Some topics about nematic and smectic-A liquid crystals

Some topics about Nematic and Smectic-A Liquid Crystals Chill´ an, enero de 2010 Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 1/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 2/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 2/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 2/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 2/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 2/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 3/25 Introduction http://moebius.physik.tu-berlin.de/lc/lcs.html Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 4/25 Introduction http://www.doitpoms.ac.uk/ Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 5/25 The models Penalized Oseen–Frank energy: (NC) Ee=ZΩ (1 2|∇d|2+F(ϕ)) (SAC) Ee=ZΩ (1 2|∆ϕ|2+F(∇ϕ)) where f(n) = ∇nF(n). F(n) = 1 4ε2(|n|2−1)2potential function of f(n) = 1 ε2(|n|2−1)n. Minimization problem −→ Euler-Lagrange equation (NC) ω≡∆d−f(d) = 0,(SAC) ω≡∆2ϕ−∇·f(∇ϕ) = 0, Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 11/25 The models Penalized Oseen–Frank energy: (NC) Ee=ZΩ (1 2|∇d|2+F(ϕ)) (SAC) Ee=ZΩ (1 2|∆ϕ|2+F(∇ϕ)) where f(n) = ∇nF(n). F(n) = 1 4ε2(|n|2−1)2potential function of f(n) = 1 ε2(|n|2−1)n. Minimization problem −→ Euler-Lagrange equation (NC) ω≡∆d−f(d) = 0,(SAC) ω≡∆2ϕ−∇·f(∇ϕ) = 0, Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 11/25 The models Ω⊂IRN(N=2 or 3), ∂Ωregular, Q= Ω ×(0,+∞) ( Ericksen-Leslie, Lin,E) Angular momentum (NC) ∂td+u· ∇d+γω =0(SAC) ∂tϕ+u· ∇ϕ+γω =0 Linear momentum ρ(∂tu+ (u· ∇)u)−∇·(σd+λ σe) + ∇p=0,∇ · u=0 where Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 12/25 The models Ω⊂IRN(N=2 or 3), ∂Ωregular, Q= Ω ×(0,+∞) ( Ericksen-Leslie, Lin,E) Angular momentum (NC) ∂td+u· ∇d+γω =0(SAC) ∂tϕ+u· ∇ϕ+γω =0 Linear momentum ρ(∂tu+ (u· ∇)u)−∇·(σd+λ σe) + ∇p=0,∇ · u=0 where Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 12/25 The models (NC) σd=µ4D(u), σe=λ∇ · (∇d ∇d) (SAC) σd=µ1(ntD(u)n)n⊗n+µ4D(u) + µ5(D(u)n⊗n+n⊗D(u)n), σe=−f(n)⊗n+∇(∇ · n)⊗n−(∇ · n)∇n Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 13/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 14/25 Nematic Model The equations        ∂tu+ (u· ∇)u−ν∆u+∇p=−∇dt∆d, ∇ · u=0, ∂td+ (u· ∇)d= (∆d−f(d)),|d| ≤ 1, in Q + time-dependent (bc) on Σ=(0,∞)×∂Ω. +(iv) or (tp) in Ω. Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 15/25 Smectic Model The equations              ∂tu+ (u· ∇)u−ν∆u−∇·σd nl −(∆2ϕ−∇·f(∇ϕ))∇ϕ+∇q=0, ∇ · u=0, ∂tϕ+u· ∇ϕ+ (∆2ϕ−∇·f(∇ϕ)) = 0, in Q where σd nl := (ntD(u)n)n⊗n+D(u)n⊗n+n⊗D(u)n + time-dependent (bc) on Σ=(0,∞)×∂Ω. +(iv) or (tp) in Ω. Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 16/25 Lifting functions Boundary condition depending on the time (NC) e d=e d(t) stationary (for weak norms) or non-stationary (for regular norms) lifting function b d(t) = d(t)−e d(t),b d=0 on ∂Ω Unknows: u,p,b d (SAC) eϕ=eϕ(t) non stationary lifting function bϕ(t) = ϕ(t)−eϕ(t),bϕ=0 on ∂Ω Unknows: u,p,bϕ Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 17/25 Table of contents 1Introduction 2The Models 3Statement of the Problems 4Nematic Case 5Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 18/25 Smectic-A Case Asymptotic Stability: u(t)→0 in H1 0(Ω) (∆2ϕ−∇·f(∇ϕ))(t)→0 in L2(Ω) when t↑+∞. Moreover, ϕ→ϕfor ”sequences” in H4(Ω)-weak, where ϕis a solution of a Euler-Lagrange problem. Stability: If initial data are small, u,ϕ, and ωare small for each t≥0 Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 24/25 Smectic-A Case Previous result: (iv)-Problem, time-independent boundary conditions. Existence of weak solutions in [0,T], global regularity of weak solutions (for big enough viscosity). [Liu’00] Goal Time-dependent (bc) case [Climent,Guill´ en] 1Uniqueness weak/strong solutions (iv)-Problem, 2Existence of global weak solutions (iv)-Problem, “bounded” up to infinity time, 3Existence of weak time-periodic solutions, 4Existence of regular solutions for both previous cases (dominant viscosity coefficient). 5Stability and asymptotic stability Time-independent (bc) case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 25/25 Smectic-A Case Previous result: (iv)-Problem, time-independent boundary conditions. Existence of weak solutions in [0,T], global regularity of weak solutions (for big enough viscosity). [Liu’00] Goal Time-dependent (bc) case [Climent,Guill´ en] 1Uniqueness weak/strong solutions (iv)-Problem, 2Existence of global weak solutions (iv)-Problem, “bounded” up to infinity time, 3Existence of weak time-periodic solutions, 4Existence of regular solutions for both previous cases (dominant viscosity coefficient). 5Stability and asymptotic stability Time-independent (bc) case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA 2010. Nematics and Smectic-A Liquid Crystals 25/25