Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two

Xiaoli Li

doi:10.3934/dcds.2017211

Article Contents

2017, Volume 37, Issue 9: 4907-4922. Doi: 10.3934/dcds.2017211

This issue Previous Article Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications Next Article Hyperbolic sets that are not contained in a locally maximal one

Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two

Xiaoli Li^,

College of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

* Corresponding author: Xiaoli Li
* Corresponding author: Xiaoli Li

Received: March 31, 2016

Revised: March 31, 2017

Published: October 2017

The author is supported in part by the National Natural Science Foundation of China under grants 11401036,11271052 and 11471050

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Abstract

This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal flows with vacuum in a bounded smooth domain of $\mathbb{R}^2$. The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous fluid, coupled with various kinematic transport equations for the molecular orientations. Assuming the initial data are sufficiently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.

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Mathematics Subject Classification: Primary: 35A05, 76D10, 76D03.

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