Uniqueness  of harmonic map heat flows and  liquid crystal flows

Junyu Lin

doi:10.3934/dcds.2013.33.739

Article Contents

2013, Volume 33, Issue 2: 739-755. Doi: 10.3934/dcds.2013.33.739

This issue Previous Article On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains Next Article Globally weak solutions to the flow of compressible liquid crystals system

Uniqueness of harmonic map heat flows and liquid crystal flows

Junyu Lin^1,

1.
Department of Mathematics, South China University of Technology, Guangzhou, 510640

Received: July 2011

Revised: January 2012

Published: February 2013

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Abstract

In this paper, we prove a limiting uniqueness criterion to harmonic map heat flows and liquid crystal flows. We firstly establish the uniqueness of harmonic map heat flows from $R^n$ to a smooth, compact Riemannian manifold $N$ in the class $C([0,T),BMO_T(R^n,N))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$ For the nematic liquid crystal flows $(v,d)$, we show that the mild solution is unique under the class $C([0,T),BMO_T^{-1}(R^n))\cap L^\infty_{loc}((0,T);L^\infty(R^n))\times C([0,T),BMO_T(R^n,S^2))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$

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Mathematics Subject Classification: 35Q35, 35K55, 76D03,76N10.

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