Strong solutions of the compressible nematic liquid crystal flow

We study strong solutions of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a domain $Ω\subset\mathbb R^3$. We first prove the local existence of unique strong solutions provided that the initial data $ρ_0, u_0, d_0$are sufficiently regular and satisfy a natural compatibility condition. The initial density function $ρ_0$ may vanish on an open subset (i.e., an initial vacuum may exist). We then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities $\|ρ\|_{L^\infty_tL^\infty_x}$ and $\|\nabla d\|_{L^3_tL^\infty_x}$.

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