Regular time-reproductive solutions for generalized Boussinesq model with Neumann boundary conditions for temperature

REGULAR TIME-REPRODUCTIVE SOLUTIONS FOR GENERALIZED BOUSSINESQ MODEL WITH NEUMANN BOUNDARY CONDITIONS FOR TEMPERATURE B. Climent-Ezquerra, F. Guill´ en-Gonz´ alez Dpto. EDAN, Universidad de Sevilla Aptdo. 1160, 41080 Sevilla, SPAIN. M.A. Rojas-Medar Dpto. Matem´atica Aplicada, IMECC-UNICAMP, C.P. 6065, 13083-970, Campinas-SP, BRAZIL. (Communicated by Aim Sciences) Abstract. The aim of this work is to prove existence of regular time reproductive solutions for a generalized Boussinesq model (with nonlinear diffusion for velocity and temperature). The main idea is to obtain higher regularity (of H3type) for temperature than for velocity (of H2type), using specifically the Neumann boundary condition for temperature. 1. Introduction Assume a bounded, regular open set Ω in IRN(N= 2 or 3). This paper is concerned with some equations governing the coupled mass and heat flow of a viscous incompressible fluid in a generalized Boussinesq approximation by assuming that viscosity and heat conductivity are explicit functions depending on temperature. The involved equations are (1)      ∂tu−∇·(ν(θ)∇u)+(u· ∇)u−αgθ+∇p=f, ∇ · u= 0, ∂tθ−∇·(k(θ)∇θ)+(u· ∇)θ= 0, in Ω ×[0,∞), where •u(x, t)∈IRNdenotes the velocity of the fluid at point x∈Ω and time t∈[0,+∞). •p(x, t)∈IR is the (hydrostatic) pressure. •θ(x, t)∈IR is the temperature. •g(x, t)∈IRNdenotes the gravitational field and α > 0 is a constant associated to the coefficient of volume expansion. •f(x, t)∈IRNdenotes the resulting of external forces. •ν(·) : IR →IR is the kinematic viscosity. •k(·) : IR →IR is the thermal conductivity. 1991 Mathematics Subject Classification. Primary: 35Q30; Secondary: 76D05, 35Q35. Key words and phrases. generalized Boussinesq, regularity, periodic solutions, reproductive solutions. The authors has been partially supported by D.G.E.S. and M.C.T. (Spain), Projet BFM200306446. Moreover, third author is also supported by the projet 301354/03-0 CNPq. (Brazil). 1 2 B. CLIMENT, F. GUILL´ EN, M.A. ROJAS We will search for a triplet {u, p, θ}regular reproductive solution of (1) in Ω×[0,∞), together the follows Dirichlet-Neumann boundary conditions: (2) u= 0, ∂nθ= 0 on [0,∞)×∂Ω, and the time reproductive condition: (3) u(0) = u(T), θ(0) = θ(T) in Ω. Existence and uniqueness of the initial value problem related to (1) and with Dirichlet’s boundary conditions for velocity and temperature, was proved by Lorca and Boldrini in [5]. The stationary problem is studied in [6] for bounded domains and in [10] for exterior domains. On the other hand, A.C. Moretti, M.A. RojasMedar and M.D. Rojas-Medar [8] proved existence of reproductive weak solutions in exterior domains. The classical Boussinesq model, where νand kare positives