On a initial-boundary Q-tensor problem related to liquid crystals

milogo Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models On a initial-boundary Q-Tensor problem related to Liquid Crystals F. Guillén-Gonzalez & M. A. Rodríguez Bellido Dpto. Ecuaciones Diferenciales y Análisis Numérico and IMUS Facultad de Matemáticas Universidad de Sevilla, Spain DIMO 2013/10-13 September, Levico Terme F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 milogo Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models 1Nematic Liquid Crystals A simplified model by F. H. Lin Some known results 2Models with Stretching Terms Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 3Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Complex Fluids It is not possible to decouple microscopic and macroscopic effects. Fluids with elastic properties. They possesses intermediate properties between solids and liquids. Examples: liquid crystals, polymers (macromolecules), ... Phase-field models. Examples: multi-fluids (mixture of fluids), multi-phases (solidification), ... These complex materials have practical utilities because its microstructure can be handled in order to produce good mechanical, optical or thermic properties. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Liquid Crystals Liquid crystals (LC) are intermediate phases between solid and liquid; at the macroscopic level, they are (viscous) liquids but their molecules have a anisotropic order due to their elastic properties. Nematic Liquid Crystals have an orientation order. Smectic Liquid Crystals have also a positional order (arranged by layers). The derivation and the analysis falls into a general energetic variational framework for complex fluids with elastic effects due to the presence of nontrivial microstructures, coupling Navier-Stokes equations for the velocity and pressure. Partial Differential Equations for the microscopic variable (called order parameter) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Figura: Types of Liquid Crystals F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models A simplified model by F. H. Lin Some known results 1Nematic Liquid Crystals A simplified model by F. H. Lin Some known results 2Models with Stretching Terms Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 3Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models A simplified model by F. H. Lin Some known results Nematic Liquid Crystals - Macroscopic model Director field d(t,x), representing the average orientation of the liquid crystal molecules. The shear stress tensor depends on elastic and viscous effects (Ericksen-Leslie theory, 1980s): σ=σd(D,d) + λσe(d), where σdis the dissipative tensor, σethe elastic tensor and λ > 0 a “balance” coefficient. Then, equations for equilibrium of forces remains as: Dtu+∇p−∇·σd−λ∇ · σe=0 in Q= (0,T)×Ω. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models A simplified model by F. H. Lin Some known results Nematic Liquid Crystals: Microscopic Model (of Allen-Canh’s type) Starting from the Ericksen-Leslie’s formulation, a penalized model is presented by [F.H. Lin]: Dtd+γδEe δd=0,in Q, where δEe δd=−∆d+∇dF(d) is the Euler-Lagrange equation associated to the elastic energy functional: Ee(d) = 1 2ZΩ |∇d|2+ZΩ F(d) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models A simplified model by F. H. Lin Some known results Ginzburg-Landau’s functional: F(d) = 1 42|d|2−12 such that f(d) = ∇d(F(d)) for every d∈R3, hence f(d) = 1 2|d|2−1d, where |d|denotes the euclidean norm in R3and >0 is a penalization parameter. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 1Nematic Liquid Crystals A simplified model by F. H. Lin Some known results 2Models with Stretching Terms Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 3Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model Free energy Free energy operator: E(Q) = ZΩ ε 2|∇Q|2+F(Q) where F(Q) = a 2|Q|2−b 3(Q2:Q) + c 4|Q|4(non-convex) (1) Let H(Q) = δE(Q) δQbe the variational derivative in L2(Ω). F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model Velocity system The variables describing the (QT)-model are (u,Q,p):(0,T)×Ω→R3×R3×3×R: (Dtu−ν∆u+∇p=∇ · τ(Q) + ∇ · σ(H,Q)in Ω×(0,T) ∇ · u=0 in Ω×(0,T) where Dtu=∂tu+ (u· ∇)uis the material derivative,          τij (Q) = −ε∂jQ:∂iQ=−ε∂jQkl ∂iQkl ,ε>0 (symmetric part) σ(H,Q) = HQ−QH (antisymmetric part when Qand Hare symmetric) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model Q-tensor system ∂tQ+ (u· ∇)Q−S(∇u,Q) = −γH(Q)in Ω×(0,T) where                          S(∇u,Q) = ∇uQt−Qt∇u,(stretching term) H(Q) = −ε∆Q+f(Q)where f(Q) = aQ−b 3Q2+QQt+QtQ+c|Q|2Q with c>0,a,b∈R (His a symmetric tensor if Qis symmetric, in fact f(Q)t=f(Qt)) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model Previous results for an initial-value problem (in the whole R3) [Paicu-Zarnescu’12] Existence of weak solution in (0,T), for each T>0. Global strong solution in 2D. Weak-Strong uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model Initial and boundary conditions Initial conditions: u|t=0=u0,Q|t=0=Q0in Ω, Boundary conditions (Γ = ∂Ω: ): For the velocity: u|Γ=0in (0,T). For the Q-tensor: ∂nQ|Γ=0 or Q|Γ=QΓin (0,T). F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model