Global well-posedness and L 2 decay estimate of smooth solutions for the 3-D incompressible Navier–Stokes–Allen–Cahn system

We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.