Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals

Global Small Solutions of Heat Conductive Compressible Navier–Stokes Equations with Vacuum: Smallness on Scaling Invariant Quantity

“…Fan et al [6] first investigated the local existence of unique strong solution to the initial boundary value problem. Guo et al [11] obtained the global existence of smooth solutions for the Cauchy problem provided that the initial datum is close to a steady state and gave the algebraic decay rate of the global solution. A blow up criterion was established in [42] for the strong solutions to the two-dimensional non-isothermal flows in a bounded domain under a geometric condition introduced by Lei et al in [24].…”

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

“…Remark 1.4 It is worth mentioning that Guo-Xi-Xie [12] established the decay rates for second order spatial derivatives of the density and the velocity provided that the initial perturbation is small in H 3norm and bounded in L q -norm (1 ≤ q < 6 5 ). However, we should point out that, for the initial density allowing vacuum states, whether the second derivatives of the velocity field decay or not remains unknown.…”

Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows

“…There are also many works on the study of liquid crystal system in the compressible case, see, e.g., [10,16,18,21,26,28] and the references therein.…”