A regularity condition of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

Abstract

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow‐up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well‐known Serrin's blow‐up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

A new blowup criterion of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

“…On the other hand, if we introduce geometric condition (1.9), then the condition ||∇d|| L s′ (0,T;L r ) can be neglected, which is the key in [9]. Thus, our results partially improve the results of [9] in some measure, which depend on geometric condition (1.9).…”

“…Finally, we need the following Beale-Kato-Majda-type inequality to estimate the term ||∇u|| L ∞ , which can be found in [14]. Lemma 2.7.…”

“…Compare with [9], we only require L s -integrability of the density, rather than the L ∞ -integrability of the density. Thus, our result is better than [9].…”

“…Compare with [9], we only require L s -integrability of the density, rather than the L ∞ -integrability of the density. Thus, our result is better than [9]. On the other hand, if we introduce geometric condition (1.9), then the condition ||∇d|| L s′ (0,T;L r ) can be neglected, which is the key in [9].…”

Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows

“…It would be interesting to consider the global existence of the strong solutions to the problem (1), for which one should study the blow-up mechanism with the structure of possible singularities to the strong solutions of (1). Along this direction, very recently, Liu-Wang [22] and Wang [25] obtained for the 2D isentropic compressible nematic liquid crystal flows (1) that if T * ∈ (0, ∞) is the maximal time of existence for strong solutions (…”