Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids

In this paper, we study the existence and long-time behavior of global strong solutions to a system describing the mixture of two viscous incompressible Newtonian fluids of the same density. The system consists of a coupling of Navier-Stokes and Cahn-Hilliard equations. We first show the global existence of strong solutions in several cases. Then we prove that the global strong solution of our system will converge to a steady state as time goes to infinity. We also provide an estimate on the convergence rate.

Keywords

Navier-Stokes equation, Cahn-Hilliard equation, convergence to equilibrium, Lojasiewicz-Simon approach

2010 Mathematics Subject Classification

35K55, 35Q35, 76D05

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Published 1 January 2009