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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 6

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

Pages: 1455 – 1486

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a1

#### Authors

#### 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\begin{flalign*}(\|\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]\end{flalign*}is sufficiently small with the smallness depending only on the parameters appearing in the system.

#### Keywords

heat conducting compressible nematic liquid crystal system, global well-posedness, vacuum, scaling invariant quantity

#### 2010 Mathematics Subject Classification

35D35, 35Q35, 76A15, 76N10

Received 23 February 2022

Received revised 4 October 2022

Accepted 3 November 2022

Published 22 September 2023