Behaviour at infinite time of Nematic Phase Problem Mouhamadou - - PowerPoint PPT Presentation
Behaviour at infinite time of Nematic Phase Problem Mouhamadou Samsidy Goudiaby + Supervisors Blanca Climent-Ezquerra , Francisco Guilln-Gonzales , + EDAN, Universidad de Sevilla, Spain. UFR SAT, LANI, Universit Gaston Berger,
Mouhamadou Samsidy Goudiaby + Supervisors Blanca Climent-Ezquerra∗, Francisco Guillén-Gonzales∗,
+ EDAN, Universidad de Sevilla, Spain. ∗ UFR SAT, LANI, Université Gaston Berger, Saint-Louis, Senegal.
What is a liquid crystal ? Nematic Model Behaviour at infinite time
1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Liquid crystals (LCs) are substances which exhibit an intermediate phase of matter that has properties between those of a conventional liquid and those of a solid crystal. Discovering of liquid crystals attributed to the Austrian botanist Friedrich Richard Reinitzer in 1888. Otto Lehmann, a German physicist solved the problem with description of a new state of matter midway between a liquid and a crystal. In 1991, Pierre-Gilles de Gennes, a French physicist received the Nobel Prize because of issues related to (LC).
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Some topics about liquid crystals ...
Some LC phases Nematic : no positional order but long-range orientational
Smectic : well-defined layers and orientational order.
What is a liquid crystal ? Nematic Model Behaviour at infinite time
LCs in nature Forming biological membrane, Protein solution generate silk of spider, DNA and polypeptides. Some technological applications Liquid crystal displays (LCD), Construction of optics blinds.
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time
Some topics about liquid crystals ...
Simplified model by [Lin’89], [Lin,Liu’95], [Lin,Liu’00] and [Coutand,Shkoller’01] Original equations by Ericksen and Leslie, during 1958-1968. Elements in LC model Studied d, an unit vectorial function modeling the orientation of the crystals molecules. Approximation by Ginzburg-Landau penalization, |d| = 1 is relaxed by |d| ≤ 1 using f (d) = 1
ǫ2 (|d|2 − 1)d
Oseen Frank energy (elastic energy) Ee =
1 2|∇d|2 + F(d)
F(n) = 1 4ǫ2 (|n|2 − 1)2 is a potential function of f (n), i.e. f (n) = ∇nF(n) Euler-Lagrange system w ≡ −∆d + f (d) = 0
We assume the liquid crystal confined in an open bounded domain Ω ⊂ RN (N = 2 or 3) with regular boundary ∂Ω. u : Ω × [0, +∞) → RN is the flow velocity, p : Ω × [0, +∞) → R is the fluid pressure, d : Ω × [0, +∞) → R3 is the orientation vector. We denote Q = (0, +∞) × Ω, = (0, +∞) × ∂Ω, the Lp norm is denoted by | · |p, 1 ≤ p ≤ ∞, and the Hm norm by || · ||m Conservation of angular momentum ∂td + (u · ∇)d − ∆d + f (d) = 0, Conservation of linear momentum ∂tu + (u · ∇)u − ν∆u + ∇p + (∇d)t(−∆d + f (d)) = 0, ∇ · u = 0
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Nematic Model ∂tu + (u · ∇)u − ν∆u + ∇p + (∇d)t(−∆d + f (d)) = 0, ∇ · u = 0 ∂td + (u · ∇)d − ∆d + f (d) = 0, (1) Initial conditions u(x, 0) = u0(x), d(x, 0) = d0(x) in Ω. (2) Boundary conditions u(x, t) = 0, d(x, t) = d0(x)
∂Ω × (0, T) (3)
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Time-independent boundary data for initial value problem F.H. Lin, C. Liu. Non-parabolic dissipative systems modeling the flow of liquid crystals, 1995. Time-dependent boundary data
Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, 2009.
Rojas-Medar, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, 2009.
Reproductivity for a nematic liquid crystal model, 2006.
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What is a liquid crystal ? Nematic Model Behaviour at infinite time
E(u(t), d(t)) = 1 2|u(t)|2
2 + 1
2|∇d(t)|2
2 +
F(d(t)) Ec(u(t)) = 1 2|u(t)|2
2
Ee(d(t)) = 1
2|∇d(t)|2 2 +
d dt E(u(t), d(t)) + ν|∇u(t)|2
2 + | − ∆d(t) + f (d(t))|2 2 = 0
(4)
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
1 What is a liquid crystal ? 2 Nematic Model 3 Behaviour at infinite time
Some topics about liquid crystals ...
