Global small solutions to heat conductive compressible nematic liquid crystal system: smallness on a scaling invariant quantity

TL;DR

This paper proves the global existence of strong solutions to a 3D heat-conducting compressible nematic liquid crystal system with vacuum, under a smallness condition on a specific scaling invariant quantity related to initial data.

Contribution

It establishes the first global well-posedness result for the system with vacuum, based on a smallness condition on a scaling invariant initial data quantity.

Findings

Global strong solutions exist under the smallness condition.

The smallness condition depends only on system parameters.

The result includes vacuum and vacuum far fields.

Abstract

In this paper, we consider the Cauchy problem to the three dimensional heat conducting compressible nematic liquid crystal system in the presence of vacuum and with vacuum far fields. Global well-posedness of strong solutions is established under the condition that the scaling invariant quantity $ (\|\rho_0\|_\infty+1)\big[\|\rho_0\|_3+(\|\rho_0\|_\infty+1)^2(\|\sqrt{\rho_0}u_0\|_2^2+ \|\nabla d_0\|_2^2)\big] \big[\|\nabla u_0\|_2^2+(\|\rho_0\|_\infty+1)(\|\sqrt{\rho_0}E_0\|_2^2 + \|\nabla^2 d_0\|_2^2)\big]$ is sufficiently small with the smallness depending only on the parameters appeared in the system.