Blow-up criterion and the global existence of strong/classical solutions to Navier–Stokes/Allen–Cahn system

In this paper, we propose a new viscosity for a coupled compressible Navier–Stokes/Allen–Cahn system because it describes the motion of a gas in a flowing liquid. The viscosity depends on two different variables (the density and the unknown function in Allen–Cahn equations). We establish blow-up criterions for strong solutions to initial-boundary value problem. We also show that the strong solutions can be improved to classical solutions. Precisely, when the viscosity only depends on the density, we prove the global existence and uniqueness of the Navier–Stokes/Allen–Cahn equations.