constants, has been analyzed in great extent, see for instance, Morimoto [9], ´ Oeda [11]. The arguments used in [5] in order to obtain regular solution (and uniqueness) are not valid to find reproductivity since the initial conditions play a fundamental role. Our contribution in this paper is to obtain higher order estimates for the temperature than in [5]; namely in [5] H2(Ω) regularity is obtained for velocity and temperature, but now we will arrive at H3(Ω) regularity for the temperature. Consequently, a reproductive condition for time derivative of temperature also holds, i.e. ∂tθ(0) = ∂tθ(T). In addition, the arguments used in this paper are remarkably simpler than the used ones in [5]. By the contrary, now the regularity obtained for the solution is not sufficient to prove uniqueness. Notation. •In general, the notation will be abridged. We set Lp=Lp(Ω), p≥1, H1 0=H1 0(Ω), etc. If X=X(Ω) is a space of functions defined in the open set Ω, we denote by Lp(X) the Banach space Lp(0, T ;X). Also, boldface letters will be used for vectorial spaces, for instance L2=L2(Ω)N. •The Lpnorm is denoted by | · |p, 1 ≤p≤ ∞. The Hmnorm is denoted by k · km •We set Vthe space formed by all fields v∈C∞ 0(Ω)Nsatisfying ∇ · v= 0. We denote H(respectively V) the closure of Vin L2(respectively H1). Hand Vare Hilbert spaces for the norms | · |2and k·k1, respectively. Furthermore, H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω}, V={u∈H1;∇ · u= 0,u= 0 on ∂Ω} •It is easy deduce that d dt ZΩ θ(x, t) = 0 from the convection-diffusion equation for θ. Then, we can fix ZΩ θ= 0.Therefore, let us consider the following spaces Hk N=½θ∈Hk;∂θ ∂n = 0 on ∂Ω,ZΩ θ= 0¾ where k= 2,3. Hence, Hk Nis a closed subspace of Hk. Consequently |∆θ|2 is equivalent to kθk2in H2 Nand |∇∆θ|2is equivalent to kθk3in H3 N([12]). REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 3 Some interpolation inequalities. We will use the following classical interpolation and Sobolev inequalities (for 3Ddomains): |v|6≤Ckvk1,|v|3≤ |v|1/2 2kvk1/2 1 and |v|∞≤Ckvk1/2 1kvk1/2 2 In this work, it will be useful to use the following result (see [5]): Lemma 1. Let u∈V∩H2and consider the Helmholtz decomposition of −∆u, i.e. −∆u=Au+∇q, where q∈H1is taken such that ZΩ q dx = 0 and Ais the Stokes operator. Then, kqk1≤C|Au|2. Moreover, for every δ > 0there exists a positive constant Cδ(independent of u) such that |q|2≤Cδ|∇u|2+δ|Au|2. 