Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 1

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

Pages: 1 – 38

DOI: https://dx.doi.org/10.4310/DPDE.2014.v11.n1.a1

Authors

Stefano Bosia (Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Italy)

Stefania Gatti (Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Modena, Italy)

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 independent of the growth of the polynomial potential considered in the Cahn-Hilliard equation.

Keywords

Cahn-Hilliard-Navier Stokes equation, pullback exponential attractor, polynomial potential

2010 Mathematics Subject Classification

35B41, 35B45, 35K25, 76Txx

Published 8 April 2014