Liquid Crystal Equations with Infinite Energy Local Well-posedness and Blow Up Criterion

TL;DR

This paper establishes local well-posedness for incompressible liquid crystal equations with potentially infinite initial energy and provides blow-up criteria based on vorticity direction, extending understanding of solution regularity and singularity formation.

Contribution

It proves local well-posedness for solutions with infinite energy initial data and introduces new blow-up criteria using vorticity direction for type I solutions.

Findings

Solutions are smooth away from initial time.

Local well-posedness holds for $L^ abla$ initial data.

Vorticity direction criteria predict blow-up scenarios.

Abstract

In this paper, we consider the Cauchy problem of the incompressible liquid crystal equations in $n$ dimensions. We prove the local well-posedness of mild solutions to the liquid crystal equations with $L^\infty$ initial data, in particular, the initial energy may be infinite. We prove that the solutions are smooth with respect to the space variables away from the initial time. Based on this regularity estimate, we employ the blow up argument and Liouville type theorems to establish vorticity direction type blow up criterions for the type I mild solutions established in the present paper.