## Advances in Differential Equations

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

### Existence of solutions for a nonlinear Boussinesq-Stefan system

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355863337

**Mathematical Reviews number (MathSciNet)**

MR2548285

**Zentralblatt MATH identifier**

1182.35004

**Subjects**

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]

#### Citation

Attaoui, Abdelatif. Existence of solutions for a nonlinear Boussinesq-Stefan system. Adv. Differential Equations 14 (2009), no. 9/10, 985--1018. https://projecteuclid.org/euclid.ade/1355863337