2. The Main Result Definition 1. It will be said that (u, p, θ)is a regular solution of (1)–(3) in (0, T ), if u∈L2(H2)∩L∞(H1)and ∂tu∈L2(L2), p∈L2(H1), θ∈L2(H3 N)∩L∞(H2 N)and ∂tθ∈L2(H1 N), satisfying (1) a.e. in (0, T)×Ω, boundary conditions (2) and time reproductivity conditions (3) in the sense of spaces Vand H2 Nrespectively. Notice that we have imposed higher regularity for θthan for u. Theorem 1. Let T > 0and Ωa bounded domain in IRN(N= 2 or 3) with a boundary of class C2,1. Let the functions ν∈C1(IR) and k∈C2(IR) such that 0< νmin ≤ν(s)≤νmax 0< kmin ≤k(s)≤kmax in IR, and ν0, k0, k00 are bounded in IR (i.e. |ν0(s)| ≤ ν0 max,|k0(s)| ≤ k0 max,|k00(s)| ≤ k00 max). Assume that f∈L2(L2)and g∈L∞(L2)and kfkL2(L2)≤δ for δsmall enough, then there exists a regular (and small) reproductive solution of (1)–(3) in (0, T). Moreover, this solution also verifies ∂tθ(0) = ∂tθ(T). Remark: The uniqueness of solutions furnished by previous Theorem remains open, because higher regularity for the velocity is necessary. To obtain H3regularity for the velocity seem complicated because the argument made in the proof of Lemma 3 in order to get H3regularity is based in the Neumann condition, but we have Dirichlet condition for u. The proof of this Theorem will be given in Section 5. The method is based on the Galerkin approximation with spectral basis (defined in Section 3) and some differential inequalities in regular norms given in Section 4. 4 B. CLIMENT, F. GUILL´ EN, M.A. ROJAS 3. The Galerkin Initial-Boundary Problem Let {φi}i≥1and {ϕi}i≥1“special” basis of Vand H1 0(Ω), respectively, formed by eigenfunctions of the Stokes and the Poisson problems following: (4) ½−∆φi=λiφiin Ω φi= 0 on ∂Ω½−∆ϕi=µiϕiin Ω ∂nϕi= 0 on ∂Ω, with kφik1= 1, kϕik1= 1 for all iand RΩϕi= 0. Let Vmand Wmbe the finite-dimensional subspaces spanned by {φ1, φ2, . . . , φm}and {ϕ1, ϕ2, . . . , ϕm}respectively. For each m≥1, given u0m∈Vmand θ0m∈Wm, we seek an approximate solution (um, θm), with um: [0, T ]7→ Vmand θm: [0, T ]7→ Wm, verifying the following variational formulation a.e. in t∈(0, T ): (5)                (∂tum(t),vm) + ((um(t)· ∇)um(t),vm)+(ν(θm(t))∇um(t),∇vm) −(αθm(t)g,vm)−(f,vm) = 0 ∀vm∈Vm (∂tθm(t), em) + ((um(t)· ∇)θm(t), em) +(k(θm(t))∇θm(t),∇em) = 0 ∀em∈Wm um(0) = u0m, θm(0) = θ0m, If we put um(t) = m X j=1 ξi,m(t)φiand θm(t) = m X j=1 ζi,m(t)ϕi, then (5) can be rewritten as a first order ordinary differential system (in normal form) associated to the unknowns (ξi,m(t), ζi,m(t)). Then, one has existence of a maximal solution (defined in some interval [0, τm)⊂[0, T ]) of the related Cauchy problem. Moreover, from a priori estimates (independent on m) which will be obtained below, in particular one has that τm=T. Finally, using regularity of the chosen spectral basis, uniqueness of approximate solution holds ([2]). 