Global well-posedness to the 3D Cauchy problem of compressible non-isothermal nematic liquid crystal flows with vacuum

We are concerned with global well-posedness of strong solutions to the Cauchy problem for compressible non-isothermal nematic liquid crystal flows with vacuum as far field density in R3. By using energy method, we establish the global existence and uniqueness of strong solutions provided that the quantity‖ρ0‖L∞+‖∇d0‖L3 is suitably small and the viscosity coefficients verify 3μ>λ. In particular, the initial density can even have compact support. When d is a constant vector and |d|=1, we also extend partially the corresponding result in Li (2020) where the global small solution of full compressible Navier–Stokes equations was obtained under the condition 2μ>λ. To our knowledge, the result in this paper could be viewed as the first one on the global existence of strong solutions to 3D Cauchy problem for compressible non-isothermal nematic liquid crystal flows with vacuum.