Advances in Differential Equations

Existence of solutions for a nonlinear Boussinesq-Stefan system

Abdelatif Attaoui

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Adv. Differential Equations, Volume 14, Number 9/10 (2009), 985-1018.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 35D05 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 76R05: Forced convection 76R50: Diffusion [See also 60J60]


Attaoui, Abdelatif. Existence of solutions for a nonlinear Boussinesq-Stefan system. Adv. Differential Equations 14 (2009), no. 9/10, 985--1018.

Export citation