Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the Ḣ⁻ˢ ( 0≤ s<3/2 ) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.