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

In this paper, we consider the global existence of strong solutions of a thermodynamically consistent diffuse interface model describing twophase 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 f 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|| H-2(3) + ||f0. ||(2) H-4|| +.||(2) H-3|| +.f20 - 1.2 L2 +..0. L1 is sufficiently small, and higher order derivatives can be arbitrarily large.