4. Differential inequalities in regular norms In the sequel, δand εwill denote some constants sufficiently small. By Cwe will denote different constants, independent on data and δand ε. Lemma 2. For each δ, ε > 0sufficiently small, there exists a constant K= K(δ, ε)>0such that d dt ZΩ (ν(θm) + 1)|∇um|2+νminkumk2 2+|∂tum|2 2≤δk∂tθmk2 1 +εkumk2 2kθmk2+K(kumk6 1+kumk2 1kθmk4 2+kgk2 L∞(L2)kθmk2 2+|f|2 2) Proof. First, taking v=Aumas test function in the u-system of (5) (Ais the Helmholtz operator mentioned in Lemma 1) one has (6) (∂tum, Aum)−(∇ · (ν(θm)∇um), Aum) + ((um· ∇)um, Aum) −α(gθm, Aum) = (f, Aum) We can write the first term as (∂tum, Aum) = 1 2 d dtkumk2 1 REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 5 The second term of (6) is split as follows (using the Helmholtz decomposition ∆u=−Au+∇q), −(∇ · (ν(θm)∇um), Aum) = (ν(θm)Aum, Aum) + (ν(θm)∇q, Aum)−(ν0(θm)∇θm∇um, Aum). Taking into account that (ν(θm)∇q, Aum) = −(q, ∇ · (ν(θm)Aum)) =−(q, ν0(θm)∇θmAum)−(q, ν(θm)∇ · Aum) =−(q, ν0(θm)∇θmAum) since ∇ · Aum= 0, hence the second term of (6) remains −(∇ · (ν(θm)∇um), Aum)=(ν(θm)Aum, Aum) −(q, ν0(θm)∇θmAum)−(ν0(θm)∇θm∇um, Aum) Then, (6) can also be written as follows (using ν(θm)≥νmin >0) (7) 1 2 d dtkumk2 1+νminkumk2 2≤ −((um· ∇)um, Aum)−α(gθm, Aum) +(q, ν0(θm)∇θmAum) + (ν0(θm)∇θm∇um, Aum)+(f, Aum) := I1+I2+I3+I4+I5 The first two terms and the last term on the right hand side of (15) are bounded respectively by I1≤δkumk2 2+Cδkumk6 1, I2≤δkumk2 2+Cδ|g|2 2kθmk2 2 and I5≤δkumk2 2+Cδ|f|2 2. In order to estimate the third term we use the Lemma 1 (and |ν0(θm)| ≤ ν0 max) |I3|=|(q, ν0(θm)∇θmAum)| ≤ ν0 max|q|3|∇θm|6|Aumk2 ≤C|q|1/2 2kqk1/2 1kθmk2kumk2≤C(Cεkumk1/2 1+εkumk1/2 2)kumk3/2 2kθmk2 ≤Cεkumk1/2 1kumk3/2 2kθmk2+εkumk2 2kθmk2 ≤δkumk2 2+Cε,δkumk2 1kθmk4 2+εkumk2 2kθmk2. In what concerns to the fourth term, |I4|=|(ν0(θm)∇θm∇um, Aum)| ≤ ν0 max|∇θm|6|∇umk3|Aum|2 ≤Ckθmk2kumk1/2 1kumk3/2 2≤δkumk2 2+Cδkumk2 1kθmk4 2. Consequently, choosing δsmall enough we obtain that (8) d dtkumk2 1+νminkumk2 2≤Cε(kumk6 1+kumk2 1kθmk4 2+|g|2 2kθmk2 2+|f|2 2) +εkumk2 2kθmk2 On the other hand, using ∂tumas a test function in the u-system of (5), one obtains (9) (∂tum, ∂tum)+(ν(θm)∇um, ∂t∇um) + ((um· ∇)um, ∂tum) −α(gθm, ∂tum) = (f, ∂tum). 6 B. CLIMENT, F. GUILL´ EN, M.A. ROJAS By taking into account that the second term in (9) can be written as (ν(θm)∇um, ∂t∇um) = 1 2 d dt(ν(θm)∇um,∇um)−1 2(∂t(ν(θm))∇um,∇um), we deduce from (9) that (10) 1 2 d dt ZΩ ν(θm)|∇um|2+|∂tum|2 2≤ −((um· ∇)um, ∂tum) +α(gθm, ∂tum) + 1 2(∂t(ν(θm))∇um,∇um)+(f, ∂tum) := J1+J2+J3+J4 The first two terms and the last term on the right side of (10) are bounded respectively, by J1≤δ(|∂tum|2 2+kumk2 2) + Cδkumk6 1, J2≤δ|∂tum|2 2+Cδ|g|2 2kθmk2 2, and J4≤δ|∂tum|2 2+Cδ|f|2 2. Lastly, we go into detail the third term: |J3|=|(ν0(θm)∂t(θm)∇um,∇um)| ≤ ν0 max|∂tθm|6|∇umk3|∇umk2 ≤Ck∂tθmk1kumk3/2 1kumk1/2 2≤δ(k∂tθmk2 1+kumk2 2) + Cδkumk6 1 Consequently, choosing δsmall enough, (11) d dt ZΩ ν(θm)|∇um|2+|∂tum|2 2≤δ(k∂tθmk2 1+kumk2 2) +Cδ(kumk6 1+|g|2 2kθmk2 2+|f|2 2) Finally, (8) and (11) prove the Lemma. Lemma 3. For each δ > 0small enough, there exists Cδ>0such that d dt(kθmk2 2+|∂tθm|2 2) + kmin(kθmk2 3+k∂tθmk2 1) ≤δ|∂tumk2 2+Cδ(kθmk6 2+kθmk4 2|∂tθm|2 2+kθmk2 2kumk4 