothesis we have zn−1≥α, which implies in particular α≤cand then we can conclude that z λ≤ ∥zn−1∥L∞(Ω). Next we prove an inferior bound for z/λ. Considering the definition of Tαand |∇z|2 Tα(z)+1 kzn−1≥α k, which comes from the hypotheses of the theorem, we can use the z-equation of (3.9) to write 1 k(z λ−α)−∆(z λ−α) + 1 2Tm 0(un)s(z λ−α)≥1 2Tm 0(u)s(α2 Tα(z)−α).(3.11) Now we test (3.11) by the negative part of (z−λα),(z−λα)−. Note that (z−λα)−≤0 and (z λ−α)−= 0 =⇒z < λα ≤α. Therefore, testing (3.11) by (z λ−α)−and reminding that λ∈(0,1], we obtain 1 k∥(z λ−α)−∥2 L2(Ω) +∥∇(z λ−α)−∥2 L2(Ω) +1 2∥Tm 0(u)s/2(z λ−α)−∥2 L2(Ω) ≤1 2ZΩ Tm 0(u)s(α2 Tα(z)−α)(z λ−α)−dx =1 2ZΩ Tm 0(u)s(α2 α−α)(z λ−α)−dx Thus we conclude that (z λ−α)−= 0, that is, z λ≥α > 0.
3.2. Study of the (u, z)-Scheme (3.3) 67 Step 3 (λ-independent bounds for any (u, z) = λS(u, z)): As we mentioned before, we consider λ∈(0,1] because if λ= 0 then we have (u, z) = (0,0). Because of the upper bound for zthat we proved in the anterior step we have z λis bounded in L∞(Ω),(3.12) independently of λ. Then we can integrate the z-equation of (3.9) and conclude that ∥∇z∥2 L2(Ω) ≤C(k, α2, m, ∥zn−1∥L∞(Ω)).(3.13) Since z > 0, we can multiply the z-equation of (3.9) by λ Tα(z)/z. This gives us Tα(z) k−λ|∇z|2 z−Tα(z) z∆z=−1 2Tm 0(u)s(Tα(z)−α2 z/λ) + Tα(z) z/λ zn−1 k.(3.14) Now we test (3.14) by −∆zand, using that Tα(z)/z ≥1,z/λ ≥αand (3.12), we obtain 1 kZΩ (Tα)′(z)|∇z|2dx + (1 −λ)ZΩ|∆z|2dx +λZΩ|∆z|2dx +λZΩ |∇z|2 z∆z dx ≤1 2ZΩ|Tm 0(u)|s|Tα(z) + α2 α| |∆z|dx +ZΩ|Tα(z) α k zn−1| |∆z|dx ≤1 δC(m, α2,∥zn−1∥L∞(Ω)) + δ∥∆z∥2 L2(Ω). Then, applying Lemma 2.10 and using (3.13) and the H2-regularity we obtain 1 kZΩ (Tα)′(z)|∇z|2dx + (1 −λ)ZΩ|∆z|2dx +C1λZΩ|∆z|2dx +ZΩ |∇z|4 z2dx ≤C2λ∥∇z∥2 L2(Ω) +1 δC(m, α2,∥zn−1∥L∞(Ω) ) + δ∥∆z∥2 L2(Ω) ≤1 δC(k, m, α2,∥zn−1∥L∞(Ω) ) + δ∥∆z∥2 L2(Ω). Taking into account that ZΩ (Tα)′(z)|∇z|2dx ≥0and denoting C= min {1, C1}we have 1+(C1−1)λ≥Cfor all λ∈[0,1] and CZΩ|∆z|2dx +Cλ ZΩ |∇z|4 z2dx ≤1 δC(m, α2,∥zn−1∥L∞(Ω) ) + δ∥∆z∥2 L2(Ω) and therefore we can choose δ > 0small enough such that ∆zis bounded in L2(Ω).(3.15) Because of the homogeneous Neumann boundary conditions, the H2-regularity and the λ-uniform bounds (3.12) and (3.15), we can conclude that zis bounded in H2(Ω).(3.16)
68 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME By Sobolev inequality in 3D domains, the latter implies that zis bounded in W1,4(Ω).(3.17) Considering the λ-independent bounds 0≤z≤ ∥zn−1∥L∞(Ω),Tm 0(u)≤mand (3.16), we can test the u-equation (3.7) by uand prove that uis bounded in H1(Ω).Using again the λ-independent bounds 0≤z≤ ∥zn−1∥L∞(Ω),Tm 0(u)≤m, (3.16) and (3.17), we have that the chemotaxis term 2∇·Tm 0(u)z∇zis bounded in L2(Ω). Then we can test the u-equation (3.7) by −∆u, obtaining uis bounded in H2(Ω), which implies, in particular, that uis bounded in W1,4(Ω).(3.18) With (3.17) and (3.18) we can finally conclude, using the Leray-Schauder fixed point theorem [19] that the auxiliary problem (3.6) has a solution (u, z). Because of the properties showed along the steps of the proof we can also conclude that (u, z)∈ H2(Ω)2,u(x)≥0and ∥zn−1∥L∞(Ω) ≥z(x)≥α a.e. x ∈Ω. Therefore we have Tα(z) = z,Tm 0(u) = Tm(u)and we conclude that the solution (u, z)of (3.6) is a solution of (3.3), finishing the proof of existence of solution. ■ 3.2.2 First uniform in time estimates The following direct estimates and the energy inequalities obtained in this subsection are valid for any solution (un, zn)of (3.3) given by Theorem 3.4. Lemma 3.5. ( (m, k, n)-uniform estimates) Let (un, zn)be a solution of (3.3). Then we have 1. ZΩ undx =ZΩ u0dx, for all n∈N; 2. ∥zn∥2 L2(Ω) + n X j=1∥zj−zj−1∥2 L2(Ω) ≤ ∥z0∥2 L2(Ω), for all n∈N; 3. k n X j=1∥∇zj∥2 L2(Ω) ≤1 4α2∥v0+α2∥2 L2(Ω), for all n∈N. Proof. The proof of item 1is achieved by integrating the un-equation of (3.3). For the items 2and 3we take the product of the zn-equation of (3.3) by zn. We obtain δt(zn)2+1 k(zn−zn−1)2−∆(zn)2+Tm(un)s((zn)2−α2)=0.(3.19) Since ((zn)2−α2)≥0and ZΩ ∆(zn)2dx = 0, by integrating (3.19) we prove item 2. On the other hand, testing (3.19) by k(zn)2leads us to kδt∥(zn)2∥2 L2(Ω) +k∥∇(zn)2∥2 L2(Ω) ≤0.
3.2. Study of the (u, z)-Scheme (3.3) 69 Then, summing up from j= 1 to ngives us k n X j=1 ∥∇(zj)2∥2 L2(Ω) ≤ ∥(z0)2∥2 L2(Ω) =∥v0+α2∥2 L2(Ω). Now using that zj≥αwe have ZΩ|∇zj|2dx =ZΩ (zj)2 (zj)2|∇zj|2dx ≤ZΩ 1 4α2|∇(zj)2|2dx, hence we obtain item 3.■ 3.2.3 Energy inequality Now we turn to the energy inequalities given for s∈[1,2) in Lemma 3.7 and s≥2 in Lemma 3.8 below. We will need the following lemma. Lemma 3.6. Any solution (un, zn)of (3.3), satisfies the inequality 1 2δt∥∇zn∥2 L2(Ω) +1 2k∥∇zn−∇zn−1∥2 L2(Ω) +C1ZΩ|D2zn|2dx +ZΩ |∇zn|4 (zn)2dx +1 2ZΩ Tm(un)s|∇zn|2dx ≤s 4ZΩ Tm(un)s−1∇(zn)2·∇Tm(un)dx +sα 2ZΩ Tm(un)s−1|∇zn||∇Tm(un)|dx +C2ZΩ|∇zn|2dx. Proof. We begin by testing the zn-equation of (3.3) by −∆zn. This gives us 1 2δt∥∇zn∥2 L2(Ω) +1 2k∥∇zn−∇zn−1∥2 L2(Ω) +∥∆zn∥2 L2(Ω) +ZΩ |∇zn|2 zn∆zndx +1 2ZΩ (1 + α2 (zn)2)Tm(un)s|∇zn|2dx =s 4ZΩ Tm(un)s−1∇(zn)2·∇Tm(un)dx +s 2α2ZΩ Tm(un)s−1 zn∇zn·∇Tm(un)dx. Then, estimating the last term on the left hand side from bellow, the last term on the right hand side from above and applying Lemma 2.10, we obtain the desired inequality. ■ Next we will obtain a local energy inequality for (un, zn), first for s∈[1,2) and then for s≥2. We consider the function fm(r) = Zr 0 f′ m(θ)dθ, where f′ m(r) = ln(Tm(r)),if s= 1,for r > 0, Tm(r)s−1 (s−1) ,if s > 1.
70 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME Lemma 3.7 (Energy inequality for s∈[1,2)). Any solution (un, zn)of the problem (3.3) satisfies, for sufficiently small α2>0, δths 4ZΩ fm(un)dx +1 2∥∇zn∥2 L2(Ω)i +1 2k∥∇zn−∇zn−1∥2 L2(Ω) +1 4ZΩ Tm(un)s|∇zn|2dx +C1ZΩ|D2zn|2dx +ZΩ |∇zn|4 (zn)2dx≤C∥∇zn∥2 L2(Ω). (3.20) Proof. The proof follows the same ideas of the analogous result that was proved in Chapter 2 for the truncated model (2.1). We are going to show the main steps of the proof, calling the attention to the differences that appear due to the fact that we are considering the time discrete scheme (3.3). In fact, we begin by considering the sequence {1/j}j∈N, the function f′ m,j(un) = f′ m(un+1/j)and testing the un-equation of (3.3) by f′ m,j(un). A difference appears in the treatment of the term of the discrete time derivative. Namely, accounting that fm,j is convex, we use Lemma 1.9 to obtain δtZΩ fm,j(un)dx +ZΩ (Tm)′(un) (Tm(un)+1/j)2−s|∇un|2dx ≤Tm(un) (Tm(un)+1/j)2−s∇(zn)2,∇Tm(un). Then we can follow the ideas in Chapter 2 until the point that we reach the inequality δths 4ZΩ fm,j(un)dx +1 2∥∇zn∥2 L2(Ω)i +1 2k∥∇zn−∇zn−1∥2 L2(Ω) +C1ZΩ|D2zn|2dx +ZΩ |∇zn|4 (zn)2dx +1 2ZΩhTm(un)s−1 2(Tm(un)+1/j)si|∇zn|2dx ≤C∥∇zn∥2 L2(Ω) +s 4ZΩhTm(un)s−1−(Tm(un)+1/j)s−1i∇Tm(un)·∇(zn)2dx. (3.21) Finally we pass to the limit as j→ ∞ in (3.21). The presence of the discrete time derivative δtinstead of ∂tis what allows us to give an unified treatment for the case s∈[1,2), differently from Chapter 2, where the cases s= 1 and s∈(1,2) are separated. We proceed with the passage to the limit term by term. We detail the passage to the limit in the term which involves the discrete time derivative, δtZΩ fm,j(un(x)) dx. (3.22) We define the functions gm,j, gm, G ∈L1(Ω) by gm,j(x) = δtfm,j(un(x)),gm(x) = δtfm(un(x)) and G(x) = |gm,1(x)|. Then, for almost every x∈Ω,gm,j(x)→gm(x) as j→ ∞ with |gm,j(x)| ≤ G(x)for all j∈N. Therefore, using the Dominated
3.3. Energy Estimates and Passage to the Limit as (m, k)→(∞,0) 71 Convergence Theorem, we can conclude that lim j→∞ δtZΩ fm,j(un(x)) dx = lim j→∞ ZΩ gm,j(x)dx =ZΩ gm(x)dx =δtZΩ fm(un(x)) dx. For the other terms of (3.21), one can again follow Chapter 2, take the limit as j→ ∞ and obtain the desired result. ■ Lemma 3.8 (Energy inequality for s≥2). The solution (un, zn)of the problem (3.3) satisfies δths 4ZΩ fm(un)dx +1 2∥∇zn∥2 L2(Ω)i+1 2k∥∇zn−∇zn−1∥2 L2(Ω) +ZΩ|∇[Tm(un)]s/2|2dx +1 4ZΩ Tm(un)s|∇zn|2dx +C1ZΩ|D2zn|2dx +ZΩ |∇zn|4 (zn)2dx≤C∥∇zn∥2 L2(Ω). (3.23) Proof. The proof follows the same ideas of the analogous result that was proved in Chapter 2 for the truncated model. Having in mind that we use Lemma 1.9 to treat the term which involves the discrete time derivative δt, we refer the reader to Chapter 2 for the details of the proof. ■ The energy inequalities (3.20) and (3.23) allow us to obtain (m, k, n)-independent estimates for the function znin the next subsection. 3.3 Energy Estimates and Passage to the Limit as (m, k)→(∞,0) Now we use again the notation (un m, zn m)for the solution of (3.3). We define the piecewise constant function uk,r mand the locally linear and globally continuous function uk mby uk,r m(t, x) = un m(x)and uk m(t, x) = un m(x) + (t−tn) kun m(x)−un−1 m(x),if t∈[tn−1, tn).(3.24) Analogously, we define the functions zk,r m,zk m,vk,r mand vk m. With these functions we can rewrite (3.3) as the differential system, a.e. in (t, x)∈(0,∞)×Ω, ∂tuk m−∆uk,r m=∇·Tm(uk,r m)∇(zk,r m)2, ∂tzk m−|∇zk,r m|2 zk,r m−∆zk,r m=−1 2Tm(uk,r m)szk,r m−α2 zk,r m.(3.25) In this subsection we are going to prove (m, k)-independent estimates for uk,r m,zk,r m, uk mand zk m, which are also uniform in time, that will allow us to pass to the limit in (3.25). First, in Subsection 3.3.1, we will obtain estimates for ∇zk,r mfrom the energy inequalities (3.20) and (3.23). Next we prove bounds for uk mand uk,r mand pass to
72 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME the limit in (3.25) as (m, k)→(∞,0), considering the cases s∈[1,2) and s≥2, separately. 3.3.1 Estimates for ∇zk,r m Let us define the energy En m=s 4ZΩ fm(un m(t, x)) dx +1 2ZΩ|∇zn m(t, x)|2dx. We use the regularity of the initial data u0and z0in order to conclude that the initial energy E0 m, is also bounded, independently of (m, k). If we consider either (3.20) or (3.23), multiply it by kand sum from j= 1 to nwe can obtain s 4ZΩ fm(un m)dx +1 2∥∇zn m∥2 L2(Ω) +1 2 n X j=1∥∇zj−∇zj−1∥2 L2(Ω) +k n X j=1ZΩ Tm(uj m)s|∇zj m|2dx +C1k n X j=1ZΩ|D2zj m|2dx +ZΩ |∇zj m|4 (zj)2dx ≤Ck n X j=1∥∇zj m∥2 L2(Ω) +s 4ZΩ fm(u0 m)dx +1 2∥∇z0 m∥2 L2(Ω), (3.26) Thus, accounting that zj≥αand that Lemma 3.5.3 and (3.26) are valid for any n∈N, we can conclude that ∇zk,r mis bounded in L∞(0,∞;L2(Ω)) ∩L4(0,∞;L4(Ω)),(3.27) Tm(uk,r m)s/2∇zk,r mand ∆zk,r mare bounded in L2(0,∞;L2(Ω)),(3.28) ∞ X j=1∥zj−zj−1∥2 H1(Ω) ≤C. (3.29) From (3.29), we can prove the following. Lemma 3.9. There is a positive constant C, independent of mand k, such that ∥zk,r m−zk m∥2 L2(0,∞;H1(Ω)) ≤C k. (3.30) Proof. From the definition of zk,r mand zk m, we observe that zk,r m(t)−zk m(t) = (tn−t) k(zn−zn−1) for t∈(tn−1, tn). If t∈(tj−1, tj)then 0≤tj−t≤kand hence ∥zk,r m−zk m∥2 L2(0,∞;H1(Ω)) = ∞ X j=1 Ztj tj−1 (tn−t) k∥zj−zj−1∥2 H1(Ω) dt ≤k ∞ X j=1 ∥zj−zj−1∥2 H1(Ω).
