Well-Posedness of Nematic Liquid Crystal Flow in $L_{uloc}^3({\mathbb{R}}^3)$

Abstract

In this paper, we establish the local well-posedness for the Cauchy problem of the simplified version of hydrodynamic flow of nematic liquid crystals (\ref{LLF}) in ${\mathbb{R}}^3$ for any initial data $(u_0,d_0)$ having small $L_{uloc}^3$-norm of $(u_0,\nabla d_0)$. Here $L_{uloc}^3({\mathbb{R}}^3)$ is the space of uniformly locally $L^3$-integrable functions. For any initial data $(u_0,d_0)$ with small $|(u_0,\nabla d_0){|}_{L^3({\mathbb{R}}^3)}$, we show that there exists a unique, global solution to (\ref{LLF}) which is smooth for $t>0$ and has monotone deceasing $L^3$-energy for $t\ge 0$.

29 pages

Cited by in corpus (8)

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