Definition (Weak solution). We say that (u, d) is a weak solution of (1)-(14) in (0, +∞) if ∇ · u = 0, u = 0, d = h, ||(u(t), d(t))||0×1 ≤ C1 ∀ t ≥ 0 (5) ∀ γ > 0, e−γt t eγs||(u(t), d(t))||2
1×2ds ≤ C2,
∀ t ≥ 0, (6) Definition (Strong solution). A weak solution (u, d) of (1)-(14) is a strong solution in (0, +∞) if ||(u(t), d(t))||1×2 ≤ C3 ∀ t ≥ 0 (7) ∀ γ > 0, e−γt t eγs||(u(t), d(t))||2
2×3ds ≤ C4,
∀ t ≥ 0, (8)
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Let Ω, d0 be regular enough, with |d0| ≤ 1 in ¯ Ω, assume (u0, d0) ∈ H1 × H2 and ||(u0, d0)||1×2 ≤ C. If (u(t), d(t)) is a strong solution of (1)-(14) in (0, +∞), then the total enery E(u(t), d(t)) ց E∞ = Ee(¯ d) when t ↑ +∞, where ¯ d is a critical point of elastic energy, that is, a solution of the stationary problem −∆¯ d + f (¯ d) = 0 in Ω ¯ d|∂Ω = h. (9) Moreover, (u(t), d(t)) satisfies u(t) − → 0 in H1
0, ∆d(t)−f (d(t)) −
→ 0 in L2 and d(t) − → ¯ d in H2.
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Let Ω, d0 be regular enough, with |d0| ≤ 1 in ¯ Ω, assume (u0, d0) ∈ H1 × H2 and ||(u0, d0)||1×2 ≤ M0. If (u(t), d(t)) is a strong solution of (1)-(14) in (0, +∞), then the total enery E(u(t), d(t)) ց E∞ ≥ 0, as t ↑ +∞ and u(t) − → 0 in H1
0(Ω), ∆d(t) − f (d(t)) −
→ 0 in L2(Ω), as t ↑ +∞
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Weak estimation d dt E(u(t), d(t)) + G(u(t), d(t)) ≤ 0 Strong estimation d dt G(u(t), d(t)) ≤ C
where G(u(t), d(t)) = ν|∇u(t)|2
2 + | − ∆d(t) + f (d(t))|2 2.
Some topics about liquid crystals ...
General Framework E(t), G(t) ≥ 0, E
′(t) + G(t) ≤ 0
a.e. t ∈ (t0, +∞) Then, E ∈ Cb[t0, +∞), is a decreasing function and there exists E∞ ≥ 0 such that lim
t→+∞ E(t) = E∞.
Let G ∈ L1(t0, +∞) be a function satisfying G
′(t) ≤ C(G(t)3 + 1). Then,
lim
t→+∞ G(t) = 0.
Rodríguez-Bellido, Stability for nematic liquid crystals with stretching terms, 2009. General Framework + Weak estimation ⇒ E(u(t), d(t)) ց E∞ ≥ 0, General Framework + Strong estimation ⇒ lim
t→+∞
What is a liquid crystal ? Nematic Model Behaviour at infinite time
E∞ = Ee(¯ d) where ¯ d is a critical point of elastic energy, that is, a solution of the stationary problem −∆¯ d + f (¯ d) = 0 in Ω ¯ d|∂Ω = h. (10)
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Let S be the set S =
d|∂Ω = h
The ω-limit set of (u0, d0) ∈ V × H2 ⊂ L2 × H1 with respect to the strong solution in (0, +∞) of problem (1)-(14) is defined as follow ω((u0, d0)) =
u(x), ¯ d(x)), / there exist {tn} ր +∞ such that (u(x, tn), d(x, tn)) → (¯ u(x), ¯ d(x)) in L2 × H1 Proposition ω((u0, d0)) is a nonempty bounded subset in H1 × H2 and w((u0, d0)) ⊂ S.