3.3. Energy Estimates and Passage to the Limit as (m, k)→(∞,0) 73 Therefore, using (3.29) we conclude the proof. ■ In particular, since zk,r m∈H2(Ω) and ∂ ∂ηzn m|Γ= 0, it stems from (3.28), the H2-regularity of the Poisson-Neumann problem (1.5) and (1.6) that ∇zk,r mis bounded in L2(0,∞;H1(Ω)).(3.31) Using the results obtained until this point we analyze the convergence towards problem (1), first for s∈[1,2) and then for s≥2. 3.3.2 Estimates for (uk,r m, zk,r m)and passage to the limit for s∈[1,2) Let ∀r > 0, f′(r) = (ln(r)if s= 1, rs−1/(s−1) if s∈(1,2), f(r) = Zr 0 f′(θ)dθ =(rln(r)−rif s= 1, rs/s(s−1) if s∈(1,2). Notice that f′′(r) = rs−2,∀r > 0, in all cases. We test the un m-equation of (3.3) by f′(un m+ 1). Using Lemma 1.9 we obtain δtZΩ f(un m+ 1) dx +1 2kZΩ f′′(cn)(un m−un−1 m)2dx +4 s2ZΩ|∇[un m+ 1]s/2|2dx+ = 2 ZΩ Tm(un m)(un m+ 1)s/2−1zn m∇zn m·∇un m(un m+ 1)s/2−1dx ≤4 sZΩ Tm(un m)1−s/2 (un m+ 1)1−s/2Tm(un m)s/2zn m∇zn m·∇[un m+ 1]s/2dx ≤4 s∥z0∥L∞(Ω)ZΩ Tm(un m)s|∇zn m|2dx1/2ZΩ|∇[un m+ 1]s/2|2dx1/2 and thus we have δtZΩ f(un m+ 1) dx +1 2kZΩ f′′(cn)(un m−un−1 m)2dx +2 s2ZΩ|∇[un m+ 1]s/2|2dx ≤CZΩ Tm(un m)s|∇zn m|2dx. Multiplying by kand summing up from 0to n, for any n∈N, we obtain ZΩ f(un m+ 1) dx +1 2 n X j=1 ZΩ f′′(cj)(uj m−uj−1 m)2dx +k n X j=1 2 s2ZΩ|∇[uj m+ 1]s/2|2dx ≤Ck n X j=1 ZΩ Tm(uj m)s|∇zj m|2dx +ZΩ f(u0 m+ 1) dx.
80 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME (∞,0), it is convenient to consider two functions of zk,r m, namely g1(zk,r m) = (zk,r m)2and g2(zk,r m)=1/zk,r m. Since 0< α ≤zk,r m≤ ∥z0∥L∞(Ω), we can use the (m, k)-independent bounds derived so far and (3.54) to show that 1/zk,r m→1/z strongly in Lp(0, T;Lp(Ω)),for each T > 0and p∈[1,∞), |∇zk,r m|2→ |∇z|2weakly in L2(0,∞;L2(Ω)), ∇(zk,r m)2→ ∇(z)2weakly in L4(0,∞;L4(Ω)). (3.55) Now we are going to use the weak and strong convergences that we proved for uk,r m, uk m,zk,r mand zk mto pass to the limit as (m, k)→(∞,0) in the equations of problem (3.3). We are going to identify the limits of the nonlinear terms Tm(uk,r m)∇(zk,r m)2, |∇zk,r m|2 zk,r m ,Tm(uk,r m)szk,r mand Tm(uk,r m)s zk,r m , respectively, with u∇(z)2,|∇z|2 z,uszand us z. In fact, considering the chemotaxis term, because of (3.52), (3.27) and (3.55), we can conclude that Tm(uk,r m)∇(zk,r m)2−→ u∇(z)2weakly in L20s/(5s+12)(0, T;L20s/(5s+12)(Ω)), for each fixed T > 0. Using (3.54) and (3.55) we can also conclude that |∇zk,r m|2 zk,r m−→ |∇z|2 zweakly in L2(0, T;L2(Ω)),for each fixed T > 0. Regarding the consumption term, considering (3.53) and (3.54) we prove that Tm(uk,r m)szk,r m−→ uszweakly in L5/3(0, T;L5/3(Ω)),for each fixed T > 0. Finally, using (3.53) and (3.55) we obtain Tm(uk,r m)s zk,r m−→ us zweakly in L5/3(0, T;L5/3(Ω)),for each fixed T > 0. With these identifications and all previous convergences, it is possible to pass to the limit as (m, k)→(∞,0) in each term of the equations of (3.3). This finishes the proof of existence of a solution (u, z)of (3.1) for s∈[1,2). In order to finish, we will prove the regularity (up to infinite time) of u. From (3.33) and (3.35) there exists a subsequence of {(uk m+ 1)s/2}, still denoted by {(uk m+ 1)s/2}, and a limit function φsuch that (uk,r m+ 1)s/2−→ φweakly* in L∞(0,∞;L2(Ω)) ∇(uk,r m+ 1)s/2−→ ∇φweakly in L2(0,∞;L2(Ω)). Then, using the strong convergence (3.51), the continuity of the function uk,r m7→ f(uk,r m) = (uk,r m+ 1)s/2and the Dominated Convergence Theorem, we prove that φ= (u+ 1)s/2.
3.3. Energy Estimates and Passage to the Limit as (m, k)→(∞,0) 81 Analogously, because of (3.28) we can conclude that, up to a subsequence, there is a limit function ϕsuch that Tm(uk,r m)s/2∇zk,r m−→ ϕweakly in L2(0,∞;L2(Ω)). And using the convergences (3.53) and (3.54) we can conclude that ϕ=us/2∇z. Therefore we have proved the global in time regularity (u+ 1)s/2∈L∞(0,∞;L2(Ω)),∇(u+ 1)s/2∈L2(0,∞;L2(Ω)), us/2∇z∈L2(0,∞;L2(Ω)).(3.56) Considering (3.56) and proceeding as in the obtaining of (3.41) and (3.42) we conclude the global in time regularity u∈L∞(0,∞;Ls(Ω)),∇u, u∇(z)2∈L2(0,∞;Ls(Ω)). This finishes the proof that (u, z)is a weak solution of (3.1) and that {(uk,r m, zk,r m)} converges to (u, z)as (m, k)→(∞,0) in the sense indicated in this section, for s∈[1,2). 3.3.3 Estimates for (uk,r m, zk,r m)and passage to the limit for s≥2 The procedure for the case s≥2is much more similar to the case s≥2in Chapter 2. In the sequel, we highlight the main steps of the proof and refer the reader to Chapter 2 for details. First we note that, multiplying the energy inequality (3.23) from Lemma 3.8 by kand summing in n, for each n∈N, we have ∇Tm(uk,r m)s/2is bounded in L2(0,∞;L2(Ω)),(3.57) Tm(uk,r m)s/2is bounded in L∞(0,∞;L2(Ω)).(3.58) From (3.58) and (3.57) we can conclude that Tm(uk,r m)is bounded in L5s/3(0, T;L5s/3(Ω)).(3.59) Analogously to Chapter 2 we use (3.28) and Lemma 3.5 to prove that Tm(uk,r m)∇(zk,r m)2is bounded in L2(0,∞;L2(Ω)).(3.60) Now we can test the un m-equation of problem (3.3) by kun mand, after bounding some terms, we sum the resulting inequality and use (3.60) to conclude that uk,r mis bounded in L∞(0,∞;L2(Ω)),(3.61) ∇uk,r mis bounded in L2(0,∞;L2(Ω)),(3.62) ∞ X j=1∥uj−uj−1∥2 L2(Ω) ≤C, (3.63)
82 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME and then, analogously to (3.30), we prove that ∥uk,r m−uk m∥2 L2(0,∞;L2(Ω)) ≤Ck. (3.64) Then, if we consider the u-equation of (3.25), by applying (3.62) and (3.60) we conclude that ∂tuk mis bounded in L2(0,∞; (H1(Ω))′).(3.65) Using the (m, k)-independent bounds obtained so far, we can follow the ideas of Chapter 2 and subsection 3.3.2 in order to prove that zk,r m−αis bounded in L2(0,∞;L2(Ω)),(3.66) ∂tzk mis bounded in L2(0,∞;L3/2(Ω)). Now, using (3.61), (3.62), (3.64) and (3.65) we can conclude that there are subsequences of {uk,r m}and {uk m}, still denoted by {uk,r m}and {uk m}, and a limit function u such that uk,r m−→ uweakly* in L∞(0,∞;L2(Ω)), ∇uk,r m−→ ∇uweakly in L2(0,∞;L2(Ω)), ∂tuk m−→ uweakly in L20,∞;H1(Ω)′. By applying the compactness result Lemma 1.14 to the sequence {uk m}and using (3.64) we have uk,r m−→ ustrongly in L2(0, T;L2(Ω)),∀T > 0. Using the Dominated Convergence Theorem and (3.59) we can also prove that Tm(uk,r m)−→ ustrongly in Lp(0, T;Lp(Ω)),∀p∈(1,5s/3), and using Lemma 1.11, Tm(uk,r m)s−→ usstrongly in Lp(0, T;Lp(Ω)),∀p∈(1,5/3). From the global in time estimate (3.58) we can conclude that, up to a subsequence, Tm(uk,r m)→uweakly* in L∞(0,∞;Ls(Ω)), hence, in particular, u∈L∞(0,∞;Ls(Ω)). For s≥2, if we consider the functions zk,r mand zk m, we have the same (m, k)- independent estimates that we had for s∈[1,2). Then we have the same convergences given in (3.54) and (3.55). Following the ideas of Subsection 3.3.2, we can identify the limits of Tm(uk,r m)∇(zk,r m)2, |∇zk,r m|2/zk,r m,Tm(uk,r m)szk,r mand Tm(uk,r m)s/zk,r mwith u∇(z)2,|∇z|2/z,uszand us/z, respectively.
3.4. Convergence of vk,r m83 This finishes the proof that (u, z)is a solution of (3.1) and that {(uk,r m, zk,r m)} converges to (u, z)as (m, k)→(∞,0) in the sense indicated in this section, for s≥2. 3.4 Convergence of vk,r m Until this point, for any s≥1fixed, we have proved that (uk,r m, zk,r m)converges to (u, z)a solution of (3.1) as (m, k)→(∞,0). Now, to conclude the proof of Theorem 3.2, we are going to prove that (uk,r m, vk,r m)converges to (u, v)a weak solution of (1) as (m, k)→(∞,0), where vk,r mis given either by (3.4) or by (3.5). For simplicity, we consider (uk,r m, zk,r m)to be the subsequence which converge to the limit function (u, z). 3.4.1 vk,r mgiven by (3.4) In this case, it is enough to show that the sequence vk,r m= (zk,r m)2−α2converges to v=z2−α2as (m, k)→(∞,0). Then, thanks to the equivalence of problems (3.1) and (1) (Lemma 3.1), we know that (u, z)is a solution of (3.1) if, and only if, (u, v) is a solution of (1), with v=z2−α2. In fact, if we consider the (m, k)-uniform bounds obtained for zk,r m(especially the pointwise estimates of Theorem 3.4, (3.27) and (3.28)) and the convergences listed in (3.54), we can prove by straightforward calculations that the sequence vk,r m= (zk,r m)2− α2converges to v=z2−α2in the same senses indicated in (3.54). Hence, by Lemma 3.1, we have that (u, v)is a solution of (1). Therefore, we conclude that (uk,r m, vk,r m)converges to (u, v), a solution of (1), as (m, k)→(∞,0). 3.4.2 vk,r mgiven by (3.5) We rewrite (3.5) as ∂tvk m−∆vk,r m+Tm(uk,r m)svk,r m= 0.(3.67) Using the (m, k)-independent bounds obtained for Tm(un m)in Subsections 3.3.2 and 3.3.3 we can test (3.5) by vn mand by −∆vn mand conclude that vk,r mis bounded in L∞(0,∞;H1(Ω)),(3.68) ∆vk,r mis bounded in L2(0,∞;L2(Ω)),(3.69) ∥vk m−vk,r m∥L2(0,∞;H1(Ω)) ≤C k. (3.70) Considering the ideas used in Subsections 3.3.2 and 3.3.3 to show that zk,r m−αis bounded in L2(0,∞;H2(Ω)) and that ∂tzk mis bounded in L2(0,∞;L3/2(Ω)), we prove that vk,r mis bounded in L2(0,∞;H2(Ω)),(3.71) ∂tvk mis bounded in L2(0,∞;L3/2(Ω)).(3.72)
84 Chapter 3. CONVERGENCE OF A TIME DISCRETE SCHEME Then, using (3.68), (3.69), (3.70), (3.71) (3.72) and Lemma 1.14 we prove that there is a function vsuch that, up to a subsequence, for each fixed T > 0, we have vk,r m→vweakly* in L∞(0,∞;L∞(Ω)) ∩L∞(0,∞;H1(Ω)), vk,r m→vweakly in L2(0,∞;H2(Ω)), vk,r m→vstrongly in L2(0, T;H1(Ω)) ∩Lp(0, T ;Lp(Ω)), p ∈[1,∞), and ∂tvk m→∂tvweakly in L2(0,∞;L3/2(Ω)). (3.73) Now, using the convergence obtained for Tm(uk,r m)in Subsections 3.3.2 and 3.3.3 and (3.73) we conclude that the limit function vis the unique solution of ∂tv−∆v+usv= 0, where uis the function of the pair (u, z), fixed in the beginning of Subsection 3.4. Therefore, thanks to the uniqueness of the limit function v, we conclude that the whole sequence vk,r mconverges towards vas (m, k)→(∞,0). In addition, combining Lemma 3.1 and the uniqueness of the function v, given u, we deduce that v=z2−α2. Thus we conclude that {(uk,r m, vk,r m)}converges to (u, v), a solution of (1), as (m, k)→(∞,0) in both cases, vn mgiven by (3.4) or (3.5), finishing the proof of Theorem 3.2.
