Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$

TL;DR

This paper investigates the conditions under which strong solutions to the Ericksen-Leslie system in three-dimensional space blow up, introducing new criteria and proving convergence of approximate solutions to the actual system.

Contribution

The paper establishes four blow-up criteria for strong solutions to the Ericksen-Leslie system and proves convergence of Ginzburg-Landau approximate solutions to the true solutions.

Findings

Four blow-up criteria including Serrin, Beal-Kato-Majda, mixed, and a new criterion.

Convergence of Ginzburg-Landau approximate solutions to the Ericksen-Leslie system.

Characterization of maximal existence time using Serrin-type norms.

Abstract

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$ for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.