On the strong solution of 3D non-isothermal Navier–Stokes–Cahn–Hilliard equations

Abstract

In this paper, we consider the global existence of strong solutions of a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. In the diffuse interface model, the evolution of the velocity u is ruled by the Navier–Stokes system, while the order parameter φ representing the difference of the fluid concentration of the two fluids is assumed to satisfy a convective Cahn–Hilliard equation. The effects of the temperature are prescribed by a suitable form of heat equation. By using a refined pure energy method, we prove the existence of the global strong solution by assuming that ‖u0‖H32+‖φ0‖H42+‖θ0‖H32+‖φ02−1‖L22+‖θ0‖L1 is sufficiently small, and higher order derivatives can be arbitrarily large.