85 Chapter 4 OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS OF CHEMOTAXIS-CONSUMPTION MODELS 4.1 Main Results Along this chapter, it will be necessary to impose the hypotheses: Ω⊂R3is a bounded domain with boundary Γof class C2,1, Q= (0, T)×Ω, f∈Lq(Q),for some q > 5/2,(4.1) (u0, v0)∈Lp(Ω) ×W2−2/q,q(Ω),(4.2) with p= 1 + ε, for some ε > 0, if s= 1, and p=s, if s > 1. Remark 4.1. In particular, Hypothesis (H1) is satisfied. □ Let us define the specific functional spaces appearing for the weak solution setting. For s∈[1,2), U=nu∈L∞(0, T;Ls(Ω)) ∩L5s/3(Q)∩L5s/(3+s)(0, T ;W1,5s/(3+s)(Ω)); ∂tu∈L5s/(3+s)(0, T; (W1,5s/(4s−3)(Ω))′)o, for s≥2, U=nu∈L∞(0, T;Ls(Ω)) ∩L5s/3(Q)∩L2(0, T ;H1(Ω)); ∂tu∈L2(0, T;H1(Ω))′)o,
86 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS and for s≥1, V=nv∈L∞(Q)∩L∞(0, T;H1(Ω)) ∩L4(0, T ;W1,4(Ω)) ∩L2(0, T;H2(Ω)); ∂tv∈L5/3(Q)o. We also introduce the bounded convex set for the control Bq(M) = nf∈Lq(Q)∥f∥Lq(Q)≤Mo. Definition 4.2 (Weak Solution of the controlled problem (2)). A pair (u, v)is called a weak solution of (2) if u(t, x), v(t, x)≥0a.e. (t, x)∈Q, with u∈ U, v ∈ V and satisfying the initial conditions for (u, v), the u-equation of (2) and the boundary condition of uin the variational sense (∂tu, φ)+(∇u, ∇φ)=(u∇v, ∇φ), for all φ∈L5s/(4s−3)(0, T;W1,5s/(4s−3)(Ω)), the v-equation pointwisely (in fact, the v-equation is satisfied in L5/3(Q)) and, since ∆v∈L2(Q), the boundary condition of vin the sense of H−1/2(Γ).□ The proof of existence of weak solution to the controlled problem (2) is based in the treatment of the uncontrolled model (1) given in Chapter 2. This is because Chapter 2 is oriented to weak solutions and then it is well suited to extend (1) to a model in which we have a nonsmooth control fas coefficient in (2). An important step in Chapter 2 is the obtaining of an energy inequality using the change of variable from (u, v)to (u, z), with z=√v+α2, where α > 0is a sufficiently small but fixed real number chosen in Lemmas 4.12, 4.13 and 4.15 below, independently of (u, v)and f. Here, the energy inequality satisfied by the constructed weak solutions of (2) will also be written in terms of (u, z). In fact, we consider the energy E(u, z)(t) = s 4ZΩ g(u(t, x)) dx +1 2ZΩ|∇z(t, x)|2dx, where g(u) = (u+ 1)ln(u+ 1) −u, if s= 1, us s(s−1),if s > 1. We have the following result of existence of weak solutions to (2). Theorem 4.3 (Existence of energy inequality weak solutions). Given f∈ Lq(Q), there is a non-negative weak solution (u, v)of (2). Moreover, this weak solution
4.1. Main Results 87 (u, v)can be built satisfying the energy inequality E(u, z)(t2) + βZt2 t1ZΩ|∇[u+ 1]s/2|2dx dt +1 4Zt2 t1ZΩ us|∇z|2dx dt +βZt2 t1ZΩ|D2z|2dx dt +Zt2 t1ZΩ |∇z|4 z2dx dt ≤E(u, z)(t1) + K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)), (4.3) for a.e. t1, t2∈[0, T], with t2> t1; where K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω))is a continuous and increasing function with respect to ∥f∥Lq(Q)and β > 0is a constant, independent of (u, v, f). Finally, in case s > 1, inequality (4.3) also holds for all t2∈(t1, T]. Remark 4.4. The existence of weak solutions satisfying an energy inequality is commonly seen, for instance for fluid models, and is used to prove either weak-strong uniqueness results [42] or large time behavior [45]. In this chapter, we use this “energy inequality weak solution setting” in order to prove the existence of a global optimal solution to an optimal control problem. To the best of our knowledge in chemotaxis PDE problems, this is the first time that the concept of weak solution with energy inequality is applied to this purpose. □ To highlight the main results of the present chapter we introduce the minimization problems. Consider the functional J:L5s/3(Q)×L2(Q)×Lq(Q)−→ R given by J(u, v, f) := 3γu 5sZT 0∥u(t)−ud(t)∥5s/3 L5s/3(Ω) dt +γv 2ZT 0∥v(t)−vd(t)∥2 L2(Ω) dt +γf qZT 0∥f(t)∥q Lq(Ω) dt, where (ud, vd)∈L5s/3(Q)×L2(Q)represents the desired states and γu, γv, γf>0 measure the costs of the states and control. In view of the existence result, Theorem 4.3, one could expect the following admissible sets Sw ad =n(u, v, f)∈ U ×V ×Lq(Q); (u, v)is a weak solution of (2) with control f} or SE ad =n(u, v, f)∈ U ×V ×Lq(Q); (u, v)is a weak solution of (2) with control fand satisfies the energy inequality (4.3)o with the corresponding minimization problems (min J(u, v, f) subject to (u, v, f)∈Sw ad,(4.4)
88 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS or (min J(u, v, f) subject to (u, v, f)∈SE ad.(4.5) Thanks to Theorem 4.3 we have that both Sw ad and SE ad are nonempty sets. However, we are not able to prove that problem (4.4) or (4.5) has a solution, as we will analyze in Remark 4.7 and Subsection 4.3.1, respectively. Hence, to prove existence of optimal control related to weak solutions of (2), we define the following admissible set for each M > 0: SM ad =n(u, v, f)∈ U ×V ×Bq(M); (u, v)is a weak solution of (2) with control fand satisfies (4.3) changing the constant K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω))for K(M, ∥v0∥W2−2/q,q(Ω))o and the corresponding minimization problem (min J(u, v, f) subject to (u, v, f)∈SM ad .(4.6) Again, from Theorem 4.3, we have SM ad =∅. But now, we are able to prove the following. Theorem 4.5 (Existence of optimal control). For each M > 0, the optimal control problem (4.6) has at least one global optimal solution, that is, there is (u, v, f)∈ SM ad such that J(u, v, f) = min (u,v,f)∈SM ad J(u, v, f). Moreover, we have the following relation between the three minimization problems (4.4), (4.5) and (4.6). Theorem 4.6. For M≥q γf inf (u,v,f)∈SE ad J(u, v, f), we have the relations inf (u,v,f)∈Sw ad J(u, v, f)≤min (u,v,f)∈SM ad J(u, v, f)≤inf (u,v,f)∈SE ad J(u, v, f). Remark 4.7. From Theorem 4.5, for each M > 0there is (uM, vM, fM)∈SM ad such that J(uM, vM, fM) = min (u,v,f)∈SM ad J(u, v, f). Let M2> M1>0. Since SM1 ad ⊂SM2 ad then J(uM, vM, fM)decreases as Mincreases. Therefore, since J(uM, vM, fM)is bounded from below, there exists lim M→∞ J(uM, vM, fM)
4.2. Existence for the Bilinear Controlled Model 89 and, accounting for Theorem 4.6, we have inf (u,v,f)∈Sw ad J(u, v, f)≤lim M→∞ J(uM, vM, fM)≤inf (u,v,f)∈SE ad J(u, v, f). In particular, {(uM, vM, fM)}Mis bounded in L5s/3(Q)×L2(Q)×Lq(Q)independently of M. Thus we conclude that there is (u∞, v∞, f∞)∈L5s/3(Q)×L2(Q)× Lq(Q), defined as the weak limit in L5s/3(Q)×L2(Q)×Lq(Q)of a subsequence of {(uM, vM, fM)}M. Then, the lower semicontinuity of Jleads to J(u∞, v∞, f∞)≤lim M→∞ J(uM, vM, fM).(4.7) In our opinion, the proof or the refutation of the following two questions are interesting open problems: 1. (u∞, v∞, f∞)∈Sw ad ? 2. lim M→∞ J(uM, vM, fM) = inf (u,v,f)∈Sw ad J(u, v, f)? If item 2 were valid, then inf (u,v,f)∈Sw ad J(u, v, f)could be approximated by the minimums min (u,v,f)∈SM ad J(u, v, f)as M→ ∞. Additionally, if items 1 and 2 were valid, then (u∞, v∞, f∞)would be an optimal solution of (4.4). Indeed, from item 1 we have inf (u,v,f)∈Sw ad J(u, v, f)≤J(u∞, v∞, f∞). On the other hand, from item 2 and (4.7), we also have J(u∞, v∞, f∞)≤inf (u,v,f)∈Sw ad J(u, v, f), which allows us to conclude that J(u∞, v∞, f∞) = inf (u,v,f)∈Sw ad J(u, v, f), that is, (u∞, v∞, f∞)∈Sw ad is a global optimal solution of (4.4). □ The rest of the chapter is organized as follows. The existence of weak solutions satisfying an energy inequality for the controlled problem given in Theorem 4.3 is established in Section 4.2 and in Section 4.3 we study the optimal control problem, proving Theorem 4.5 and Theorem 4.6. 4.2 Existence for the Bilinear Controlled Model The existence of weak solutions of the uncontrolled problem (1) is proved in Chapter 2, by means of a sequence of truncated problems. Here, we prove the existence
96 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS Remark 4.14. In Chapter 2, when the authors prove the energy inequality for s∈ (1,2), the term ZΩ|∇[Tm(u)+1/j]s/2|2dx is estimated by ZΩ|∇[Tm(u)+1/j]s/2|2dx ≥0,for all j∈N, but it can be estimated by ZΩ|∇[Tm(u)+1/j]s/2|2dx ≥ZΩ|∇[Tm(u) + 1]s/2|2dx, for all j∈N, instead, yielding (4.29). □ Lemma 4.15 (Energy inequality for s≥2). The solution (u, z)of the problem (4.20) satisfies, for sufficiently small α > 0, d dtEm(u, z)(t) + ZΩ|∇[Tm(u)]s/2|2dx +1 4ZΩ Tm(u)s|∇z|2dx +βZΩ|D2z|2dx +ZΩ |∇z|4 z2dx≤C(K2 1)∥∇z∥2 L2(Ω) +K2 1∥f∥2 L2(Ω). (4.30) 4.2.4 m-independent estimates and passage to the limit as m→ ∞ In the present subsection we go back to the notation (um, zm)and (um, vm)to the solution of problems (4.20) and (4.8), respectively. m-independent estimates for ∇vm We will integrate the energy inequalities (4.26), (4.29) and (4.30) with respect to t, from 0to T > 0. We take into account that, because of Corollary 4.10, we have the following bounds independently of m: ∇zmis bounded in L2(Q) and 0< α ≤zm(t, x)≤ K1, a.e. (t, x)∈Q. (4.31) We also use the hypothesis (4.2) on the initial data u0, v0to prove that the energy given in (4.25) at time t= 0,Em(um, zm)(0), is also bounded, independently of m. Thus we conclude that ∇zmis bounded in L∞(0, T;L2(Ω)) ∩L4(Q), Tm(um)s/2∇zmand ∆zmare bounded in L2(Q). But using the fact that zm=√vm+α2and (4.31) we can conclude that ∇vmis bounded in L∞(0, T;L2(Ω)) ∩L4(Q),(4.32)
4.2. Existence for the Bilinear Controlled Model 97 Tm(um)s/2∇vmand ∆vmare bounded in L2(Q).(4.33) Case s∈[1,2) First, following Chapter 2, to which we refer the reader that might be interested in more details, we prove the existence of weak solution (u, v)of problem (2). Next, to conclude the proof of Theorem 4.3, letting z=√v+α2, we prove the energy inequality (4.3). Existence of weak solution to (2): In order to prove the existence of a weak solution (u, v)to (2), first we obtain mindependent estimates for the solutions (um, vm)of (4.8) and then we use compactness results in weak*, weak and strong topologies to pass to the limit as m→ ∞. Let g′(r) = (ln(r)if s= 1, rs−1/(s−1) if s∈(1,2),∀r > 0. and let g(r) = Zr 0 g′(θ)dθ =(rln(r)−(r−1) if s= 1, rs/s(s−1) if s∈(1,2). Notice that g′′(r) = rs−2,∀r > 0, in all cases. We test the um-equation of (4.8) by g′(um+ 1) and obtain d dt ZΩ g(um+ 1) dx +4 s2ZΩ|∇[um+ 1]s/2|2dx =ZΩ Tm(um)(um+ 1)s/2−1∇vm·∇um(um+ 1)s/2−1dx =2 sZΩ Tm(um)1−s/2 (um+ 1)1−s/2Tm(um)s/2∇vm·∇[um+ 1]s/2dx ≤2 sZΩ Tm(um)s|∇vm|2dx1/2ZΩ|∇[um+ 1]s/2|2dx1/2 and thus we have d dt ZΩ g(um+ 1) dx +2 s2ZΩ|∇[um+ 1]s/2|2dx ≤1 4ZΩ Tm(um)s|∇vm|2dx. Integrating with respect to tfrom 0to Twe obtain ZΩ g(um(T) + 1) dx +2 s2ZT 0ZΩ|∇[um+ 1]s/2|2dx dt ≤1 4ZT 0ZΩ Tm(um)s|∇vm|2dx dt +ZΩ g(u0+ 1) dx. Then, because of (4.17), (4.2) and the definition of gand (4.33) we conclude that (um+ 1)s/2is bounded in L∞(0, T;L2(Ω)) ∩L2(0, T ;H1(Ω)).(4.34)
