A Liouville theorem for three-dimensional stationary nematic liquid crystal equations

Abstract

In this paper, we prove a Liouville type theorem for three-dimensional stationary liquid crystal equationsunder the condition $u,~\\nabla~d~\\in~L^{9/2,~q}(\\mathbb{R}^3)$,where $u:~\\mathbb{R}^3~\\to~\\mathbb{R}^3$ denotes the velocity field and $d:~\\mathbb{R}^3~\\to~\\mathbb{S}^2$ is the moleculardirection. The proof of this theorem is inspired by Galdis ideas of using local energy estimates in provingthe Liouville theorem for three-dimensional incompressible Navier-Stokes equations in $L^{9/2}(\\mathbb{R}^3)$ and Schoens description on the harmonic maps. We generalize Galdis results to Lorentz spaces and nematic liquid crystal systems.