Global existence and optimal time decay rates of 3D non-isentropic compressible Navier-Stokes-Allen-Cahn system.

In this paper, we study the global existence and large time behavior of the classical solution for the non-isentropic compressible Navier-Stokes-Allen-Cahn (NSAC) system describing the mixture motion of a two-phase immiscible viscous compressible heat-conducting flow in R 3. We first establish the global well-posedness of the Cauchy problem to the non-isentropic compressible NSAC system when the initial data is a small regular perturbation near the constant state. Then by a delicate spectral analysis and the damping structure for the perturbation equation of the phase field, we prove that the phase field decays to the phase separation state at an exponential rate while the density, velocity and the temperature converge to the constant state at the optimal time rate (1 + t) − 3 / 4 in L 2 norm. Furthermore, for well-chosen initial data, we also establish the lower bounds of time decay rate. [ABSTRACT FROM AUTHOR]