98 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS Using the Sobolev inequality H1(Ω) ⊂L6(Ω) and interpolation inequalities we obtain (um)s/2is bounded in L10/3(Q). The latter and (4.34) imply that umis bounded in L∞(0, T;Ls(Ω)) ∩L5s/3(Q).(4.35) From (4.35) we can conclude, using the vm-equation of (4.8) that ∂tvmis bounded in L5/3(Q). Reminding that s∈[1,2), if we use (4.34) and (4.35) in the relation ∇um=∇(um+ 1) = ∇(um+ 1)s/22/s =2 s(um+ 1)1−s/2∇(um+ 1)s/2. then we also have umis bounded in L5s/(3+s)(0, T;W1,5s/(3+s)(Ω)).(4.36) Considering the chemotaxis term of the um-equation of (4.8), we can write Tm(um)∇vm as Tm(um)∇vm=Tm(um)1−s/2Tm(um)s/2∇vm. Then, we have Tm(um)1−s/2bounded in L10s/(6−3s)(Q), because of (4.34), and Tm(um)s/2∇vm bounded in L2(Q), because of (4.33), and hence we can conclude that Tm(um)∇vmis bounded in L5s/(3+s)(Q).(4.37) Then, if we consider the um-equation of (4.8), from (4.36) and (4.37) we conclude that ∂tumis bounded in L5s/(3+s)(0, T; (W1,5s/(4s−3)(Ω))′).(4.38) Now we are going to obtain compactness for {um}which is necessary to pass to the limit as m→ ∞ in the nonlinear terms of the equations of (4.8). We observe that W1,5s/(3+s)(Ω) ⊂Lq(Ω), with continuous embedding for q= 15s/(9 −2s)and compact embedding for q∈[1,15s/(9 −2s)). Then, since s∈[1,2), we have 5s/3<15s/(9 −2s)and therefore the embedding W1,5s/(3+s)(Ω) ⊂L5s/3(Ω) is compact. Note also that q= 5s/3≥5/3>1. Now we can use Lemma 1.14 with X=W1,5s/(3+s)(Ω), B =L5s/3(Ω), Y =W1,5s/(4s−3)(Ω)′
4.2. Existence for the Bilinear Controlled Model 99 and q= 5s/3, to conclude that there is a subsequence of {um}(still denoted by {um}) and a limit function usuch that um−→ uweakly in L5s/(3+s)(0, T;W1,5s/(3+s)(Ω)), and um−→ ustrongly in Lp(Q),∀p∈[1,5s/3).(4.39) Using the Dominated Convergence Theorem we can conclude from (4.39) that Tm(um)→ustrongly in Lp(Q),∀p∈[1,5s/3).(4.40) It stems from the convergence (4.40) and Lemma 1.11 that (Tm(um))s→usstrongly in Lq(Q),∀q∈[1,5/3).(4.41) The convergence of vmis better. There is a subsequence of {vm}(still denoted by {vm}) and a limit function vsuch that vm→vweakly* in L∞(Q)∩L∞(0, T;H1(Ω)), vm→vweakly in L2(0, T;H2(Ω)), ∇vm→ ∇vweakly in L4(Q), and ∂tvm→∂tvweakly in L5/3(Q). (4.42) Now we are going to use the weak and strong convergences obtained so far to pass to the limit as m→ ∞ in the equations of problem (4.8). We are going to identify the limits of the nonlinear terms related to chemotaxis and consumption, Tm(um)∇vmand Tm(um)svm, respectively, with u∇vand usv. In fact, considering the chemotaxis term, from (4.40), (4.32) and (4.42), we can conclude that Tm(um)∇vm−→ u∇vweakly in L20s/(5s+12)(Q). Considering now the consumption term, considering (4.41) and (4.42) we conclude that Tm(um)svm−→ usvweakly in L5/3(Q). With these identifications and all previous convergences, it is possible to pass to the limit as m→ ∞ in each term of the equations of (4.8). Energy inequality (4.3):
100 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS In order to finish we must prove the energy inequality (4.3). First we obtain an integral inequality for the solution (um, zm)of (4.20), where we remind that zm= √vm+α2, for small enough but fixed α > 0, being (um, vm)the solution of (4.8). According to Lemmas (4.12) and (4.13), (um, zm)satisfies Em(um, zm)(t2) + βZt2 t1ZΩ|∇[Tm(um) + 1]s/2|2dx dt +1 4Zt2 t1ZΩ Tm(um)s|∇zm|2dx dt +βZt2 t1ZΩ|D2zm|2dx dt +Zt2 t1ZΩ |∇zm|4 z2 m dx dt ≤Em(um, zm)(t1) + C(K2 1)Zt2 t1∥∇zm∥2 L2(Ω) dt +K2 1Zt2 t1∥f∥2 L2(Ω) dt, (4.43) where Emis given by (4.25). Next we collect some convergences that can be obtained from the m-independent bounds and the weak and strong convergences proved so far and that will be useful to pass to the limit in the inequality (4.43). Recalling that we denote z=√v+α2, we have, in particular, (um+ 1)s/2−→ (u+ 1)s/2weakly* in L∞(0, T;L2(Ω)), (um+ 1)s/2−→ (u+ 1)s/2weakly in L2(0, T;H1(Ω)), ∇Tm(um)s/2−→ ∇us/2weakly in L2(Q), Tm(um)s/2∇zm−→ us/2∇zweakly in L2(Q), ∇zm √zm−→ ∇z √zweakly in L4(Q), D2zm−→ D2zweakly in L2(Q), ∇zm−→ ∇zstrongly in L2(Q). (4.44) Recalling that we are dealing with the case s∈[1,2), let E(u, z)(t) = s 4ZΩ g(u(t, x)) dx +1 2ZΩ|∇z(t, x)|2dx, where g(u) = (u+ 1)ln(u+ 1) −u, if s= 1, us s(s−1),if s∈(1,2). Then the following convergence will be also necessary. Lemma 4.16. Em(um, zm)−→ E(u, z)in L1(0, T). Proof. From (4.44) we have that ∇zm→ ∇zin L2(Q)which, in particular, leads us to ZΩ|∇zm(t, x)|2dx −→ ZΩ|∇z(t, x)|2dx in L1(0, T). Then, it remains to prove that ZΩ gm(um)dx −→ ZΩ g(u)dx in L1(0, T).(4.45)
4.2. Existence for the Bilinear Controlled Model 101 We begin by rewriting gm(um)−g(u)as gm(um)−g(u) = gm(um)−gm(u) + gm(u)−g(u).(4.46) For the first difference in (4.46), gm(um)−gm(u), using that g′ mand g′are monotone increasing functions and that g′ m(r)≤g′(r)for all r≥0, we have |gm(um)−gm(u)|=|Zum u g′ m(θ)dθ| ≤ |um−u|(g′ m(um) + g′ m(u)) ≤ |um−u|(g′(um) + g′(u)). Then, for s= 1, we have |gm(um)−gm(u)| ≤ C|um−u|(ln(um+ 1) + ln(u+ 1)) and, for s∈(1,2), we have |gm(um)−gm(u)| ≤ C|um−u|(|um|s−1+|u|s−1). Considering the case s∈(1,2), since from (4.35) we have (|um|s−1+|u|s−1)bounded in L5s/(3s−3)(Q)and, from (4.39), we have |um−u| → 0in L5s/(2s+3)(Q), we conclude that gm(um)−gm(u)−→ 0in L1(Q).(4.47) Considering now the case s= 1, from (4.35) we have ln(um+ 1) + ln(u+ 1) bounded in Lp(Q), for all p∈[1,∞). Then, analogously to the case s∈(1,2), we use (4.39) and obtain (4.47) also for s= 1. From (4.47) we conclude, in particular, that ZΩ gm(um)dx −ZΩ gm(u)dx −→ 0in L1(0, T).(4.48) For the second difference in (4.46), gm(u)−g(u), we use the Dominated Convergence Theorem. In fact, we write gm(u)−g(u) = Zu 0 (g′ m(θ)−g′(θ)) dθ. Using this expression one can verify that gm(u)−g(u)−→ 0, a.e. (t, x)∈Q. Next we note that gm(u)−g(u)is bounded by 2g(u)∈L1(Q). Therefore we conclude, by using the Dominated Convergence Theorem, that gm(u)−g(u)→0in L1(Q)and, in particular ZΩ gm(u)dx −ZΩ g(u)dx −→ 0in L1(0, T).(4.49)
102 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS With (4.48) and (4.49) we obtain (4.45), finishing the proof. ■ Lemma 4.17. For s≥1we have v∈Cw([0, T]; H1(Ω)) and u∈Cw([0, T]; Ls(Ω)). Proof. For any s≥1, we have v∈L∞(Q)⊂L5/3(Q)and vt∈L5/3(Q), which implies that v∈C([0, T]; L5/3(Ω)) and, in particular, v∈Cw([0, T]; L5/3(Ω)). Since we also have v∈L∞(0, T;H1(Ω)), we use Lemma 1.5 to conclude that v∈Cw([0, T]; H1(Ω)). Considering uwe have u∈L5s/(3+s)(0, T;W1,5s/(3+s)(Ω)) ⊂L5s/(3+s)(0, T; (W1,5s/(4s−3)(Ω))′) and ∂tu∈L5s/(3+s)(0, T; (W1,5s/(4s−3)(Ω))′), which implies that u∈C([0, T]; (W1,5s/(4s−3)(Ω))′)and, in particular, u∈Cw([0, T]; (W1,5s/(4s−3)(Ω))′). Since from (4.44), u∈L∞(0, T ;Ls(Ω)), we conclude, using Lemma 1.5, that u∈Cw([0, T]; Ls(Ω)).■ Next we pass to the limit in inequality (4.43). Because of Lemma 4.16, we conclude that, up to a subsequence, Em(um, zm)(t)−→ E(u, z)(t)a.e. t ∈[0, T].(4.50) Therefore, using the convergences (4.44), the weak lower semicontinuity of the norm (Lemma 1.12) and the almost everywhere pointwise convergence (4.50), we are able to pass to the limit in (4.43) and conclude that, for a.e. t1, t2∈[0, T], with t2> t1, we have E(u, z)(t2) + βZt2 t1ZΩ|∇[u+ 1]s/2|2dx dt +1 4Zt2 t1ZΩ us|∇z|2dx dt +βZt2 t1ZΩ|D2z|2dx dt +Zt2 t1ZΩ |∇z|4 z2dx dt ≤E(u, z)(t1) + C(K2 1)Zt2 t1∥∇z∥2 L2(Ω) dt +K2 1Zt2 t1∥f∥2 L2(Ω) dt. Accounting for the m-independent bound for ∇zmgiven in (4.24) and the strong convergence of ∇zmto ∇zgiven in (4.44), we have ∥∇z∥L2(Q)≤ K1. And since K1=K1(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)), we conclude that there is other constant K= K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)), which is increasing and continuous with respect to ∥f∥Lq(Q), such that, for a.e. t1, t2∈[0, T], with t2> t1, we have E(u, z)(t2) + βZt2 t1ZΩ|∇[u+ 1]s/2|2dx dt +1 4Zt2 t1ZΩ us|∇z|2dx dt +βZt2 t1ZΩ|D2z|2dx dt +Zt2 t1ZΩ |∇z|4 z2dx dt ≤E(u, z)(t1) + K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)). (4.51)
4.2. Existence for the Bilinear Controlled Model 103 To finish, we consider the case s > 1. For simplicity, let us write (4.51) in terms of the energy E(u, z)(·)and the dissipative term D(u, z)(t1, t2) = βZt2 t1ZΩ|∇[u+ 1]s/2|2dx dt +1 4Zt2 t1ZΩ us|∇z|2dx dt +βZt2 t1ZΩ|D2z|2dx dt +Zt2 t1ZΩ |∇z|4 z2dx dt, for a.e. t1, t2∈[0, T], with t2> t1, as E(u, z)(t2) + D(u, z)(t1, t2)≤E(u, z)(t1) + K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)). Now, let t2∈(t1, T]and let {tn 2}nbe a sequence such that tn 2→t2as n→ ∞ and such that, for all n, we have E(u, z)(tn 2) + D(u, z)(t1, tn 2)≤E(u, z)(t1) + K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)).(4.52) If we take the lim inf in both sides of (4.52), we obtain lim inf n→∞ E(u, z)(tn 2) + D(u, z)(t1, t2)≤E(u, z)(t1) + K(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)). Then, from the definition of E(u, z)for s > 1and Lemma 1.5, we have E(u, z)(t2)≤lim inf n→∞ E(u, z)(tn 2) and therefore we conclude that, for s > 1, the energy inequality (4.51) is satisfied for a.e. t1∈[0, T], and for all t2∈(t1, T ]. Case s≥2 Existence of weak solution to (2): The procedure for the case s≥2is slightly different. First we note that, integrating the energy inequality (4.30) from Lemma 4.15 with respect to t, we have ∇Tm(um)s/2is bounded in L2(Q).(4.53) We also remind that we defined g′ m(r) = Tm(r)s−1/(s−1), for s≥2. Then we have Tm(r)s=sZr 0 (Tm)′(θ)Tm(θ)s−1dθ ≤Cs Zr 0 Tm(θ)s−1dθ =Cs(s−1)gm(r). Therefore it also stems from integrating the energy inequality (4.30) with respect to tthat Tm(um)s/2is bounded in L∞(0, T;L2(Ω)).(4.54)
104 Chapter 4. OPTIMAL CONTROL RELATED TO WEAK SOLUTIONS From (4.54) and (4.53) we can conclude that Tm(um)s/2is bounded in L10/3(Q), that is, Tm(um)is bounded in L5s/3(Q).(4.55) For each fixed m∈N, consider the zero measure set N ⊂ (0, T)such that um(t∗,·), vm(t∗,·)∈H1(Ω),∀t∗∈(0, T)\N. Then, for each fixed t∗∈(0, T)\N, let us consider the sets {0≤um≤1}=nx∈Ω0≤um(t∗, x)≤1o and {um≥1}=nx∈Ωum(t∗, x)≥1o. Now note that, since s≥2, we have ZΩ Tm(um(t∗, x)2|∇vm(t∗, x)|2dx ≤Z{0≤um≤1}|∇vm(t∗, x)|2dx +Z{um≥1} Tm(um(t∗, x))s|∇vm(t∗, x)|2dx ≤ZΩ|∇vm(t∗, x)|2dx +ZΩ Tm(um(t∗, x))s|∇vm(t∗, x)|2dx. The last inequality is valid for all t∗∈(0, T)\N, then if we integrate in the variable twe obtain Z∞ 0ZΩ Tm(um(t, x)2|∇vm(t, x)|2dx dt ≤Z∞ 0ZΩ|∇vm(t, x)|2dx dt +Z∞ 0ZΩ Tm(um(t, x))s|∇vm(t, x)|2dx dt. Therefore by (4.17) and (4.33) we can conclude that Tm(um)∇vmis bounded in L2(Q).(4.56) Now we test the um-equation of problem (4.8) by um. This gives us 1 2 d dt∥um∥2 L2(Ω) +∥∇um∥2 L2(Ω) =ZΩ Tm(um)∇vm·∇umdx ≤1 2ZΩ Tm(um)2|∇vm|2dx +1 2∥∇um∥2 L2(Ω), hence we have d dt∥um∥2 L2(Ω) +∥∇um∥2 L2(Ω) ≤ZΩ Tm(um)2|∇vm|2dx.
