Well‐posedness and decay for 3D non‐isothermal Navier–Stokes–Allen–Cahn system in ℝ3

Abstract

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in . This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with “good” linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small.