Strong solutions of the compressible nematic liquid crystal flow

AbstractWe study strong solutions of the simplified Ericksen–Leslie system modeling compressible nematic liquid crystal flows in a domain Ω⊂R3. We first prove the local existence of a unique strong solution provided that the initial data ρ0,u0,d0 are sufficiently regular and satisfy a natural compatibility condition. The initial density function ρ0 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‖Lt∞Lx∞ and ‖∇d‖Lt3Lx∞