Approximation of the trajectory attractor for a  3D model of incompressibletwo-phase-flows

Ciprian G. Gal; T. Tachim Medjo

doi:10.3934/cpaa.2014.13.2229

Article Contents

2014, Volume 13, Issue 6: 2229-2252. Doi: 10.3934/cpaa.2014.13.2229

This issue Previous Article Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping Next Article Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case

Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows

Ciprian G. Gal^1, and
T. Tachim Medjo^2,

1.
Department of Mathematics, Florida International University, Miami, FL, 33199
2.
Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199

Received: June 2013

Revised: April 2014

Published: November 2014

Abstract / Introduction Related Papers Cited by

Abstract

In this article we study the relations between the long-time dynamics of the 3D Allen-Cahn-LANS-$\alpha$ model and the exact 3D Allen-Cahn-Navier-Stokes system. Following the idea of [26], we prove that bounded set of solutions of the Allen-Cahn-LANS-$\alpha$ model converge to the trajectory attractor $\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes system as time goes to $+ \infty $ and $\alpha$ approaches $0^+. $ In particular we show that the trajectory attractors $\mathcal{U}_{\alpha} $ of the 3D Allen-Cahn-LANS-$\alpha$ model converges to the trajectory attractor $\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes as $ \alpha $ approaches $0^+. $ Let us mention that the strong nonlinearity that results from the coupling of the convective Allen-Cahn system and the LANS-$\alpha$ equations makes the analysis of the problem considered in this article more involved.

Keywords:

Allen-Cahn system,
Lagrange averaged Navier-Stokes-$\alpha$,
phase transition,
trajectory attractors.

Mathematics Subject Classification: 35Q30, 35Q35, 35Q72.

Citation:

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