1) tu Proof. Differenting respect to the time the θm-equation of (5) and using ∂tθmas test function, one obtains (12) 1 2 d dt|∂tθm|2 2+ (∂t(k(θm)∇θm), ∂t∇θm)+(∂tum· ∇θm, ∂tθm) = 0 since (um· ∇∂tθm, ∂tθm) = 0. By taking into account that the second term in (12) can be split as (∂t(k(θm)∇θm), ∂t∇θm) = (k0(θm)∂tθm∇θm, ∂t∇θm)+(k(θm)∂t∇θm, ∂t∇θm), we deduce from (12) that (13) 1 2 d dt|∂tθm|2 2+kmin|∂t∇θm|2 2≤ −(k0(θm)∂tθm∇θm, ∂t∇θm) −(∂tum· ∇θm, ∂tθm). REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 7 Bounding both terms on the right hand side of (13) (k0 max = max |k0|): |(k0(θm)∂tθm∇θm, ∂t∇θm)| ≤ k0 max|∇θm|6|∂tθm|3|∂t∇θm|2 ≤Ckθmk2|∂tθm|1/2 2k∂tθmk3/2 1≤δk∂tθmk2 1+Cδkθmk4 2|∂tθm|2 2 and |(∂tum· ∇θm, ∂tθm)| ≤ |∂tum|2|∇θm|6|∂tθm|3 ≤C|∂tum|2kθmk2|∂tθm|1/2 2k∂tθmk1/2 1 ≤δ(k∂tθmk2 1+|∂tum|2 2) + Cδkθmk4 2|∂tθm|2 2 we obtain, for δsmall enough, (14) d dt|∂tθm|2 2+kmink∂tθmk2 1≤δ|∂tum|2 2+Cδkθmk4 2|∂tθm|2 2 Now, using ∆2θmas test function (∆2θm∈Wmthanks to the election of spectral basis) and integrating by parts in all terms (boundary terms vanish since (∇∆θm· n)|∂Ω= 0), one obtains: (15) −(∂t∇θm,∇∆θm)+(∇[∇·(k(θm)∇θm)],∇∆θm)−(∇(u·∇θm),∇∆θm) = 0. Notice that if Dirichlet boundary condition is imposed for the temperature θ, the boundary terms do not vanish in the integration by parts and we can not obtain the previous inequalities. Integrating by parts the first term of (15) (again the boundary term vanishes since (∂t∇θm·n)|∂Ω= 0), the term remains 1 2 d dt|∆θm|2 2. The second term is (∇[∇ · (k(θm)∇θm)],∇∆θm) = (k00(θm)(∇θm)3,∇∆θm) +2(k0(θm)∇2θm∇θm,∇∆θm) +(k0(θm)∇θm∆θm,∇∆θm) + (k(θm)∇∆θm,∇∆θm). Hence, we deduce of (15) that (|k00(θm)| ≤ k00 max = max |k00|) 1 2 d dt|∆θm|2 2+k0|∇∆θm|2 2≤k00 max|((∇θm)3,∇∆θm)| +2k0 max|(∇2θm∇θm,∇∆θm)|+k0 max|(∇θm∆θm,∇∆θm)| +|(∇um∇θm,∇∆θm)|+|(um∇2θm,∇∆θm)| := L1+L2+L3+L4+L5. Replacing in the above inequality the following estimations L1≤C|∇θm|3 6|∇∆θm|2≤δkθmk2 3+Cδkθmk6 2 L2≤C|∇2θm|3|∇θm|6|∇∆θm|2≤Ckθmk3/2 2kθmk3/2 3≤δkθmk2 3+Cδkθmk6 2 L3≤C|∇θm|6|∆θm|3|∇∆θm|2≤Ckθmk3/2 2kθmk3/2 3≤δkθmk2 3+Cδkθmk6 2 L4≤C|∇um|2|∇θm|∞|∇∆θm|2≤Ckumk1kθmk1/2 2kθmk3/2 3 ≤δkθmk2 3+Cδkumk4 1kθmk2 2 L5≤C|um|6|∇2θm|3|∇∆θm|2≤Ckumk1kθmk1/2 2kθmk3/2 3 ≤δkθmk2 3+Cδkumk4 1kθmk2 2 8 B. CLIMENT, F. GUILL´ EN, M.A. ROJAS we get, taking δsmall enough, (16) d dtkθmk2 2+kminkθmk2 3≤Cδ(kθmk6 2+kumk4 1kθmk2 2) Finally, (14) added to (16) proves the Lemma. 5. Proof of Theorem 2.2 If we denote Φm(t) = ZΩ (ν(θm) + 1)|∇um|2+kθmk2 2+|∂tθm|2 2 Ψm(t) = kumk2 2+|∂tum|2 2+kθmk2 3+k∂tθmk2 1 taking an adequate balance between inequalities from Lemmas 2 and 3 in order to vanish the term ||g||2 L∞(L2)||θm||2 2at the right hand-side, one has (17) (Φ0 m+CΨm≤εΨmΦ1/2 m+C0(t) + DΦ3 m Φm(0) = Φm0 where C, D > 0 are constant and C0(t) is a positive function depending on data f. Concretely, C0(t) = C0|f|2 2. Let