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

Abstract

This paper studies the local existence of strong solutions to the Cauchy problem of the 2D simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows, coupled via ρ (the density of the fluid), u (the velocity of the field), and d (the macroscopic/continuum molecular orientations). Notice that the technique used for the corresponding 3D local well-posedness of strong solutions fails treating the 2D case, because the L pnorm (p > 2) of the velocity u cannot be controlled in terms only of ρ 1 2 u and ∇u here. In the present paper, under the framework of weighted approximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (2014) 640-671] for Navier-Stokes equations, we obtain the local existence of strong solutions to the 2D compressible nematic liquid crystal flows.

A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow

“…In dimension one, Ding-Lin-Wang-Wen [4] and Ding-Wang-Wen [5] have proven the existence of global strong solution and weak solution respectively. In dimension two, Liu-Zheng-Li-Liu [15] obtained the local existence of strong solution to the Cauchy problem. In dimension two or three, Jiang-Jiang-Wang [11] has proven the global existence of weak solution to the initial-boundary problem with large initial energy.…”

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

“…In this section, some elementary lemmas will be used later. First, the local existence of strong solutions when the initial density may not be positive can be founded in [11].…”