On well-posedness and large time behavior for smectic-A liquid crystals equations in $$\mathbb {R}^3$$ R 3
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
On well-posedness and large time behavior for smectic-A liquid crystals equations in R3 Xiaopeng Zhao and Yong Zhou Abstract. The main purpose of this manuscript is to study the well-posedness and decay estimates for strong solutions to the Cauchy problem of 3D smectic-A liquid crystals equations. First, applying Banach fixed point theorem, we prove the local existence and uniqueness of strong solutions. Then, by establishing some nontrivial estimates with energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the initial data are sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solutions. Mathematics Subject Classification. 35D35, 35Q35, 76A15. Keywords. Smectic-A liquid crystals equations, Local well-posedness, Global well-posedness, Decay estimates.
1. Introduction Smectic liquid crystal is a liquid crystalline phase, which possesses not only some degree of orientational order like the nematic liquid crystal, but also some degree of positional order (layer structure). The study of smectic-A liquid crystals has a long history, see for instance, [4,10,11,26]. In 1997, derived from the theory of hydrodynamics motivated by the Ericksen–Leslie system [13,23] for the nematic liquid crystal flow, E [35] proposed the following density dependent incompressible smectic-A liquid crystals equations ⎧ ρt + ∇ · (ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ρ(ut + u · ∇u) = ∇ · (−pI + σ e + σ d ), (1) 2 ⎪ ⎪ ϕt + u · ∇ϕ = λ(∇ · (ξ∇ϕ) − KΔ ϕ), ⎪ ⎪ ⎩ ∇ · u = 0, where ρ(x, t) denotes the density of the material, u(x, t) is the flow velocity, p ∈ R describes a potential function, which dependent on the fluid pressure, ϕ(x, t) means the layer variable, whose level sets represent the layer structure, the positive constant K arises in the free energy and means a comparable in magnitude quantity, and λ > 0 is elastic relaxation time. Moreover, let D represent the symmetric part of the derivative of the velocity: D = 12 (∇u + ∇T u); then, the viscous stress tensor σ d and the elastic stress tensor (Ericksen tensor) satisfy σ d = μ1 (nT Dn)n ⊗ n + μ4 D + μ5 (Dn ⊗ n + n ⊗ Dn), σ e = −ξn ⊗ n + K∇(∇ · n) ⊗ n − K(∇ · n)∇2 ϕ,
(2)
where ξ denotes the corresponding Lagrange multiplier, n = ∇ϕ represents the molecule orientational direction. Besides, μ1 ≥ 0, μ4 > 0 and μ5 ≥ 0 are dissipative coefficients in the stress tensor. The equation |∇ϕ| = 1,
(3)
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X. Zhao and Y. Zhou
ZAMP
also holds, which represents the incompressibility of the layer. In system (1), Eq. (1)1 represents the transporting relation (conservation of mass), Eq. (1)2 is the conservation of linear momentum, and Eq. (1)3 denotes the angular momentum equation, respectively. It is worth pointing out that system (1) also exhibits the property of the anisotropy of the material. In [25], Liu relaxed the constraint (3) and intro
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