Convergence to equilibrium for smectic-A liquid crystals in 3D domains

Convergence to equilibrium for smectic-A liquid crystals in 3D domains F. Guillén-González, Depto. EDAN & IMUS, Universidad de Sevilla. Spain. AIMS, 07-11 July 2014, Madrid. In collaboration: B. Climent-Ezquerra (Univ. Sevilla), .[NA-TMA’14]. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 1 / 25 Liquid Crystals Liquid crystals (LC) exhibit properties between liquids and solids. A LC can behave like a liquid (macroscopically), but their molecules have a preferential orientation (microscopically), due to elasticity effects (anisotropic liquids). The time-dynamics interaction between macroscopic and microscopic is modeled by nonlinear parabolic PDE + gradient flow, involving: Macroscopic: fluid dynamics (Navier-Stokes) Microscopic: order parameter Different phases in LC, for instance: nematic phases, with orientational order of molecules, smectic phases, with moreover positional order (arranged in layers). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 2 / 25 Liquid Crystals Liquid crystals (LC) exhibit properties between liquids and solids. A LC can behave like a liquid (macroscopically), but their molecules have a preferential orientation (microscopically), due to elasticity effects (anisotropic liquids). The time-dynamics interaction between macroscopic and microscopic is modeled by nonlinear parabolic PDE + gradient flow, involving: Macroscopic: fluid dynamics (Navier-Stokes) Microscopic: order parameter Different phases in LC, for instance: nematic phases, with orientational order of molecules, smectic phases, with moreover positional order (arranged in layers). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 2 / 25 Liquid Crystals Liquid crystals (LC) exhibit properties between liquids and solids. A LC can behave like a liquid (macroscopically), but their molecules have a preferential orientation (microscopically), due to elasticity effects (anisotropic liquids). The time-dynamics interaction between macroscopic and microscopic is modeled by nonlinear parabolic PDE + gradient flow, involving: Macroscopic: fluid dynamics (Navier-Stokes) Microscopic: order parameter Different phases in LC, for instance: nematic phases, with orientational order of molecules, smectic phases, with moreover positional order (arranged in layers). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 2 / 25 F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 3 / 25 Static Osseen-Frank’s theory for nematic LC Ω⊂Rd(d=2,3): domain filled by the LC, with boundary ∂Ω Equilibrium states: minimum of a free energy Oseen-Frank free energy derive to the Static (Minimization) problem: min |d|=1Eela(d) where Eela is the elastic energy functional: Eela(d) = ZΩk1 2(∇ · d)2+k2 2(d·(∇ × d))2+k3 2|d×(∇ × d)|2 ki>0 elastic constants. Remark: Problem with convex functional but non-convex constraints. