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September/October 2009 Existence of solutions for a nonlinear Boussinesq-Stefan system
Abdelatif Attaoui
Adv. Differential Equations 14(9/10): 985-1018 (September/October 2009).

Abstract

In this paper, we consider a class of nonlinear Boussinesq-Stefan type systems: a Navier-Stokes equation for the velocity $u$ and the pressure $p$ with second member $F(\theta)$ where $\theta$ is the temperature field, the incompressibility condition and a scalar equation for $\theta$ having a convection term and a nonlinear diffusion operator, in which the right-hand side $\mu(\theta) |D u|^2$ is the dissipation energy. The function $F(\theta)$ is the buoyancy force which satisfies a growth assumption in dimension $2$ and is bounded in dimension $3$. We present some existence results through a fixed-point argument. We use the traditional results of Navier-Stokes equations and those of renormalized solutions. One of the difficulties is the coupling between the two equations for $u$ and $\theta$ through the dissipation energy $\mu(\theta) |D u|^2$. This prevents us from showing compactness, at least if we use the classical results of renormalized solutions for a Stefan problem with $L^1$ data.

Citation

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Abdelatif Attaoui. "Existence of solutions for a nonlinear Boussinesq-Stefan system." Adv. Differential Equations 14 (9/10) 985 - 1018, September/October 2009.

Information

Published: September/October 2009
First available in Project Euclid: 18 December 2012

zbMATH: 1182.35004
MathSciNet: MR2548285

Subjects:
Primary: 35B30, 35D05, 35K55, 76R05, 76R50

Rights: Copyright © 2009 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.14 • No. 9/10 • September/October 2009
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