Pullback exponential attractors for a Cahn-Hilliard-Navier-Stokes system in 2D

TL;DR

This paper proves the existence of pullback exponential attractors for a 2D coupled Navier-Stokes and Cahn-Hilliard system modeling two incompressible fluids, with results on long-term behavior and regularity.

Contribution

It establishes the existence of pullback exponential attractors for the 2D Navier-Stokes/Cahn-Hilliard system with uniform regularity estimates.

Findings

Existence of pullback exponential attractors for the system.

Regularity estimates depend only on initial data norms.

Uniform bounds with respect to polynomial potential growth.

Abstract

We consider a model for the evolution of a mixture of two incompressible and partially immiscible Newtonian fluids in two dimensional bounded domain. More precisely, we address the well-known model H consisting of the Navier-Stokes equation with non-autonomous external forcing term for the (average) fluid velocity, coupled with a convective Cahn-Hilliard equation with polynomial double-well potential describing the evolution of the relative density of atoms of one of the fluids. We study the long term behavior of solutions and prove that the system possesses a pullback exponential attractor. In particular the regularity estimates we obtain depend on the initial data only through fixed powers of their norms and these powers are uniform with respect to the growth of the polynomial potential considered in the Cahn-Hilliard equation.