4.2. Existence for the Bilinear Controlled Model 105 Integrating with respect to t, we conclude from (4.56) that umis bounded in L∞(0, T;L2(Ω)) (4.57) and ∇umis bounded in L2(Q).(4.58) Then, if we consider the um-equation of (4.8), by applying (4.58) and (4.56) we conclude that ∂tumis bounded in L2(0, T; (H1(Ω))′).(4.59) Considering (4.33), (4.17) and (4.55) we conclude from the vm-equation of (4.8) that ∂tvmis bounded in L2(0, T;L3/2(Ω)). Now, using (4.57), (4.58) and (4.59) we can conclude that there is a subsequence of {um}, still denoted by {um}, and a limit function usuch that um−→ uweakly* in L∞(0, T;L2(Ω)), ∇um−→ ∇uweakly in L2(Q), ∂tum−→ uweakly in L20,∞;H1(Ω)′. By applying the compactness result Lemma 1.14, one has um−→ ustrongly in L2(Q). Using the Dominated Convergence Theorem and (4.55) we can also prove that Tm(um)−→ ustrongly in Lp(Q),∀p∈(1,5s/3), and using Lemma 1.11, Tm(um)s−→ usstrongly in Lp(Q),∀p∈(1,5/3). From the global in time estimate (4.54) we can conclude that, up to a subsequence, Tm(um)→uweakly* in L∞(0, T;Ls(Ω)), hence, in particular, u∈L∞(0, T;Ls(Ω)). For s≥2, if we consider the functions vm, we have the same m-independent estimates that we had for s∈[1,2). Then we have the same convergences given in (4.42). Following the ideas of Subsection 4.2.4, for s∈[1,2), we can identify the limits of Tm(um)∇vmand Tm(um)svmwith u∇vand usv, respectively.
112 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS q≤5/2. We also recall that the hypothesis q > 5/2is essential in Lemma 4.8 and therefore in the proof of existence of weak solutions to (2) with control f∈Lq(Q). Moreover, if q > 5/2then Xq,→L∞(Q). Then Theorem 5.1 also gives an additional regularity hypothesis over a weak solution of the controlled problem (2) such that the cell density udoes not blow up at finite time. □ The second main contribution is the existence of optimal solution to the following optimal control problem. Let Fbe a closed and convex subset of Lq(Q), for a given q > 5/2. Consider the cost functional J:Lsq(Q)×L2(Q)×F −→ R given by J(u, v, f) := γu sq ZT 0∥u(t)−ud(t)∥sq Lsq dt +γv 2ZT 0∥v(t)−vd(t)∥2 L2dt +γf qZT 0∥f(t)∥q Lq(Ωc)dt, (5.2) where (ud, vd)∈Lsq(Q)×L2(Q)represents the desired states and the parameters γu, γv, γf≥0measure the costs of the states and control. In addition, we will consider γu, γfsatisfying γu>0and γf>0or Fis bounded in Lq(Q).(5.3) We are going to minimize J(u, v, f)subject to the admissible set of the triples (u, v, f)satisfying the controlled problem (2) in the strong setting, that is Sad ={(u, v, f)∈Xq×Xq×F; (u, v)is the strong solution of (2) with control f}. Then, the following minimization problem is considered: (min J(u, v, f) subject to (u, v, f)∈Sad.(5.4) Since, given f∈ F, one can not assure the existence of a strong solution (u, v) associated to f, we are going to assume that Sad =∅.(5.5) Remark 5.3. Analogously to [30] and [44], if Ωc= Ω, that is, if the control acts in the whole domain, then (5.5) holds. In addition, when Ωis a 2D domain then (5.5) also holds. Indeed, from Theorem 2.5 we have the existence and uniqueness of weak solution (u, v)with u∈L∞(Q), of the uncontrolled problem, that is (2) with f= 0. Since (u, v)and f= 0 satisfy the hypotheses of Theorem 5.1, we conclude that (u, v)∈Xq×Xqis the strong solution of (2) with f≡0. In particular, (u, v, 0) ∈Sad and hence Sad =∅.□
5.1. Main Results 113 Theorem 5.4 (Existence of optimal control). Assume Sad =∅. Then the optimal control problem (5.4) has at least one global optimal solution (u, v, f)∈Sad. The third main contribution of this chapter is the existence and uniqueness of Lagrange multipliers associated to any local optimal solution of (5.4). Let q′=q/(q− 1), the conjugate exponent of q, let (u, v, f)∈Sad be a local optimal solution of (5.4) and consider the following Lagrange multiplier problem for (λ, η)associated to (u, v, f): −∂tλ−∆λ−∇v·∇λ+sus−1vη =gλ, −∂tη−∆η+usη−fη 1Ωc+∇·(u∇λ) = gη, ∂nλ|Γ=∂nη|Γ= 0, λ(T, x) = η(T, x) = 0, (5.6) where gλ=γusgn(u−ud)|u−ud|sq−1, gη=γv(v−vd).(5.7) Definition 5.5. (Very weak solution of (5.6))Let s≥1,q > 5/2and f∈Lq(Q). A pair (λ, η)∈Lp′(Q)×Lp′(Q)is called a very weak solution of (5.6) if (λ, η)satisfies (5.6) in the sense of the dual space of Xp×Xp, that is, the following variational formulation holds for any U, V ∈Xqwith ∂nU|Γ=∂nV|Γ= 0 and U(0) = V(0) = 0: ZT 0ZΩ λ∂tU−∆U+∇·(U∇v)dx dt +ZT 0ZΩ sus−1vη U dx dt =ZT 0ZΩ gλU dx dt, (5.8) ZT 0ZΩ η∂tV−∆V+usV−fV 1Ωcdx dt +ZT 0ZΩ λ∇·(u∇V)dx dt =ZT 0ZΩ gηV dx dt. (5.9) □ Theorem 5.6 (Existence of Lagrange multipliers). Let (u, v, f)∈Sad be a local optimal solution of (5.4). Then there exists a unique Lagrange multiplier (λ, η)∈ Lq′(Q)×Lq′(Q)which is a very weak solution of the optimality system (5.6) and satisfies the optimality condition ZT 0ZΩc (γfsgn(f)|f|q−1+vη)(f−f)dx dt ≥0,∀f∈ F.(5.10) Remark 5.7. If γf>0and there is no convex constraint on the control, that is F=Lq(Q), then (5.10) is equivalent to γfsgn(f)|f|q−1+vη = 0.
114 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS Since v≥0, we conclude that f=−sgn(η)1 γf v|η|1/(q−1) . □ Remark 5.8. The key to establish the existence of a Lagrange multiplier is to prove the existence of solution to the linearized problem given in (5.38) below. To help with this proof, in the Appendix C, we provide a result of existence of solution to an adequate generic parabolic linear system. This result is also useful in the study of the regularity of the Lagrange multiplier provided by Theorem 5.6 that is carried out in Theorem 5.9 below. □ Finally, we establish a result on the additional regularity of the Lagrange multiplier (λ, η)given by Theorem 5.6 depending on the Lpregularity of the right hand side term gλ. Theorem 5.9. Let (u, v, f)∈Sad be a local optimal of problem (5.4). We have: 1. if gλ∈Lp(Q), for p∈[10/9,10/7), then the Lagrange multiplier (λ, η)∈ L2(Q)×L2(Q)and satisfies (5.6) in the very weak sense; 2. if gλ∈Lp(Q), for p∈[10/7,2], then the Lagrange multiplier (λ, η)∈Xp×Xp and satisfies (5.6) in the strong sense. Remark 5.10. Since we consider vd∈L2(Q), which implies gη∈L2(Q), the previous analysis for p > 2does not seem to lead to more relevant conclusions. □ Remark 5.11. To guarantee that the terms of the functional Jgiven in (5.2) make sense it is enough that ud∈L˜q(Q), with ˜q≥sq, and vd∈L2(Q). With this regularity, gη∈L2(Q)and gλ∈Lp(Q), where p=p(s, q, ˜q) = ˜q/(sq −1). Hence the regularity of gλdepends on s≥1,q > 5/2and ˜q≥sq, and is decreasing with respect to swith p(s, q, ˜q)→1as s→ ∞. For instance if ˜q=sq, we have p=sq/(sq −1). In this case, since s≥1and q > 5/2, then p∈(1,5/3). Let us fix q > 5/2close to 5/2and vary the values of s. Then, if s∈[1,10/3q]we are in item 2 of Theorem 5.9, and if s∈(10/3q, 10/q]we are in item 1 of Theorem 5.9. But, if s > 10/q then p∈(1,10/9) and we only have the existence and uniqueness given in Theorem 5.6 because, in this case, additional regularity for the Lagrange multiplier is not clear. □ The rest of the chapter is organized as follows. In Section 5.2 we prove Theorem 5.1 after establishing some preliminary results. In Section 5.3 we prove Theorem 5.4. In Section 5.4 we prove Theorem 5.6 and study the additional regularity of the Lagrange multiplier demonstrating Theorem 5.9. 5.2 Regularity Criterion The main objective of the present section is to prove Theorem 5.1. To do it, we first introduce and prove a series of useful results.
5.2. Regularity Criterion 115 Lemma 5.12. Let Ω⊂R3be a bounded Lipschitz domain. Then we have: 1. Xp,→L5p/(5−2p)(Q), if p∈[1,5/2); 2. Xp,→L∞(Lq), for all q∈[1,∞), if p= 5/2; 3. Xp,→L∞(Q)if p > 5/2. Proof. By definition of Xp, if w∈Xpthen we have w∈C(W2−2/p,p)∩Lp(W2,p). If p∈[1,5/2), this implies w∈C(W2−2/p,p)∩Lp(W2,p),→L∞(L3p/(5−2p))∩Lp(W2,p). Then, using Lemma 1.4 yields the desired result. For p= 5/2we use the continuous injection W2−2/p,p(Ω) ,→Lq(Ω), for all q∈[1,∞), and for p > 5/2, the continuous injection W2−2/p,p(Ω) ,→L∞(Ω).■ Lemma 5.13. Let Ω⊂R3be a bounded Lipschitz domain and p∈(1,5). If w∈Xp then ∇w∈L5p/(5−p)(Q). Moreover, there is a constant C > 0such that, ∥∇w∥L5p/(5−p)(Q)≤C∥w∥Xp,∀w∈Xp. Proof. Suppose p∈[2,5). Since w∈Xp, we have by definition that ∇w∈L∞(W1−2/p,p)∩Lp(W1,p),→L∞(L3p/(5−p))∩Lp(W1,p) and using Lemma 1.4 we conclude that ∇w∈L5p/(5−p)(Q). Now we must deal with the case p∈(1,2). In this case we have, from the definition of Xp, w∈L∞(W2−2/p,p)∩Lp(W2,p(Ω)). Then we have D2−2/pw∈L∞(Lp)∩Lp(W2/p,p) and this implies that D2−2/pw∈L∞(Lp)∩Lp(Wβ,3p/(1+βp)),for any β∈(1,2/p). Now, using Lemma 1.2 with α=2 p−1, β =β, ˜p=3p 1 + βp, γ = 0 and ˜q=p we obtain λ=2 p−1and r=3βp2 −βp2+ (5β+ 2)p−4
116 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS with ∥D2−2/pw∥r W2/p−1,r(Ω) ≤C∥D2−2/pw∥(2/p−1)r Wβ,3p/(1+βp)(Ω)∥D2−2/pw∥(2−2/p)r Lp(Ω) . For the right hand side of this inequality to be integrable, we must choose βsuch that (2/p −1)r=p. Therefore, choosing β=10 −5p 5p−p2 we conclude that ∇w∈Lr(Q), with r=10p−5p2 p2−7p+ 10 =5p 5−p. ■ Lemma 5.14. Let (u, v)be a weak solution of (2). Suppose u∈Lp(Q), for some p > 5/3, and v∈Xq, for some q > 5/2. We conclude that u∈Xpq/(p+q)and that there is C=C(∥u∥Lp(Q),∥∇u∥L5/4(Q),∥v∥Xq)>0, which is continuous and increasing with respect to each entry, ∥u∥Lp(Q),∥∇u∥L5/4(Q)and ∥v∥Xq, such that ∥u∥Xpq/(p+q)≤C(∥u∥Lp(Q),∥∇u∥L5/4(Q),∥v∥Xq).(5.11) The result is also valid for p=∞and, in this case, we conclude that u∈Xqwith ∥u∥Xq≤C(∥u∥L∞(Q),∥∇u∥L5/4(Q),∥v∥Xq).(5.12) Proof. The basic idea of the proof is a bootstrapping in the u-equation of (2) that allows one to arrive at the desired regularity in a finite number of iterations. We are going to consider the case p < ∞and, with small adaptations, one can follow the same reasoning to prove the result for p=∞. Also, we are going to prove that u∈Xpq/(p+q). The proofs of (5.11) and (5.12) come from the fact that all the results used along this proof, such as Lemmas 5.13 and 1.13, for example, give us continuous injections. Indeed, since the number of steps of the procedure of gaining regularity, to be presented in what follows, is finite, one can follow the estimates furnished by Lemmas 5.13 and 1.13 each time they are applied and, at the end, conclude (5.11) and (5.12). Bearing that in mind, we proceed with the proof of u∈Xpq/(p+q), for finite p. Using Lemma 5.13 for vwe conclude that ∇v∈L5q/(5−q)(Q). Since q > 5/2we have, in particular, that there is β > 1such that ∇v∈L5β(Q).(5.13)
5.2. Regularity Criterion 117 And since (u, v)is a weak solution of (2) we have, in particular, ∇u∈L5/4(Q). By hypothesis and by the definition of Xqwe have u∈Lp(Q),with p > 5/3and ∆v∈Lq(Q)with q > 5/2. Considering these regularities, we have the u-equation satisfied in the strong sense ∂tu−∆u=−u∆v−∇u·∇v(5.14) with u∆v∈Lpq/(p+q)(Q),with pq p+q>1,(5.15) and ∇u·∇v∈Lr0,with r0=5β 4β+ 1 >1.(5.16) Hence, using Lemma 1.13 for (5.14) we conclude that u∈Xr,with r= min nr0,pq p+qo>1. If r0≥pq p+qthen r=pq p+qand the proof is finished. Therefore let us treat the case in which r0<pq p+q. Since for u∆vwe already have (5.15), we focus on enhancing the regularity of the term ∇u·∇v. In this case, we have u∈Xr0. Using Lemma 5.13 we obtain ∇u∈L5r0/(5−r0)(Q) Considering this regularity and (5.13), where β > 1, and (5.16), where r0>1, the new regularity of ∇u·∇vis Lγ(Q), with γ=5β 5β−(β−1)r0 r0>5β 4β+ 1 r0. Define α= 5β/(4β+ 1). Note that α=r0>1and γ > αr0. Then, let us define r1=αr0. Since α > 1, we have r1> r0>1. Now, if r1<pq p+qthen, from Lemma 1.13, we have u∈Xr1. Proceeding by induction, if we have ∇u·∇v∈Lrn−1, with rn−1=αn−1r0<pq p+q, then we have u∈Xrn−1and, by Lemma 5.13 we obtain ∇u∈L5rn−1/(5−rn−1)(Q). And using again (5.13), where β > 1, and (5.16), where r0>1, the new regularity of ∇u·∇vis Lγ(Q), with γ=5β 5β−(β−1)rn−1 rn−1>5β 4β+ 1 rn−1=αrn−1=αnr0.