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Using Poincaré Inequality and the strong estimation u(tn) − → 0 in L2, −∆d(tn)+f (d(tn)) − → 0 in L2, tn → +∞ Strong solution definition. Convergence in H1 ω(u0, d0)) is a nonempty bounded subset in H1 × H2 and it consist of a single point (0, ¯ d) Weak convergence in H2 and Convergence in H1 , then −∆d + f (d) = 0 Convergence in H1 + Convergence in L2 + Unicity of the limit shows that E∞ = Ee(¯ d)
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
d(t) − → ¯ d in H2 when t ↑ +∞.
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
sous-ensembles analytiques réels, 1963. Lemma (Lojasiewicz-Simon type inequality) Let ¯ d be a critical point of Ee(¯ d) subject to d ∈ H1 with the boundary condition d|∂Ω = h. There exist constant θ ∈ (0, 1/2) and β > 0 depending on ¯ d such that for any d ∈ H1 satisfying d|∂Ω = h and ||d(t) − ¯ d||1 < β, there holds |Ee(d) − Ee(¯ d)|1−θ ≤ || − ∆d + f (d)||−1. (12)
system modeling the nematic liquid crystal flows 2010.
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Proposition Let (0, ¯ d) ∈ ω((u0, d0)). Assume that for all t ∈ [t0, t1) one has ||d(t) − ¯ d||1 < β and Ee( ¯ d) < E(u(t), d(t)), then the following differential inequality holds L-S I C θ d dt
d) θ +G(u(t), d(t))1/2 ≤ 0, ∀ t ∈ [t0, t1), and IPD t1
t0
|∂td(t)|2dt ≤ C θ
d) θ.
Some topics about liquid crystals ...
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Proposition Assume Ee( ¯ d) < E(u(t), d(t)) for all t ≥ 0 and let (0, ¯ d) ∈ ω((u0, d0)) and {tn} ր +∞ such that d(tn) → ¯ d in H1, then there exists n0 big enough such that for all t ≥ tn0 > 0, ||d(t) − ¯ d||1 < β.
Some topics about liquid crystals ...
d(tn) → ¯ d in H1 and E(u(tn), d(tn)) → E∞ = Ee(¯ d) then for any ε ∈ (0, β), there exists N(ε) / n ≥ N ||d(·, tn) − ¯ d||1 < ε, C θ
d) θ < ε. (13) n ≥ N, ¯ tn = sup{t > tn/||d(·, s) − ¯ d||1 < β, ∀ s ∈ [tn, t]}. ¯ tn > tn, for all n ≥ N. Assume that ¯ tn is finite. Use the former differential inequality lim
n→+∞ |d(¯
tn) − ¯ d|2 = 0 Relative compactness of d(t) in H1 n is sufficiently large, ||d(¯ tn) − ¯ d||1 < β Contradiction because ||d(¯ tn) − ¯ d||1 = β
Proposition d(t) → ¯ d in H2 as t → +∞. Proof: Case 1 : If there is a t0 > 0 such that E(u(t0), d(t0)) = Ee(¯ d), then d(t) = ¯ d for t > t0 Case 2 : t ≥ tn0, L-S I and IPD implies ,
sequence in L2 lim
t→+∞ |d(t) − ¯
d|2 = 0. d(t) is uniformly bounded in H2, lim
t→+∞ ||d(t) − ¯
d||1 = 0. Finally lim
t→+∞ ||d(t) − ¯
d||2 = 0
What is a liquid crystal ? Nematic Model Behaviour at infinite time
system modeling the nematic liquid crystal flows 2010.
Cahn-Hilliard model with the dynamic boundary conditions, 2004.
for a parabolic-hyperbolic phase-field system with Neumann boundary conditions, 2007.
hydrodynamic system in the nematic liquid crystal flows, 2009.
for a parabolic-hyperbolic time dependent Ginzburg-Landau-Maxwell equations, 2009.
Some topics about liquid crystals ...
1 Neumann boundary condition 2 Numerical analysis on the Nematic model 3 Behaviour at infinite time of a smectic A model
∂tu + (u · ∇)u − ν∆u − ∇ · σd
nl − w∇ϕ + ∇p = 0,
∇ · u = 0 ∂tϕ + (u · ∇)ϕ + w = 0, w = ∆2ϕ − ∇ · f (∇ϕ) Initial conditions u(x, 0) = u0(x), ϕ(x, 0) = ϕ0(x) in Ω. Boundary conditions u|∂Ω = 0, ϕ|∂Ω = ϕ1(x), ∂nϕ|∂Ω = ϕ2(x)
What is a liquid crystal ? Nematic Model Behaviour at infinite time
Some topics about liquid crystals ...