Main results for the initial-boundary QT-model existence of global in time weak solution (without using maximum principle). Modification of the model to enforce traceless and symmetry constraints for Q. maximum principle uniqueness criteria for weak solutions local existence (and uniqueness) of a “intermediate” regular solution F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness 1Nematic Liquid Crystals A simplified model by F. H. Lin Some known results 2Models with Stretching Terms Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 3Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Step 1. Energy equality (Lyapunov functional) d dt 1 2kuk2 L2(Ω) +E(Q)+νk∇uk2 L2(Ω) +γkH(Q)k2 L2(Ω) =0, with E(Q) = ZΩ ε 2|∇Q|2+F(Q)dx6≥ 0 Using that: (S(∇u,Q),H(Q))L2= (σ(H,Q),∇u))L2 (u· ∇Q,H(Q))L2= (∇ · τ(Q),u)L2 Q-system u-system F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Step 2. Lower bound of the potential But the term F(Q)could be negative. Observe that: F(Q)≥         a 2|Q|2+c 8|Q|4−α1,for α1=α1(b,c)>0 if a>0, c 8|Q|4−α2−β, for α2=α2(b,c), β =β(a,c)>0 if a<0, Defining e F(Q) = F(Q) + µwith µ=α1if a≥0 and µ=α2+β if a<0, then: e F(Q)≥     a 2|Q|2+c 8|Q|4≥0 if a>0, c 8|Q|4≥0 if a<0, F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Maximum Principle Based on: 1S(∇u,Q) : Q=0 2f(Q) : Q≥c 2|Q|2|Q|2−βfor β=b2 c2−2a c. Then, ∂t|Q|2+u·∇ |Q|2−γε∆|Q|2+γc 2|Q|2|Q|2−β≤0 If kQ0kL∞(Ω) ≤α(and kQΓkL∞(Γ) ≤α) with α≥β, then: kQ(t)kL∞(Ω) ≤α∀t≥0. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness 1Nematic Liquid Crystals A simplified model by F. H. Lin Some known results 2Models with Stretching Terms Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 3Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Problems with the strong regularity Prodi’s (space) estimates (taking −∆ufor u-system and −∆(−ε∆Q+f(Q)) for Q-system) only works for 1periodic-space boundary conditions for Q, 2large enough viscosity. Modified Ladyzhenskaya’s (time) estimates works for Neumann and Dirichlet boundary conditions for Q. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness The key point for Prodi’s estimates Due to the boundary condition for Q(non space-periodic): (S(∇u,Q),−∆H(Q))L26= (σ(H,Q),∇(−∆u)))L2 and (∇ · τ(Q),−∆u)L26= (u· ∇Q,−∆H(Q))L2 because some (high nonlinear) boundary terms don’t vanish. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness The bad terms (only bounded for large viscosity) 1 2 d dt k∇uk2 L2(Ω) +k∆Qk2 L2(Ω)+νkAuk2 L2(Ω) +γk∇(∆Q)k2 L2(Ω) =−(u· ∇u,Au)+(Au· ∇Q,∆Q)+(∇ · σ, Au) + . . . −(∇(u· ∇Q),∇(∆Q)) + (∇S(∇u,Q),∇(∆Q)) ≤ · · · +ZΩ |Q| |∇(∆Q)| |Au|dx≤.... • F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness 1Nematic Liquid Crystals A simplified model by F. H. Lin Some known results 2Models with Stretching Terms Nematic Liquid Crystals with Stretching Terms A generic Q-tensor model 3Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Modified Ladyzhenskaya’s estimates Weak estimates for (∂tu, ∂tQ) Deriving in time u-system and Q-system, and taking ∂tuand −∆(∂tQ)as test functions: d dt k∂tuk2 L2(Ω) +εk∂tQk2 H1(Ω) +νk∂tuk2 H1(Ω) +γε2k∂tQk2 H2(Ω) ≤a(t)k∂tuk2 L2(Ω) +k∂tQk2 H1(Ω) +Cν,γ,εk∇uk4 L2(Ω) +kHk4 L2(Ω)k∂tuk2 L2(Ω) +k∂tQk2 H1(Ω) (2) where a∈L1(0,T)(due to weak estimates). F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Intermediate strong estimates for Q Taking ∂tH=∂t(−ε∆Q+f(Q)) as test function in the Q-system: γ 2 d dt kHk2 L2(Ω) +εk∂t(∇Q)k2 L2(Ω) ≤Cδ1+kQkH2(Ω)k∂tQk2 L2(Ω) +δk∂tHk2 L2(Ω) +CδkQkH2(Ω) k∇uk2 L2(Ω) (3) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Intermediate strong estimates for u Taking ∂tuas test function in the equation for u: νd dt k∇uk2 L2(Ω) +k∂tuk2 L2(Ω) ≤Ck∇uk3 L2(Ω) +kQkH2(Ω)kHk2 L2(Ω)(4) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Weak-t estimates Putting together (2)-(3)-(4), we get: y0(t) + z(t)≤e a(t)y(t) + C y(t)3 where    y(t) = k∂tuk2 L2(Ω) +k∂tQk2 H1(Ω) +k∇uk2 L2(Ω) +kHk2 L2(Ω) z(t) = k∂tuk2 H1(Ω) +k∂tQk2 H2(Ω) F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness F.H. Lin & C. Liu. Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537, 1995. C. Liu, H. Wu & X. Xu. Asymptotic behavior of a hydrodynamic system in the nematic liquid crystal flows Calc. Var. Partial Differential Equations 45 (2012), no. 3-4, 319–345. M. Paicu & A. Zarnescu. Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System. Arch. Ration. Mech. Anal. 203 (1), 45–67, 2012. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness H. Petzeltová, E. Rocca & G. Schimperna. On the long-time behavior of some mathematical models for nematic liquid crystals. Calc. Var. Partial Differential Equations 46 (2013), no. 3-4, 623–639. F.Guillén-González, M.A.Rodríguez-Bellido. Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals, Submitted (2013). F.Guillén-González, M.A.Rodríguez-Bellido.Partial regularity and uniqueness of the reduced Q-tensor model, In preparation. F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013 Nematic Liquid Crystals Models with Stretching Terms Some analytical results for Q-tensor models Weak existence Weak/strong uniqueness Maximum Principle Strong solution ? Local weak regularity for (∂tu, ∂tQ)and uniqueness Thank you very much! F. Guillén-Gonzalez, EDAN and IMUS, Univ. Sevilla DIMO2013