118 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS Therefore we can define rn=αnr0and again using Lemma 1.13 for (5.14) we conclude that u∈Xr,with r= min nαnr0,pq p+qo. Since α > 1, there is an index n0such that αn0r0<pq p+qbut αn0+1r0≥pq p+q. Therefore we conclude, using the result proved by induction, that in fact we have u∈Xpq/(p+q), as we wanted to prove. ■ Theorem 5.15. Let (u, v)be a weak solution of (2) given f∈Lq(Q),q > 5/2. If, additionally, us∈Lq(Q)then v∈Xqand u∈L∞(Q). This implies, in particular, that ∇v∈L5q/(5−q)(Q),→L5(Q),u∈Xqand that (u, v)is the unique strong solution of (2). Moreover, there is C=C(∥us∥Lq(Q),∥f∥Lq(Q),∥∇u∥L5/4(Q))>0, which is continuous and increasing with respect to each entry, ∥us∥Lq(Q),∥f∥Lqand ∥∇u∥L5/4(Q), such that ∥(u, v)∥Xq×Xq≤C(∥us∥Lq(Q),∥f∥Lq(Q),∥∇u∥L5/4(Q)).(5.17) Proof. Analogously to Lemma 5.14, we are going to prove that (u, v)∈Xq×Xqand, since the number of steps of the procedure of gaining regularity, to be presented in what follows, is finite, the proof of (5.17) is a consequence of the estimates furnished by Lemmas 1.13, 5.12 and 5.14. Considering the regularity v∈L∞(Q)given by the regularity of the weak solution (u, v)of (2), if we have us, f ∈Lq(Q)then, by applying Lemma 1.13 to the v-equation of (2), we conclude that v∈Xq.(5.18) Now denote p0=sq > 5/2. Then we have u∈Lp0(Q),∆v∈Lq(Q)and we can apply Lemma 5.14 to conclude that u∈Xqp0/(q+p0). At least in this first iterations, we assume that we are in the case in which we have qp0/(q+p0)<5/2. Then, if now we apply Lemma 5.12 we obtain u∈Lr(Q), with r=5q 5q+ 5p0−2qp0 p0. Since q > 5/2we can say that that there is α > 1such that q= 5α/2. Then using the fact that p0≥q, we obtain r=5q 5q−5(α−1)p0 p0=q q−(α−1)p0 p0=1 + (α−1)p0 q−(α−1)p0p0 >1 + (α−1)p0 qp0≥(1 + α−1)p0=αp0
5.2. Regularity Criterion 119 Define p1=αp0. Since r≥p1we have, in particular u∈Lp1(Q). Proceeding by induction, if we have u∈Lpn−1(Q), with pn−1=αn−1p0≥qsatisfying qpn−1/(q+pn−1)≤5/2then we can apply Lemmas 5.14 and 5.12 and conclude that u∈Lpn(Q),with pn=αnp0. As a consequence, we can apply Lemma 5.14 and obtain u∈Xqpn/(q+pn). Since α > 1,pn=αnp0=αnsq grows as nincreases in such a way that there is an index n0such that qpn0−1/(q+pn0−1)≤5/2 but applying the result proved by induction we conclude that u∈Xqpn0/(q+pn0),with qpn0 (q+pn0)>5/2. Hence, applying Lemma 5.12 we obtain u∈L∞(Q). Finally, once we have u∈L∞(Q), we use (5.18) and Lemma 5.14 to conclude that u∈Xq, finishing the proof. ■ Now let z=√v+α2, for some α > 0, and consider the energy E(u, z)(t) = s 4ZΩ g(u(t, x)) dx +1 2ZΩ|∇z(t, x)|2dx, where g(u) = (u+ 1)ln(u+ 1) −u, if s= 1, us s(s−1),if s > 1. We have the following. Lemma 5.16. Let (u, v)be the strong solution of (2) given f∈Lq(Q), with q > 5/2, let αbe a positive real number and z=√v+α2. There is α0>0, independent of (u, v, f), such that if 0< α ≤α0then (u, v)satisfies 0< α ≤z(t, x)≤ K1(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω))(5.19)
120 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS and the energy inequality E(u, z)(t2) + βZt2 t1ZΩ|∇[u+ 1]s/2|2dx dt +1 4Zt2 t1ZΩ us|∇z|2dx dt +βZt2 t1ZΩ|D2z|2dx dt +Zt2 t1ZΩ |∇z|4 z2dx dt ≤E(u, z)(t1) + K1(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)), (5.20) for a.e. t1, t2∈[0, T], with t2> t1; where K1(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω))is a continuous and increasing function with respect to ∥f∥Lq(Q)and β > 0is a constant, independent of (u, v, f). Sketch of the proof. In Chapter 4, given f∈Lq(Q), the existence of weak solutions of (2) satisfying the pointwise bound (5.19) and the energy inequality (5.20) is proved. In the present lemma, we state that the unique strong solution of (2) satisfies (5.19) and (5.20). For the proof of this statement we refer the reader to Chapter 4, where the authors first prove the pointwise bound and the energy inequality for the solution of a truncated problem, depending on a parameter m, and then pass to the limit as m→ ∞, proving (5.19) and (5.20) for the weak solution of (2) obtained through this limit. In the present case, due to the strong regularity, the ideas of Chapter 4 can be applied directly to the strong solution (u, v)of (2), yielding the desired result. ■ Now the idea is to eliminate the dependence on ∇uin (5.17). Using the results developed in the present section we are finally in position of proving Theorem 5.1. 5.2.1 Proof of Theorem 5.1 Let (u, v)be a weak solution of (2) given f∈Lq(Q),q > 5/2, satisfying, additionally, us∈Lq(Q). It stems from Theorem 5.15 that (u, v)∈Xq×Xqis the strong solution of (2) satisfying (5.17). Therefore it suffices to prove that ∥∇u∥L5/4(Q)can be estimated in terms of ∥f∥Lq(Q)to obtain (5.1), finishing the proof. We analyze the cases s∈[1,2) and s≥2separately. Case s∈[1,2):From (5.20) with t1= 0 (and by integrating the u-equation of (2), in the case s= 1), we have ∥(u+ 1)s/2∥L∞(L2)≤E(u, z)(0) + K1(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)), ∥∇[u+ 1]s/2∥L2(Q)≤E(u, z)(0) + K1(∥f∥Lq(Q),∥v0∥W2−2/q,q(Ω)). (5.21) This implies that there exists C1=C1(∥f∥Lq(Q))>0(C1also depends on (u0, v0), but since the initial data are fixed we omit it from now on) which is continuous and increasing with respect to ∥f∥Lq(Q)and such that ∥(u+ 1)s/2∥L∞(L2)∩L2(H1)≤C2(∥f∥Lq(Q)).
5.2. Regularity Criterion 121 In particular, by interpolation we have ∥(u+ 1)∥L5s/3(Q)≤C3(∥f∥Lq(Q)).(5.22) Now, reminding that s∈[1,2), if we use (5.21), (5.22) and the relation ∇u=∇(u+ 1) = ∇(u+ 1)s/22/s =2 s(u+ 1)1−s/2∇(u+ 1)s/2, we conclude that there exists C4=C4(∥f∥Lq(Q))>0which is continuous and increasing with respect to ∥f∥Lq(Q)satisfying ∥∇u∥L5s/(3+s)(Q)≤C4(∥f∥Lq(Q)). Since s≥1, we have 5s/(3 + s)≥5/4and this implies, in particular, that there is C > 0such that ∥∇u∥L5/4(Q)≤C C4(∥f∥Lq(Q)).(5.23) Therefore, using (5.23) in (5.17) we conclude (5.1). Case s≥2:From (5.19) and (5.20) with t1= 0 we conclude, in particular, that there is C1=C1(∥f∥Lq(Q))>0such that ∥z∥L∞(Q),∥∇z∥L∞(L2),∥us/2∇z∥L2(Q)≤C1(∥f∥Lq(Q)).(5.24) Now, let us consider the sets {0≤u≤1}=n(t, x)∈Ω0≤u(t, x)≤1o and {u≥1}=n(t, x)∈Ωu(t, x)≥1o. Note that, since s≥2, we have ZT 0ZΩ u(t, x)2|∇z(t, x)|2dx dt ≤ZT 0Z{0≤u≤1}|∇z(t, x)|2dx dt +ZT 0Z{u≥1} u(t, x))s|∇z(t, x)|2dx dt ≤ZT 0ZΩ|∇z(t, x)|2dx +ZΩ u(t, x))s|∇z(t, x)|2dx. Thus, by (5.24) we conclude that there is C > 0such that ∥us/2∇z∥L2(Q)≤C C1(∥f∥Lq(Q)).(5.25)
128 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS L10q/(10n+(5−4n)q)(Q)then we have Z2∈L10q/(10(n+1)+(5−4(n+1))q)(Q). Recalling that q > 5/2, if we study the function n7→ 10q/(10n+ (5 −4n)q), we conclude that there exists n0, n1∈Nsuch that, 10q/(10n0+(5−4n0)q)< q and 10q/(10n1+(5−4n1)q)≥q and thus we proved that the right hand side of the U-equation of (5.38) belongs to Lq(Q). From Lemma 1.13, we conclude that U∈Xq.■ Now we are in position of proving Theorem 5.6, that is, we are able to prove that, for any r= (u, v, f)∈Sad local optimal solution of problem (5.35), there exists a unique Lagrange multiplier for (5.35) at r. 5.4.3 Proof of Theorem 5.6 The proof is divided in two steps: the existence of Lagrange multiplier and the uniqueness. Step 1: Existence From Lemmas 5.21, 5.22 and Theorem 5.23 we have all the hypotheses of Theorem 5.20 fulfilled. Therefore there exists a Lagrange multiplier ξ= (λ, η)∈Lq′(Q)×Lq′(Q) satisfying, according to (5.34), the inequality L′ r(r, λ, η)[c] = J′(r)[c]−⟨λ, G′ 1(r)[c]⟩Lq′(Q)−⟨η, G′ 2(r)[c]⟩Lq′(Q)≥0,(5.44) for all c= (U, V, F)∈ˆ Xq׈ Xq× C(f). Then, using (5.36) and (5.37) in (5.44) we conclude that there exists a Lagrange multiplier ξ= (λ, η)∈Lq′(Q)×Lq′(Q)such that, for all (U, V, F)∈ˆ Xq׈ Xq×C(f), we have γuZT 0ZΩ sgn(u−ud)|u−ud|sq−1U dx dt +γvZT 0ZΩ (v−vd)V dx dt +γfZT 0ZΩc sgn(f)|f|q−1F dx dt −ZT 0ZΩ∂tU−∆U+∇·(U∇v) +∇·(u∇V)λ dx dt −ZT 0ZΩ∂tV−∆V+sus−1Uv +usV −fV 1Ωc−Fv 1Ωcη dx dt ≥0 (5.45) Since (5.45) is valid for all (U, V, F)∈ˆ Xq׈ Xq×C(f), we can deduce the optimality system (5.6) and the optimality condition (5.10). In fact, since ˆ Xqis a vectorial space, if U, V ∈ˆ Xqthen −U, −V∈ˆ Xq. With this in mind, if we take (V, F ) = (0,0) in (5.45) we obtain ZT 0ZΩ∂tU−∆U+∇·(U∇v)λ dx dt +ZT 0ZΩ sus−1Uvη dx dt =γuZT 0ZΩ sgn(u−ud)|u−ud|sq−1U dx dt, ∀U∈ˆ Xq. (5.46)
5.4. First Order Necessary Conditions for a Local Optimal Solution 129 On the other hand, if we take (U, F) = (0,0) in (5.45) we obtain ZT 0ZΩ∂tV−∆V+usV−fV 1Ωcη dx dt +ZT 0ZΩ∇·(u∇V)λ dx dt =γvZT 0ZΩ (v−vd)V dx dt, ∀V∈ˆ Xq. (5.47) Note that if, considering formal computations, we integrate by parts the terms of (5.46) and (5.47), passing the derivatives from (U, V )to (λ, η), we see that (λ, η)∈ Lq′(Q)×Lq′(Q)satisfying (5.46) and (5.47) are actually very weak solutions of (5.6). Also, choosing (U, V ) = 0 in (5.45) leads us to γfZT 0ZΩc sgn(f)|f|q−1F dx dt +ZT 0ZΩc vηF dx dt ≥0,∀F∈ C(f). Hence, taking F=θ(f−f), with θ≥0and for all f∈ F finally gives (5.10). Step 2: Uniqueness Now, to prove the uniqueness, we suppose that there are two Lagrange multipliers (λ1, η1),(λ2, η2)∈Lq′(Q)×Lq′(Q)satisfying (5.46) and (5.47). Let (˜ λ, ˜η)=(λ2, η2)− (λ1, η1), we will prove that ˜ λ= ˜η= 0. Subtracting the equation satisfied by (λ1, η1) from the respective equation that is satisfied by (λ2, η2)we conclude that (˜ λ, ˜η)satisfies ZT 0ZΩ∂tU−∆U+∇·(U∇v)˜ λ+sus−1Uv ˜η dx dt = 0,∀U∈ˆ Xq,(5.48) ZT 0ZΩ∂tV−∆V+usV−fV 1Ωc˜η+∇·(u∇V)˜ λ dx dt = 0,(5.49) for all V∈ˆ Xq. Summing (5.48) and (5.49) we obtain ZT 0ZΩ∂tU−∆U+∇·(U∇v) + ∇·(u∇V)˜ λ dx dt +ZT 0ZΩ∂tV−∆V+usV+sus−1Uv −fV 1Ωc˜η dx dt = 0, (5.50) for all (U, V )∈ˆ Xq׈ Xq. Now let gU=sgn(˜ λ)|˜ λ|1/(q−1) and gV=sgn(˜η)|˜η|1/(q−1). Since (˜ λ, ˜η)∈Lq′(Q)×Lq′(Q), with q′=q/(q−1), we have gU, gV∈Lq(Q). Take (U, V )∈ˆ Xq׈ Xqas the unique strong solution of (5.39) for this choice of gUand gV, therefore we have from (5.50) ∥˜ λ∥q′ Lq′(Q)+∥˜η∥q′ Lq′(Q)= 0, which implies that ˜ λ= ˜η= 0.
