Optimal time-decay rates of the 3D compressible nematic liquid crystal flows with discontinuous initial data and large oscillations

Abstract

The global existence of low-energy weak solutions for the 3D compressible nematic liquid crystal flows with discontinuous initial data and large oscillations has been proved by Wu and Tan (2018) under the assumptions that the initial energy is small and the initial density has positive lower and upper bounds. However, up to now, the time-decay rate of these solutions has remained an open problem since the solutions have low regularity, and particularly the density has no regularity. We resolve this problem by proving time-decay rates of the solutions in Lr-norm with 2≤r≤∞. Moreover, if additionally the initial data satisfies some low-frequency assumption, the optimal lower bound decay rates of solution are also obtained. Therefore, our decay rates are optimal in this sense.