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 4 / 25 Static Osseen-Frank’s theory for nematic LC Ω⊂Rd(d=2,3): domain filled by the LC, with boundary ∂Ω Equilibrium states: minimum of a free energy Oseen-Frank free energy derive to the Static (Minimization) problem: min |d|=1Eela(d) where Eela is the elastic energy functional: Eela(d) = ZΩk1 2(∇ · d)2+k2 2(d·(∇ × d))2+k3 2|d×(∇ × d)|2 ki>0 elastic constants. Remark: Problem with convex functional but non-convex constraints. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 4 / 25 Static Osseen-Frank’s theory for nematic LC Ω⊂Rd(d=2,3): domain filled by the LC, with boundary ∂Ω Equilibrium states: minimum of a free energy Oseen-Frank free energy derive to the Static (Minimization) problem: min |d|=1Eela(d) where Eela is the elastic energy functional: Eela(d) = ZΩk1 2(∇ · d)2+k2 2(d·(∇ × d))2+k3 2|d×(∇ × d)|2 ki>0 elastic constants. Remark: Problem with convex functional but non-convex constraints. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 4 / 25 Smectic-A LC Smectic LC: the molecules have an orientational order vector dand a positional order (layer structure of normal vector n). Assume: ∃a potential function ϕs.t. layers =level sets of ϕ: ∇ × n=0⇐⇒ n=∇ϕ Incompressibility of the layers: |n|=1. Smectic-A LC: Assume d=n. Since d=nand n=∇ϕ, the elastic energy can be rewritten (reduced form) as Eela =1 2ZΩ |∇ · n|2=1 2ZΩ |∆ϕ|2 F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 5 / 25 L2(Ω)-Identification of the variational derivative 1Assuming ∆ϕ|∂Ω=0(N1) or ∇ϕ·m|∂Ω=0(D2) (mis the normal vector): δEe(ϕ) δϕ , ϕ=−ZΩ w· ∇ϕ, w:= ∇∆ϕ−1 ε2f(∇ϕ). 2Assuming w·m|∂Ω=0(N2) or ϕ|∂Ω=0(D1): δEe δϕ ≡ ∇ · w= ∆2ϕ−1 ε2∇ · f(∇ϕ)in Ω. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 8 / 25 Adimissible B.C. to fourth-order Euler-Lagrange ∇ · w=0 [D1−D2]ϕ|∂Ω=ϕ1,∇ϕ·m|∂Ω=ϕ2, [D1−N1]ϕ|∂Ω=ϕ1,∆ϕ|∂Ω=0, [D2−N2]∇ϕ·m|∂Ω=ϕ2,w·m|∂Ω=0, [N1−N2]∆ϕ|∂Ω=0,w·m|∂Ω=0. Non admissible B.C.: [D1-N2] and [D2-N1] F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 9 / 25 Adimissible B.C. to fourth-order Euler-Lagrange ∇ · w=0 [D1−D2]ϕ|∂Ω=ϕ1,∇ϕ·m|∂Ω=ϕ2, [D1−N1]ϕ|∂Ω=ϕ1,∆ϕ|∂Ω=0, [D2−N2]∇ϕ·m|∂Ω=ϕ2,w·m|∂Ω=0, [N1−N2]∆ϕ|∂Ω=0,w·m|∂Ω=0. Non admissible B.C.: [D1-N2] and [D2-N1] F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 9 / 25 Adimissible B.C. to fourth-order Euler-Lagrange ∇ · w=0 [D1−D2]ϕ|∂Ω=ϕ1,∇ϕ·m|∂Ω=ϕ2, [D1−N1]ϕ|∂Ω=ϕ1,∆ϕ|∂Ω=0, [D2−N2]∇ϕ·m|∂Ω=ϕ2,w·m|∂Ω=0, [N1−N2]∆ϕ|∂Ω=0,w·m|∂Ω=0. Non admissible B.C.: [D1-N2] and [D2-N1] F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 9 / 25 Adimissible B.C. to