130 Chapter 5. OPTIMAL CONTROL SUBJECT TO STRONG SOLUTIONS 5.4.4 Proof of Theorem 5.9 Case gλ∈Lp(Q), with p∈[10/9,10/7): Let t=T−t, then the backward problem (5.6) is equivalent to the forward one ∂tλ−∆λ=∇v·∇λ−sus−1vη +γusgn(u−ud)|u−ud|sq−1, ∂tη−∆η=−usη+fη 1Ωc−∇·(u∇λ) + γv(v−vd), ∂nλ|Γ=∂nη|Γ= 0, λ(0, x) = η(0, x)=0. (5.51) Then, applying Theorem C.1, item 1b, with U=η,V=λ,a1=us−f1Ωc,b1= 0, c1= 0,d=u,gU=γv(v−vd),a2= 0,b2=sus−1v,c2=∇vand gV=γusgn(u− ud)|u−ud|sq−1we conclude that there is a very weak solution (˜ λ, ˜η)∈L2(Q)×L2(Q) of (5.51) and, therefore, of (5.6). since q > 5/2>2we have q′<2and hence (˜ λ, ˜η)∈Lq′(Q)×Lq′(Q). Then, from the uniqueness result of Theorem 5.6 we conclude that (˜ λ, ˜η)is equal to the Lagrange multiplier (λ, η)furnished by theorem 5.6 and (λ, η)∈L2(Q)×L2(Q). Case gλ∈Lp(Q), with p∈[10/7,2): Using the same argument of the previous case, this time applying Theorem C.1, item 1a, with gU=γv(v−vd)∈L2(Q),→L10/7(Q)and gV=γusgn(u−ud)|u−ud|sq−1∈ Lp(Q),→L10/7(Q), we conclude that the Lagrange multiplier (λ, η)furnished by Theorem 5.6 is a weak solution of (5.6) with regularity (λ, η)∈W2×W2. Now we enhance the regularity of (λ, η)by means of a bootstrap procedure analogous to the used in the proof of Theorem 5.23. We first enhance the regularity of λ. Since in the right hand side of the λ-equation we have −sus−1vη +γusgn(u−ud)|u−ud|sq−1∈ L10/3(Q) + Lp(Q), with p≤2, we apply the procedure and conclude that λ∈Xp. Next we apply the bootstrap argument to the η-equation. Since in the right of the η-equation we have −∇·(u∇λ+γv(v−vd)∈Lp(Q) + L2(Q), with p≤2, we apply the procedure and conclude that η∈Xp, finishing the proof.
131 CONCLUSION In this thesis we have focused on the chemotaxis-consumption models (∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv, ∂nu|Γ=∂nv|Γ= 0, u(0) = u0, v(0) = v0,(5.52) where ∇·(u∇v)is the chemotaxis term and usvis the consumption term, with s≥1, and on optimal control problems subject to the controlled model ∂tu−∆u=−∇·(u∇v), ∂tv−∆v=−usv+fv1Ωc, ∂nu|Γ=∂nv|Γ= 0, u(0) = u0, v(0) = v0, (5.53) where f: (0, T)×Ω→Ris the control, being T > 0a fixed and finite final time, Ωc⊂Ωis the control domain and 1Ωcis its characteristic function. Reviewing the available literature about the models (5.52) we found works addressing only the case s= 1 that, in addition, were developed using classical in time solution tools and therefore considering smooth coefficients and smooth domains. This is not the most adequate framework to study the numerical approximation of (5.52) or optimal control problems subject to the controlled problem (5.53). In fact, when studying the numerical approximation of PDEs, one usually employs weak formulations of the problem posed in more general domains. Moreover, in the controlled problem (5.53), the control f=f(t, x)is usually a Lq-function, acting as a nonsmooth coefficient in the chemical equation. Therefore we identified the opportunity and the need of extending the existing theory about problem (5.52) and, in Chapter 2, we studied the existence and regularity of solutions of the models (1) in a weak setting, varying the power s≥1. We developed the results in terms of the regularity of the Neumann-Poisson problem (1.5) and, when necessary, of Hypothesis (H1). This allowed us to enlarge the class of the considered domains, when compared to the previous literature. By means of a regularization procedure using adequate truncated models and the cancellation between the attraction and consumption effects (see Subsection 2.3.1), we established the existence of uniform in time weak solutions in 3Ddomains, and uniqueness and regularity in 2D(or 1D) domains. The results of Chapter 2 were published in [11]. Another novelty, when compared to the available literature, was the study of the chemotaxis-consumption models considering all the powers s≥1. The regularity of the chemical concentration vdoes not depend on this power s, while the regularity of
132 CONCLUSION the density of cells uincreases as s∈[1,2] increases. For s > 2, only the Lpregularity of uincreases with s. This was possible, in part, because of the dissipative term ZΩ us|∇v|2dx that appears in the formal energy inequality (2.22) and its rigorous version in Subsection 2.3.2. This dissipative term seems to be an interesting feature of the present chemotaxis-consumption models, specially when compared to the close related chemorepulsionproduction model. Indeed, concerning the chemorepulsion-production model, in [24, 21, 22], the authors also rely on the cancellation of the repulsion and production effects to show existence of weak solutions and introduce a potential production term up, for p∈(1,2] (the case p= 1 was studied previously in [10]). On the other hand, it was not possible to control the effect of the production term upwhen p > 2for large values of uand prove existence of weak solutions for the chemorepulsion model. Based on the theory developed in Chapter 2, in Chapter 3, we designed a time discrete scheme for the chemotaxis-consumption models (5.52). Using the change of variables z=√v+α2and a upper truncation of uin the nonlinear chemotaxis and consumption terms we proposed a Backward Euler scheme for the (u, z)problem and two different ways of retrieving an approximation for the function v. We proved the existence of solution to the time discrete scheme, uniform in time a priori estimates and convergence of the scheme towards a weak solution (u, v)of the chemotaxisconsumption model. We remark that, although the existence of solution was proved in Chapter 2 (and published in [11]), the design of a convergent time discrete scheme was not straightforward. Indeed, in order to obtain a time discrete scheme satisfying an energy law, independently of the time step size, it was essential to propose the time discrete scheme in terms variable z=√v+α2instead of the variable v. In addition, Lemma 3.10 was decisive to prove convergence in the case s∈[1,2). Indeed, regarding the backward Euler method, in the study of the convergence, we define the approximations ukr m,uk m, zkr mand zk mas in (3.24) and we have to prove that ukr m−uk mand zkr m−zk mgo to zero in some norm as (m, k)→(∞,0). This is clear when s≥2, because, for zkr m−zk m, one can conclude (3.30) directly from (3.29) and, for ukr m−uk m, one can conclude (3.64) directly from (3.63). On the other hand, when s∈[1,2), we must rely on the term (3.32), where f′′ is not strictly positive and therefore the desired conclusion is not immediate, hence the analysis carried out in Lemma 3.10 was necessary. In Chapter 4, we studied optimal control problems related to weak solutions of (5.53). To do it, we introduced the concept of weak solutions of the controlled model (5.53) satisfying an energy inequality. To the best of our knowledge, this was the first time that the concept of weak solution with energy inequality was applied to this purpose. Next, we considered an optimal control problem for which we proved existence of global optimal solution and discussed its relation with two other related
133 optimal control problems that might be of interest. In Chapter 5, we studied an optimal control problem subject to strong solutions of (5.53). We introduced the appropriate concept of strong solution of the controlled problem (5.53), given the control f, and then proved a more generic and sharp regularity criterion, when compared to the available literature, that allows us to get existence and uniqueness of global-in-time strong solutions. Using this regularity criterion, we showed the existence of a global optimal solution, under the hypothesis that the admissible set is nonempty. Next, we established first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers. We also would like to remark that we have done a great effort in order to ensure that, along the whole work, all of our computations are rigorous. In what follows, based in what has been done in this thesis, we present some possible perspectives of future works. Future works In Chapter 2 we analyzed problem (5.52) using a regularization procedure and we wrote the results in terms of the regularity of the Poisson-Neumann problem (1.5) and Hypothesis (H1). Hypothesis (H1) is used in the proof of Lemma 2.10, in Appendix B, because it is sufficient to guarantee that we can apply Lemma B.3 to a certain boundary integral. In Appendix A we showed a large class of domains satisfy this hypothesis. Therefore, if one finds a less restrictive condition that makes the application of Lemma B.3 possible (the functions do not need to be pointwisely defined on the whole boundary, but only to be pointwisely defined over the regular components of the boundary, for example) or a less restrictive lemma to deal with the boundary integral term, then it is probable that one can enlarge even more the class of considered domains. On the other hand, the regularity of the Poisson-Neumann problem (1.5) is considered because of the the self-diffusion operator and the boundary conditions that are present in problem (5.52): the Laplacian operator and the homogeneous Neumann boundary conditions. The effort of putting the results of Chapter 2 in terms of the regularity of an adequate boundary-value problem related to the self-diffusion operator is probably a useful structure to the extension of the analysis of Chapter 2 to chemotaxis-consumption models with more general self-diffusion operators and their corresponding boundary conditions. As an example we refer the reader to [23], where a chemorepulsion-production model with variable diffusion coefficients is studied. In the previous literature we also find a considerable number of works about chemotaxis-consumption models coupled with models for fluids, namely, the (Navier-) Stokes equations. Hence, another interesting and open question is whether all the aforementioned technique used in Chapter 2 to study problem (5.52) can be extended to the chemotaxis-fluids models. It is probable that the present approach to (5.52)
134 CONCLUSION could be combined with the regularization for the fluid equations used in [62], for example. In Chapter 3 we proposed a conservative, energy stable, positivity preserving and convergent time discrete scheme to approximate (5.52). Now, an interesting original work on the numerical approximation of (5.52) should propose a fully discrete scheme preserving properties of the solutions in the discrete level, such as conservation of the total population of cells, positivity and energy estimates, and which converges to a weak solution of (5.52). The convergence is probably one of the most important features of a discrete scheme, but in the case of the chemotaxis models, there is evidence that the preservation of the positivity in the discrete level is also relevant. Indeed, in [25], although the considered schemes are convergent, the authors show through numerical simulations that the scheme with “approximated” positivity of the discrete solutions prevents spurious oscillations. As far as we know, four techniques are used with the objective of preserving positivity of the discrete solutions of chemotaxis models: 1. the estimate of a singular functional and Finite Element Method (FEM) (implying approximate positivity); 2. “upwind” schemes in the context of Finite Volume Method (FVM), Discontinuous Galerkin method (DG) or FEM; 3. mass lumping combined with FEM in 2D domains; 4. and FEM schemes with stabilization terms used to preserve pointwise bounds. Concerning the chemotaxis model (5.52), with s= 1, the singular functional approach is addressed in [27], where, among other schemes, the authors defined a fully discrete scheme possessing a kind of energy inequality and used a singular functional to obtain approximate positivity. The drawback of this scheme is that the energy inequality has a residual term whose estimation is not clear in 2D and 3D domains, yielding a decreasing energy only in the one dimensional case. Consequently, in the 2D or 3D cases, convergence is not clear. The upwind approaches are possibly the better suited to preserve (exact) positivity of the discrete solutions and in our opinion, their application to the chemotaxisconsumption models (5.52) may lead to original works. For an example of these approaches in biology related models, we refer the reader to [1]. On the other hand, accounting for the complex procedures involved in obtaining energy estimates that can be observed in Chapters 2 and 3, we do not expect that these upwind approaches will lead to fully discrete energy stable schemes. Since energy stability is a key ingredient to prove convergence, the design of a convergent upwind fully discrete scheme may also remain as an open question in this case. In [32] the authors design a fully discrete scheme for a Keller-Segel model applying the mass lumping with FEM approach to obtain conditional positivity of the discrete solutions in 2D domains. The drawback of this approach is that the positivity is
Future works 135 attained provided that the space mesh size is small enough. This condition may lead to the need of very small mesh sizes, increasing the computational cost. In a future work, we intend to combine the regularization procedure used in Chapters 2 and 3 with this mass lumping approach to design an energy stable and convergent fully discrete scheme to approximate (5.52), preserving positivity and total population of cells. For a FEM scheme with stabilization terms we cite [2]. The authors propose two schemes to approximate the Keller-Segel model which preserve the lower bounds of the solutions, yielding positivity in the discrete level. In addition, they prove a discrete energy law that is satisfied by the discrete solutions. In Chapter 4 we studied an optimal control problem subject to problem (5.53) in the weak setting. We defined the adequate concept of weak solution of (5.53) satisfying the energy inequality (4.3) and used it to define control problems with bounded controls for which we proved existence of optimal solution. Nevertheless, the existence of optimal solution for the optimal control problem subject to weak solutions (5.53) which satisfy (4.3), but without bounded controls, that is, the minimization problem (4.5), remains as an open question. In addition, in Remark 4.7 we have already pointed out some other related questions which have the potential of being the focus of future research. To the best of our knowledge, there is not any study of a optimal control problem related to chemotaxis-consumption-fluid models in a weak setting. Then, a future work could deal with this problem, possibly extending the ideas of Chapter 4 to chemotaxis-consumption-fluid models. As it was mentioned in the introduction, it was not clear how to deduce some type of optimality system associated to local optimal solutions using only the weak regularity that is available in 3D domains. On the other hand, analogously to the studies about optimal control problems related to chemotaxis models in 2D (and 1D) domains [51, 29, 31, 5, 65, 54], it is probable that, in 2D (and 1D) domains, using the ideas of these cited previous works, one has more regularity, which will possibly allow one to prove the existence of global optimal solution and to derive an optimality system, establishing existence and regularity of Lagrange multipliers for any local optimum. In Chapter 5, in order to study an optimal control problem related to strong solutions of (5.53) in 3D domains, we proved a regularity criterion that allows us to get existence and uniqueness of global-in-time strong solutions. The uniqueness of solution (u, v), given the control f=f(t, x), allows us to define the state (u, v)=(u(f), v(f)) in terms of the control f. Then the functional (5.2) can be written in terms of fas J(f) := γu sq ZT 0∥u(f)(t)−ud(t)∥sq Lsq dt +γv 2ZT 0∥v(f)(t)−vd(t)∥2 L2dt +γf qZT 0∥f(t)∥q Lq(Ωc)dt. (5.54)
136 CONCLUSION An interesting work should investigate under which conditions we can calculate the derivative of Jwith respect to f, exhibit its expression and use it to propose descent methods. For an example of this kind of work we refer the reader to [9]. Once one computes the derivative of Jwith respect to f, we expect that it is possible to use it to obtain the optimality conditions of Chapter 5, similarly to the procedure adopted in [9, 36] and, in the context of chemotaxis-Navier-Stokes-consumption models, in [44]. In our opinion, this should be the subject of further study and, if this is indeed possible, another question that arises is whether, using this procedure, one can establish the same optimality conditions under the same hypothesis of Chapter 5. Finally, we would like to remark that, even when compared to other chemotaxis models, the chemotaxis-consumption models (with attraction and consumption) studied in this thesis showed to be very challenging due to the complex procedures needed to obtain of energy estimates that can be observed in Chapter 2, probably mainly because of the test functions and the treatment of a boundary integral that are involved in these procedures. This is why we expect that the treatment given to the present chemotaxis-consumption models can be useful to the approach of other chemotaxis models.
