Well-posedness for the three-dimensional compressible liquid crystal flows

Boling Guo,Xiaoli Li

doi:10.3934/dcdss.2016078

Abstract

This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.

Highlights

We establish the well-posedness of a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals formulated in [7]-[9] and [15] in the 1960’s
When the Ossen-Frank energy configuration functional reduces to the Dirichlet energy functional, the hydrodynamic flow equation of liquid crystals in R3 can be written as follows: ρt + ∇ · = 0, (1.1a)
|∇d|2t + ∇ · + ∇p(ρ) = μ∆u − λ∇ · ∇d ∇d − 2 I3, (1.1b) dt + u · ∇d = θ ∆d + |∇d|2d, (1.1c) where u ∈ R3 denotes the velocity, d ∈ S2 is the unit-vector field that represents the macroscopic molecular orientations, p(ρ) is the pressure with p = p(·) ∈ C1[0, ∞), p(0) = 0; and they all depend on the spatial variable x = (x1, x2, x3) ∈ R3 and the time variable t > 0. σ = μ∇u−p I3 is the Cauchy stress tensor given by Stokes’ law and μ∇u stands for the

Summary

Introduction

We establish the well-posedness of a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals formulated in [7]-[9] and [15] in the 1960’s. Established the global existence of strong solutions under the smallness conditions on the initial data in Sobolev spaces in dimension three. Concerning the compressible case, Hu-Wu considered the Cauchy problem for the three-dimensional compressible flow of nematic liquid crystals and obtained the existence and uniqueness of the global strong solution in critical Besov spaces provided that the initial data is close to an equilibrium state in a recent work [11]. Under the hypotheses of Theorem 2.1, there exists a global unique solution u of (2.1c) with the initial data uδ0 and the boundary condition (1.3) such that u ∈ C([0, T ]; H01 ∩ H2) ∩ L2(0, T ; W 2,q), ut ∈ C([0, T ]; L2) ∩ L2(0, T ; H01), utt ∈ L2(0, T ; H−1).

Lq v 2 3q

L2 ρ L3