fourth-order Euler-Lagrange ∇ · w=0 [D1−D2]ϕ|∂Ω=ϕ1,∇ϕ·m|∂Ω=ϕ2, [D1−N1]ϕ|∂Ω=ϕ1,∆ϕ|∂Ω=0, [D2−N2]∇ϕ·m|∂Ω=ϕ2,w·m|∂Ω=0, [N1−N2]∆ϕ|∂Ω=0,w·m|∂Ω=0. Non admissible B.C.: [D1-N2] and [D2-N1] F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 9 / 25 Adimissible B.C. to fourth-order Euler-Lagrange ∇ · w=0 [D1−D2]ϕ|∂Ω=ϕ1,∇ϕ·m|∂Ω=ϕ2, [D1−N1]ϕ|∂Ω=ϕ1,∆ϕ|∂Ω=0, [D2−N2]∇ϕ·m|∂Ω=ϕ2,w·m|∂Ω=0, [N1−N2]∆ϕ|∂Ω=0,w·m|∂Ω=0. Non admissible B.C.: [D1-N2] and [D2-N1] F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 9 / 25 Smectic-A Liquid Crystal Model [E, ARMA’97] Ω⊂R3,Q= Ω ×(0,T)and Σ = ∂Ω×(0,T). Linear momentum,(u,p)-system in Q: ∂u ∂t+ (u· ∇)u+∇p− ∇ · (σd+λ σe) = 0,∇ · u=0, σd=σd(D(u),n): dissipative (symmetric) stress tensor: σd=µ1(ntDn)n⊗n+µ4D+µ5(Dn⊗n+n⊗Dn),D=∇u+∇ut 2 σe=σe(ϕ): elastic (non-symmetric) stress tensor, s.t. ∇ · σe=δEe δϕ ∇ϕ+∇Ee Angular momentum,ϕ-equation in Q(of Allen-Cahn’s type): ∂ϕ ∂t+u· ∇ϕ+γδEe δϕ =0, γ > 0 constant (elastic relaxation time). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 10 / 25 Smectic-A Liquid Crystal Model [E, ARMA’97] Ω⊂R3,Q= Ω ×(0,T)and Σ = ∂Ω×(0,T). Linear momentum,(u,p)-system in Q: ∂u ∂t+ (u· ∇)u+∇p− ∇ · (σd+λ σe) = 0,∇ · u=0, σd=σd(D(u),n): dissipative (symmetric) stress tensor: σd=µ1(ntDn)n⊗n+µ4D+µ5(Dn⊗n+n⊗Dn),D=∇u+∇ut 2 σe=σe(ϕ): elastic (non-symmetric) stress tensor, s.t. ∇ · σe=δEe δϕ ∇ϕ+∇Ee Angular momentum,ϕ-equation in Q(of Allen-Cahn’s type): ∂ϕ ∂t+u· ∇ϕ+γδEe δϕ =0, γ > 0 constant (elastic relaxation time). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 10 / 25 Smectic-A Liquid Crystal Model [E, ARMA’97] Ω⊂R3,Q= Ω ×(0,T)and Σ = ∂Ω×(0,T). Linear momentum,(u,p)-system in Q: ∂u ∂t+ (u· ∇)u+∇p− ∇ · (σd+λ σe) = 0,∇ · u=0, σd=σd(D(u),n): dissipative (symmetric) stress tensor: σd=µ1(ntDn)n⊗n+µ4D+µ5(Dn⊗n+n⊗Dn),D=∇u+∇ut 2 σe=σe(ϕ): elastic (non-symmetric) stress tensor, s.t. ∇ · σe=δEe δϕ ∇ϕ+∇Ee Angular momentum,ϕ-equation in Q(of Allen-Cahn’s type): ∂ϕ ∂t+u· ∇ϕ+γδEe δϕ =0, γ > 0 constant (elastic relaxation time). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 10 / 25 Smectic-A Liquid Crystal Model [E, ARMA’96] Initial-boundary problem (with Dirichlet [D1-D2] B.C. for ϕ) in Q: (P)                                Du Dt − ∇ · σd+∇e p−λδEe δϕ ∇ϕ=0,u ∇ · u=0,p ∂ϕ ∂t+u· ∇ϕ+γδEe δϕ =0,δEe δϕ ∆2ϕ−1 ε2∇ · f(∇ϕ) = δEe δϕ ,∂ϕ ∂t u|Σ=0, ϕ|Σ=ϕ1, ∂nϕ|Σ=ϕ2, u|t=0=u0, ϕ|t=0=ϕ0 e p=p+λEe(Lagrange multiplier ∼constraint ∇ · u=0) Other B.C. are possible. For instance, w·m|∂Ω=0 implies the conservation property d dt ZΩ ϕ=0. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 11 / 25 Previous results [Liu, DCDS’00] Time-independent [D1-D2] B.C.: Existence of global in time weak solutions. Existence (and uniqueness) of global in time regular solutions for large viscosity (µ4>>), and its ω-limit is a nonempty set furnished by “equilibrium points”. Global minimizers of the elastic energy Eeare stable. [Climent-Ezquerra & GG, CPAA’10] Extension for time-dependent [D1-D2] B.C. Moreover, Existence of time-periodic weak solutions and regularity for µ4>>. [Segatti & Wu, SIMA’11] Long-time behaviour with periodic B.C.: Finite dimensional attractor in 2Ddomains. Convergence to a single equilibrium (and convergence rate) In particular, ϕ(t)→ϕ∞as t→ ∞, with δEe(ϕ∞) δϕ =0. Moreover, local minimizers of the elastic energy Eeare stable. RK: It may exists a “continuum" of equilibrium solutions with the same energy. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 14 / 25 Previous results [Liu, DCDS’00] Time-independent [D1-D2] B.C.: Existence of global in time weak solutions. Existence (and uniqueness) of global in time regular solutions for large viscosity (µ4>>), and its ω-limit is a nonempty set furnished by “equilibrium points”. Global minimizers of the elastic energy Eeare stable. [Climent-Ezquerra & GG, CPAA’10] Extension for time-dependent [D1-D2] B.C. Moreover, Existence of time-periodic weak solutions and regularity for µ4>>. [Segatti & Wu, SIMA’11] Long-time behaviour with periodic B.C.: Finite dimensional attractor in 2Ddomains. Convergence to a single equilibrium (and convergence rate) In particular, ϕ(t)→ϕ∞as t→ ∞, with δEe(ϕ∞) δϕ =0. Moreover, local minimizers of the elastic energy Eeare stable. RK: It may exists a “continuum" of equilibrium solutions with the same energy. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 14 / 25 Main result Let Sbe the set of equilibrium points: S={(0, ϕ) : ϕ∈H4(Ω),∆2ϕ− ∇ · 1 ε2f(ϕ) = 0, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2}. and the ω-limit set of a global weak solution: ω(u, ϕ) = {(u∞, ϕ∞)∈V×H4:∃{tn} ↑ +∞s.t. (u(tn), ϕ(tn)) →(u∞, ϕ∞)in H1×H4}. Theorem ω(u, ϕ)⊂S ω(u, ϕ) = {(0, ϕ∞)} F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 15 / 25 Goal: Long-time behavior with admissible B.C. Generic situation: Let E(t),D(t)≥0satisfying the “weak estimates": (WE)E0(t) + D(t)≤0,a.e. t∈(0,+∞). Then, E∈L∞(0,+∞)and E(t)&E∞≥0. Moreover, integrating (WE), one has D∈L1(0,+∞). Lemma ([Climent-Ezquerra,Rodriguez-Bellido & GG, JBC’10]) Let D(t)≥0,D∈L1(0,+∞)satisfying the “strong estimate”: (SE)D0(t)≤K(D(t)3+1) (K>0). Then, there exists T∗≥0(large enough) such that D∈L∞(T∗,+∞)and ∃lim t→+∞D(t) = 0. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 16 / 25 Goal: Long-time behavior with admissible B.C. Generic situation: Let E(t),D(t)≥0satisfying the “weak estimates": (WE)E0(t) + D(t)≤0,a.e. t∈(0,+∞). Then, E∈L∞(0,+∞)and E(t)&E∞≥0. Moreover, integrating (WE), one has D∈L1(0,+∞). Lemma ([Climent-Ezquerra,Rodriguez-Bellido & GG, JBC’10]) Let D(t)≥0,D∈L1(0,+∞)satisfying the “strong estimate”: (SE)D0(t)≤K(D(t)3+1) (K>0). Then, there exists T∗≥0(large enough) such that D∈L∞(T∗,+∞)and ∃lim t→+∞D(t) = 0. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 16 / 25 Extension for sequences Goal: Convergence uniformly with respect to the Galerkin sequence’s index. Theorem ([Climent-Ezquerra & GG, NA-TMA’14]) Let Em(t),Dm(t)≥0, satisfying (WE)(Em)0(t) + Dm(t)≤0, (SE)(Dm)0(t)≤K(Dm(t)3+1)K>0independent of m. Then, ∀ε < 1,∃T∗=T∗(ε)≥0(large enough), independent of m, such that kDmkL∞(T∗,+∞)≤ε. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 17 / 25 Main steps to prove convergence to a single steady equilibrium 1To obtain (WE) and (SE) for E(t) = Ekin(u(t)) + λEe(ϕ(t)) and D(t) := ku(t)k2 H1+    δEe δϕ     2 L2 ∼ kuk2 H1+k∂tϕk2 L2 2Existence of weak solutions (u, ϕ)in (0,+∞)which are strong solutions in (T∗ reg,+∞)for some T∗ reg >0. 3E(u(t), ϕ(t)) &E∞in R,u(t)→0in H1 0and δEe(ϕ(t)) δϕ →0in L2(as t↑+∞). 4Recall the ω-limit set: ω={(u∞, ϕ∞)∈H1×H4:∃{tn} ↑ +∞s.t. (u(tn), ϕ(tn)) →(u∞, ϕ∞)in H1×H4} Then, u∞=0and Ee(ϕ∞) = E∞. Moreover: ω6=∅. If (0, ϕ∞)∈ωthen δEe(ϕ∞) δϕ =0in Ω, + B.C. for ϕ∞(i.e. ϕ∞is an “equilibrium point”). 5(Application of Lojasiewicz-Simon). ϕ(t)→ϕ∞in H4as t↑+∞. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 18 / 25 Main steps to prove convergence to a single steady equilibrium 1To obtain (WE) and (SE) for E(t) = Ekin(u(t)) + λEe(ϕ(t)) and D(t) := ku(t)k2 H1+    δEe δϕ     2 L2 ∼ kuk2 H1+k∂tϕk2 L2 2Existence of weak solutions (u, ϕ)in (0,+∞)which are strong solutions in (T∗ reg,+∞)for some T∗ reg >0. 3E(u(t), ϕ(t)) &E∞in R,u(t)→0in H1 0and δEe(ϕ(t)) δϕ →0in L2(as t↑+∞). 4Recall the ω-limit set: ω={(u∞, ϕ∞)∈H1×H4:∃{tn} ↑ +∞s.t. (u(tn), ϕ(tn)) →(u∞, ϕ∞)in H1×H4} Then, u∞=0and Ee(ϕ∞) = E∞. Moreover: ω6=∅. If (0, ϕ∞)∈ωthen δEe(ϕ∞) δϕ =0in Ω, + B.C. for ϕ∞(i.e. ϕ∞is an “equilibrium point”). 5(Application of Lojasiewicz-Simon). ϕ(t)→ϕ∞in H4as t↑+∞. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 18 / 25 Main steps to prove convergence to a single steady equilibrium 1To obtain (WE) and (SE) for E(t) = Ekin(u(t)) + λEe(ϕ(t)) and D(t) := ku(t)k2 H1+    δEe δϕ     2 L2 ∼ kuk2 H1+k∂tϕk2 L2 2Existence of weak solutions (u, ϕ)in (0,+∞)which are strong solutions in (T∗ reg,+∞)for some T∗ reg >0. 3E(u(t), ϕ(t)) &E∞in R,u(t)→0in H1 0and δEe(ϕ(t)) δϕ →0in L2(as t↑+∞). 4Recall the ω-limit set: ω={(u∞, ϕ∞)∈H1×H4:∃{tn} ↑ +∞s.t. (u(tn), ϕ(tn)) →(u∞, ϕ∞)in H1×H4} Then, u∞=0and Ee(ϕ∞) = E∞. Moreover: ω6=∅. If (0, ϕ∞)∈ωthen δEe(ϕ∞) δϕ =0in Ω, + B.C. for ϕ∞(i.e. ϕ∞is an “equilibrium point”). 5(Application of Lojasiewicz-Simon). ϕ(t)→ϕ∞in H4as t↑+∞. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 18 / 25 Main steps to prove convergence to a single steady equilibrium 1To obtain (WE) and (SE) for E(t) = Ekin(u(t)) + λEe(ϕ(t)) and D(t) := ku(t)k2 H1+    δEe δϕ     2 L2 ∼ kuk2 H1+k∂tϕk2 L2 2Existence of weak solutions (u, ϕ)in (0,+∞)which are strong solutions in (T∗ reg,+∞)for some T∗ reg >0. 3E(u(t), ϕ(t)) &E∞in R,u(t)→0in H1 0and δEe(ϕ(t)) δϕ →0in L2(as t↑+∞). 4Recall the ω-limit set: ω={(u∞, ϕ∞)∈H1×H4:∃{tn} ↑ +∞s.t. (u(tn), ϕ(tn)) →(u∞, ϕ∞)in H1×H4} Then, u∞=0and Ee(ϕ∞) = E∞. Moreover: ω6=∅. If (0, ϕ∞)∈ωthen δEe(ϕ∞) δϕ =0in Ω, + B.C. for ϕ∞(i.e. ϕ∞is an “equilibrium point”). 5(Application of Lojasiewicz-Simon). ϕ(t)→ϕ∞in H4as t↑+∞. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 18 / 25 Mixed second-order reformulation of (P) By using the auxiliary variable ψ=−∆ϕ:                  Du Dt ,u+σd,D(u)−p,∇ · u+λ γ(∂ϕ ∂t+u· ∇ϕ)∇ϕ, u=0, ∇ · u,p=0, λ γ∂ϕ ∂t+u· ∇ϕ, ϕ+λ∇ψ, ∇ϕ+λ ε2f(∇ϕ),∇ϕ=0 ∂tψ, ¯ ψ−∇∂tϕ, ∇¯ ψ=0. Energy law: Choosing (u,p, ϕ, ψ) = (u,p, ∂tϕ, λψ) d dt Etot u, ψ, ∇ϕ+σd,D(u)+λ γk∂ϕ ∂t+u· ∇ϕk2 L2=0, where Etot (u, ψ, ∇ϕ) := Ekin(u) + λEela(ψ) + Epen(∇ϕ), Ekin(u) = 1 2kuk2 L2,Eela(ψ) = 1 2kψk2 L2,Epen(∇ϕ) = 1 ε2ZΩ F(∇ϕ). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 21 / 25 Generic second order scheme Given (un, ϕn, ψn), compute (un+1,pn+1 2, ϕn+1, ψn+1)∈Uh×Ph×Φh×Ψhsuch that for any (¯ u,¯ p,¯ϕ, ¯ ψ)∈Uh×Ph×Φh×Ψh: δtun+1,¯ u+ce u,un+1 2,¯ u+σd(D(un+1 2),∇eϕ),D(¯ u) −pn+1 2,∇ · ¯ u+λ γ(δtϕn+1+un+1 2·∇eϕ)∇eϕ,¯ u=0, ∇ · un+1 2,¯ p=0, λ γδtϕn+1+un+1 2·∇eϕ,¯ϕ+λ∇ψn+1 2,∇ϕ+λ ε2fk(∇ϕn+1,∇ϕn),∇¯ϕ=0, δtψn+1,¯ ψ−δt∇ϕn+1,∇¯ ψ=0 where un+1 2= (un+1+un)/2 etc. F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 22 / 25 Unconditional energy stability Lemma The following discrete energy inequality holds: δtEtot (un+1,∇ϕn+1,wn+1) + µ4kD(un+1 2)k2 L2(Ω) +λ γkδtϕn+1+un+1 2· ∇eϕk2 L2(Ω) +λ ε2ND ≤0, where ND =ZΩ fk(∇ϕn+1,∇ϕn)·δt∇ϕn+1−δtZΩ F(∇ϕn+1). F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 23 / 25 Conclusions and open problems Conclusions: 1Regularity and uniqueness for large times 2Convergence to single equilibrium steady solutions 3Unconditional fully discrete energy-stable schemes Open problems: 1Long-time behaviour for other B.C. 2Splitting energy-stable schemes 3Numerical simulations F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 24 / 25 Conclusions and open problems Conclusions: 1Regularity and uniqueness for large times 2Convergence to single equilibrium steady solutions 3Unconditional fully discrete energy-stable schemes Open problems: 1Long-time behaviour for other B.C. 2Splitting energy-stable schemes 3Numerical simulations F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 24 / 25 THANK YOU FOR YOUR ATTENTION F. Guillén-González () Smectic-A liquid crystals AIMS, 2014 25 / 25