Global well-posedness and large time behavior for compressible non-isothermal nematic liquid crystal flows with vacuum at infinity

We investigate the Cauchy problem of three-dimensional compressible non-isothermal nematic liquid crystal flows in $\mathbb{R}^{3}$. We derive the global existence and uniqueness of strong solutions with both interior and far field vacuum states provided that the initial data are of small total energy. This improves our previous work [Nonlinear Anal. Real World Appl. 58:103219, 2021] in the sense that the initial data allow possibly large oscillations. The key analysis is based on delicate energy estimates and the structural characteristic of the system under consideration. Moreover, we also show the algebraic decay estimates of the solution. The results could be viewed as an extension of the studies in Li-Xu-Zhang [J. Math. Fluid Mech. 20:2105-2145, 2018] for the isothermal case to the non-isothermal situation.