Φm(0) ≤δfor δa small enough constant (that we will specify latter). First step: If Φm(0) ≤δand kfkL2(L2)≤δ, then Φm(t)<2δ∀t∈[0, T ]. Indeed, by an absurd argument, let T∗be the first value in [0, T ] such that Φm(T∗) = 2δ, hence Φm(T∗) = 2δand Φm(s)<2δ∀s∈[0, T ∗). Moreover, there exists a Poincar´e constant Cp>0 such that Φm(t)≤CpΨm(t). Then for εsmall enough we have CΨm−εΨmΦ1/2 m≥CΨm−εΨm(2δ)1/2≥¯ CΨm≥¯ C Cp Φm≡˜ CΦm. The above inequality together (17) lead: (18) (Φ0 m+˜ CΦm≤C0(t) + DΦ3 m Φm(0) = Φm0. in [0, T∗]. Then, Φ0 m+˜ CΦm≤C0(t)+4δ2DΦmin [0, T∗]. We can find δsuch that ˜ C−4δ2D≥¯ Cbeing ¯ Ca positive constant. Therefore, Φ0 m+¯ CΦm≤C0(t) in [0, T∗] hence (19) (e¯ CtΦm)0≤e¯ CtC0(t) in [0, T ∗]. Integrating in [0, T ∗] one finds: e¯ CT ∗Φm(T∗)≤Φm(0) + ZT∗ 0 e¯ CtC0(t), hence Φm(T∗)≤δe−¯ CT ∗+ZT∗ 0 C0(t). REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 9 We can choose ZT∗ 0 C0(t) small enough such that the right side is smaller that 2δ (for example ZT∗ 0 C0(t)≤δ). Thus, we arrive at a contradiction. Second step: If Φm(0) and kfkL2(L2)are small enough, then Φm(T)≤Φm(0) Now, as Φm(t)<2δ∀t∈[0, T ], we can repeat the above argument and to obtain (18) in [0, T]. Therefore, integrating (19) in [0, T ] we arrive at Φm(T)≤Φm(0)e−¯ CT +ZT 0 C0(s). Again, for Zt 0 C0(t) small enough (for example ZT∗ 0 C0(t)≤δ(1 −e−¯ CT )) one obtains that Φm(T)≤Φm(0). Third step: Existence of approximate reproductive solution Given (um0, θm0)∈Vm×Wm, we define the map Lm: [0, T]7→ IRm×IRm t7→ (ξ1m(t), ..., ξmm(t), ζ1m(t), ..., ζmm(t)) where (ξ1m(t), ..., ξmm(t)) and (ζ1m(t), ..., ζmm(t)) are coefficients of um(t) and θm(t) respect to Vmand Wmrespectively, being (um(t), θm(t)) the (unique) approximate solution of (5) corresponding to the initial data (um0, θm0). Now, varying the initial data (um0, θm0), we are going to define a new map Rm:¯ B⊂IRm×IRm7→ IRm×IRm as follows: given Lm 0∈IRm×IRm, we define Rm(Lm 0) = Lm(T), where Lm(t) is related to the solution of problem (5) with initial data Lm 0(= Lm(0)) and ¯ B={(ξ1m, ..., ξmm, ζ1m, ..., ζmm) := Lm 0: Φm(0) ≤δ}. By uniqueness of approximate solution of problem (5), this map is well-defined. Moreover, using regularity of the corresponding ordinary differential system (equivalent to (5)), this map is continuous. By the second step, Rmapply ¯ Binto ¯ Band ¯ Bis a closed, convex and compact set. Consequently, Brouwer Theorem implies the existence of fixed point of Rm, which give us existence of reproductive Galerkin solution. Four step: Pass to the limit in reproductive approximate solutions If the data of the problem are small, thanks to the first step we have Φm(t) = ZΩ (ν(θm) + 1)|∇um|2+kθmk2 2+|∂tθm|2 2≤2δ and (19). Therefore, the following uniformly bounds hold: (um) in L∞(H1)∩L2(H2), (θm) in L∞(H2 N)∩L2(H3 N), (∂tum) in L2(L2), (∂tθm) in L∞(L2)∩L2(H1).