Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one

Abstract

We consider the equation modeling the compressible hydrodynamic flow of liquidcrystals in one dimension. In this paper, we establish the existence of a weaksolution $(\rho, u,n)$ of such a system when the initial density function $0\le \rho_0\in L^\gamma$ for $\gamma>1$, $u_0\in L^2$, and $n_0\in H^1$. This extends aprevious result by [12], where the existence of a weaksolution was obtained under the stronger assumption thatthe initial density function $0$<$c\le \rho_0\in H^1$,$u_0\in L^2$, and $n_0\in H^1$.