137 Appendix A Hypothesis (H1) In the proof of Lemma 2.10 we will need Hypothesis (H1) (see page 25). Therefore, in order to show that this hypothesis is not too restrictive, we show that Hypothesis (H1) holds if the Poisson-Neumann problem (1.5) has the W3,p-regularity (see definition 1.6 in page 18), for p>N. According to [20], this is true if Γis at least C2,1, for example. Up to our knowledge, the validity of Hypothesis (H1) in other domains of practical interest, such as polyhedra and polygons, is an open question. Lemma A.1. Suppose that the Poisson-Neumann problem (1.5) has the W3,p-regularity, for some p>N, and let z∈H2(Ω) such that ∂ηzΓ= 0. Then there is a sequence {ρn} ⊂ C2(Ω), with ∂ηρnΓ= 0, which converges to zin H2(Ω). Proof. For any fixed z∈H2(Ω) such that ∂ηzΓ= 0, define f=−∆z+z. Note that f∈L2(Ω) and z∈H2(Ω) is the solution of −∆z+z=f ∂ηzΓ= 0.(A.1) Let {fn}n∈Nbe a sequence of mollifiers of f, that is, fn∈C∞ c(Ω) and fn→f in L2(Ω) as n→ ∞. Then, for each fixed n∈N, consider the following regularized problem: Find ρn: Ω →Rsuch that −∆ρn+ρn=fn ∂ηρnΓ= 0.(A.2) Considering the hypothesis that the Poisson-Neumann problem (1.5) has the W3,pregularity, for some p > N, we can conclude that, for each n∈N, there is one, and only one, function ρn∈W3,p(Ω) ⊂C2(Ω) such that ∂ηρnΓ= 0 which solves problem (A.2). Since the functions zand ρnsolve the problems (A.1) and (A.2), respectively, the functions z−ρnsolve the problem −∆(z−ρn)+(z−ρn) = (f−fn) ∂η(z−ρn)Γ= 0.(A.3)
144 Appendix C. Existence of solution for a generic linear system Next, among other things, we must use Young’s inequality and we highlight the following estimation: ∥gU∥L10/7∥Un∥L10/3≤C1∥gU∥L10/7∥Un∥2/5 L2∥Un∥3/5 H1 ≤C1∥gU∥2/7 L10/7∥Un∥2/5 L2∥gU∥5/7 L10/7∥Un∥3/5 H1 ≤C2∥gU∥10/7 L10/7∥Un∥2 L2+C3∥gU∥25/28 L10/7∥Un∥15/20 H1 ≤C2∥gU∥10/7 L10/7∥Un∥2 L2+C4∥gU∥10/7 L10/7+C5∥Un∥2 H1. Then, applying the properties of the mollified sequences, the interpolation inequality (1.2) and Young’s inequality with the appropriate weights we conclude that for any δ > 0there are C, ˜ β > 0such that 1 2 d dt∥Un∥2 L2+˜ β∥∇Un∥2 L2≤C(∥a1∥5/2 L5/2+∥b1∥5/2 L5/2 +∥c1∥5 L5+∥gU∥10/7 L10/7)∥Un∥2 L2+C∥Vn∥2 L2 +C∥gU∥10/7 L10/7+C(∥d∥2 L∞+ 1)∥∇Vn∥2 L2. (C.5) Now we take φ=Vn∈Xnin (C.3), which gives us 1 2 d dt∥Vn∥2 L2+∥∇Vn∥2 L2≤ ∥an 2∥L5/2∥Vn∥2 L10/3 +∥bn 2∥L5/2∥Un∥L10/3∥Vn∥L10/3 +∥cn 2∥L5∥∇Vn∥L2∥Vn∥L10/3+∥gV∥L10/7∥Vn∥L10/3. Applying the properties of the mollified sequences, the interpolation inequalities (1.2) and Young’s inequality with the appropriate weights we conclude that for any δ > 0 there are C, ˜ β > 0such that 1 2 d dt∥Vn∥2 L2+˜ β∥∇Vn∥2 L2≤C∥Un∥2 L2 +C(∥a2∥5/2 L5/2+∥b2∥5/2 L5/2+∥c2∥5 L5+∥gV∥10/7 L10/7)∥Vn∥2 L2 +C∥gV∥10/7 L10/7+δ∥∇Un∥2 L2. (C.6) Let C0= 2C(∥d∥2 L∞+ 1)/˜ β, summing (C.5) and C0times (C.6) and choosing δ > 0 small enough then the terms δ∥∇Un∥2 L2and C(∥d∥2 L∞+ 1)∥∇Vn∥2 L2can be absorbed and we conclude that there is β > 0such that 1 2 d dt(∥Un∥2 L2+C0∥Vn∥2 L2) + β(∥∇Un∥2 L2+∥∇Vn∥2 L2) ≤C(∥a1∥5/2 L5/2+∥b1∥5/2 L5/2+∥c1∥5 L5 +∥gU∥10/7 L10/7+ 1)∥Un∥2 L2+C(∥a2∥5/2 L5/2+∥b2∥5/2 L5/2 +∥c2∥5 L5+∥gV∥10/7 L10/7+ 1)∥Vn∥2 L2 +C∥gU∥10/7 L10/7+C∥gV∥10/7 L10/7. (C.7)
Appendix C. Existence of solution for a generic linear system 145 Since ∥a1∥5/2 L5/2,∥b1∥5/2 L5/2,∥c1∥5 L5,∥gU∥10/7 L10/7,∥a2∥5/2 L5/2,∥b2∥5 L5,∥c2∥5 L5,∥gV∥10/7 L10/7∈ L1(0, T), we are able to apply Gronwall’s Lemma to (C.7) and conclude that (Un, Vn)is bounded in L∞(L2)×L∞(L2)∩L2(H1)×L2(H1). Using this bound in the equations (C.2) and (C.3) we also obtain n-independent bounds for ∂tUnand ∂tVn, which leads us to (Un, Vn)is bounded in W2×W2.(C.8) Next, we skip the standard procedures of the application of the Galerkin’s method to linear equations and state that with (C.8) we are able to pass to the limit as n→ ∞ in (C.2) and (C.3), concluding that there is (U, V )∈W2×W2solution of problem (C.1). Case 1b:The n-independent a priori estimates for this case are similar to those of the case 1a, but now, instead of choosing φ=Un∈Xnin (C.2) and φ=Vn∈Xn in (C.3), we take φ= (−∆ + I)−1Un∈Xnin (C.2) and φ= (−∆ + I)−1Vn∈Xnin (C.3), where Φ=(−∆ + I)−1Unis well defined as the function −∆Φ + Φ = Un, ∂nΦ|Γ= 0. We also use the fact that there is a constant C > 0such that ∥ϕ∥H2≤C∥(−∆ + I)ϕ∥L2,∀ϕ∈H2(Ω). Another relevant change is that we integrate by parts to reduce the order of the space derivatives of Unand Vnin (C.2) and (C.3) and we highlight the term (dn∇Vn,∇φ) of the (C.2) that, in this very weak solution setting, is written as −(Vn∇dn,∇φ)− (Vndn,∆φ). Case 2a:We take φ=Vn−∆Vn∈Xnin (C.3), which gives us 1 2 d dt∥Vn∥2 H1+∥∇Vn∥2 L2+∥∆Vn∥2 L2≤ ∥an 2∥L5/2∥Vn∥2 L10/3 +∥an 2∥L5/2∥Vn∥L10 ∥∆Vn∥L2+∥bn 2∥L5∥Un∥L10/3∥Vn∥L2 +∥bn 2∥L5∥Un∥L10/3∥∆Vn∥L2+C∥cn 2∥L5∥∇Vn∥L10/3∥Vn∥L2 +C∥cn 2∥L5∥∇Vn∥L10/3∥∆Vn∥L2+∥gV∥L2(∥Vn∥L2+∥∆Vn∥L2). Applying the properties of the mollified sequences, the interpolation inequalities (1.2) and (1.3) and Young’s inequality with the appropriate weights we conclude that for
146 Appendix C. Existence of solution for a generic linear system any δ > 0there are C, ˜ β > 0such that 1 2 d dt∥Vn∥2 H1+˜ β(∥∇Vn∥2 L2+∥∆Vn∥2 L2)≤C(∥a2∥5/2 L5/2 +∥b2∥5 L5+∥c2∥5 L5+∥gV∥2 L2+ 1)∥Vn∥2 H1+C∥Un∥2 L2 +C∥gV∥2 L2+δ∥∇Un∥2 L2. (C.9) Now we take φ=Un∈Xnin (C.2) and obtain 1 2 d dt∥Un∥2 L2+∥∇Un∥2 L2≤ ∥an 1∥L5/2∥Un∥2 L10/3 +∥bn 1∥L5/3∥Vn∥L10 ∥Un∥L10/3+∥Un∥L10/3∥cn 1∥L5∥∇Un∥L2 +∥dn∥L5∥∇Vn∥L10/3∥∇Un∥L2+∥gU∥L10/7∥Un∥L10/3. Applying the properties of the mollified sequences, the interpolation inequality (1.2) and Young’s inequality with the appropriate weights we conclude again that for any δ > 0there are C, ˜ β > 0such that 1 2 d dt∥Un∥2 L2+˜ β∥∇Un∥2 L2≤C(∥a1∥5/2 L5/2+∥b1∥5/3 L5/3 +∥c1∥5 L5+∥d∥5 L5+∥gU∥10/7 L10/7+ 1)∥Un∥2 L2 +C∥Vn∥2 H1+C∥gU∥10/7 L10/7+δ∥∆Vn∥2 L2. (C.10) Summing (C.9) and (C.10) and choosing δ > 0small enough so that the terms δ∥∇Un∥2 L2+δ∥∆Vn∥2 L2on the right hand side can be absorbed by the corresponding terms on the left hand side, we conclude that there is β > 0such that 1 2 d dt(∥Un∥2 L2+∥Vn∥2 H1) + β(∥∇Un∥2 L2+∥∇Vn∥2 L2) +β∥∆Vn∥2 L2≤C(∥a1∥5/2 L5/2+∥b1∥5/3 L5/3+∥c1∥5 L5 +∥d∥5 L5+∥gU∥10/7 L10/7+ 1)∥Un∥2 L2 +C(∥a2∥5/2 L5/2+∥b2∥5 L5+∥c2∥5 L5+∥gV∥2 L2+ 1)∥Vn∥2 H1 +C∥gU∥10/7 L10/7+C∥gV∥2 L2. (C.11) Since ∥a1∥5/2 L5/2,∥b1∥5/3 L5/3,∥c1∥5 L5,∥d∥5 L5,∥gU∥10/7 L10/7,∥a2∥5/2 L5/2,∥b2∥5 L5,∥c2∥5 L5,∥gV∥2 L2∈ L1(0, T), we are able to apply Gronwall’s Lemma to (C.11) and conclude that (Un, Vn)is bounded in L∞(L2)×L∞(H1)∩L2(H1)×L2(H2). Using this bound in the equations (C.2) and (C.3) we also obtain n-independent bounds for ∂tUnand ∂tVn, which leads us to (Un, Vn)is bounded in W2×X2.(C.12)
Appendix C. Existence of solution for a generic linear system 147 Again, we skip the standard procedures of the application of the Galerkin’s method to linear equations and state that with (C.12) we are able to pass to the limit as n→ ∞ in (C.2) and (C.3), concluding that there is (U, V )∈W2×X2solution of problem (C.1). Case 2b:The n-independent a priori estimates for this case are similar to those of the case 2aand one can obtain them based on the previous cases. ■
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