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Global strong solution of the 3D inhomogeneous liquid crystal flows with density-dependent viscosity and large velocity

Jiaxu Li, Yu Mei, Rong Zhang

Abstract

This paper concerns the initial boundary value problem of three-dimensional inhomogeneous incompressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficient $μ(ρ)$ is a power function of the density with the power larger than $1$, that is $μ(ρ)=μρ^α$ with $α>1$, it is proved that the system exists a unique global strong solution as long as the initial density is sufficiently large and $L^3$-norm of the derivative of the initial director is sufficiently small. This is the first result concerning the global strong solution for three-dimensional inhomogeneous liquid crystal flows without smallness of velocity.

Global strong solution of the 3D inhomogeneous liquid crystal flows with density-dependent viscosity and large velocity

Abstract

This paper concerns the initial boundary value problem of three-dimensional inhomogeneous incompressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficient is a power function of the density with the power larger than , that is with , it is proved that the system exists a unique global strong solution as long as the initial density is sufficiently large and -norm of the derivative of the initial director is sufficiently small. This is the first result concerning the global strong solution for three-dimensional inhomogeneous liquid crystal flows without smallness of velocity.

Paper Structure

This paper contains 4 sections, 16 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A priori estimates
  4. Proof of Theorem \ref{['global']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^3$. Assume that Given constants Suppose that the initial data $(\rho_0,u_0)$ satisfies Then there exists positive constants $\Lambda_0, \varepsilon_0$ depending only on $C_0,\mu, \nu, \lambda, \alpha,$$\|\nabla\rho_0\|_{L^q}, \| u_0\|_{H^1},$$\|\nabla d_0\|_{H^1},$ and $\Omega$ such that if then the inhomogeneous incompressible simplified

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 17 more