Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems

TL;DR

This paper establishes the existence and uniqueness of strong solutions for the 2D nonlocal Cahn-Hilliard-Navier-Stokes system, showing finite-time regularization and convergence to equilibrium under certain conditions.

Contribution

It proves the existence of unique strong solutions in two dimensions and analyzes their long-term behavior, advancing understanding of the nonlocal model with regular potentials.

Findings

Existence of unique strong solutions in 2D

Finite-time regularization of weak solutions

Convergence of solutions to equilibrium with analytic potentials

Abstract

A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier-Stokes system coupled with a convective Cahn-Hilliard equation. In some recent contributions the standard Cahn-Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.