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

The Cauchy problem for the three-dimensional non-isothermal model for compressible nematic liquid crystals is considered. Existence of global-in-time smooth solutions is established provided that the initial datum is close to a steady state (ρ¯,0,d¯,θ¯). By using the Lq-Lp estimates and the Fourier splitting method, if the initial perturbation is small in H3-norm and bounded in Lq (q∈[1,65)) norm, we obtain the optimal decay rates for the first and second order spatial derivatives of solutions. In addition, the third and fourth order spatial derivatives of director field d in L2-norm are achieved.