Weak-renormalized solution for a nonlinear Boussinesq system. (English) Zbl 1240.35401

Attaoui, Abdelatif; Blanchard, Dominique; Guibé, Olivier

Differ. Integral Equ. 22, No. 5-6, 465-494 (2009).

The article is devoted to the problem of existence of a weak-renormalized solution for a class of nonlinear Boussinesq systems. The viscosity is a function of temperature and the buoyance force satisfies a growth assumption in 2D case and is bounded in 3D. The author defines weak solution for Navier-Stokes equations and renormalized solution for the additional transport-diffusion equation for the temperature. The framework of renormalized solution has been introduced by R. DiPerna and P. L. Lions for the Boltzmann equations. This notion was adapted to the parabolic version with \(L^1\) data. The existence of solution of the coupled system is proved for small initial data and under growth assumption on force in 2D case and for small initial data and force in 3D case.

Reviewer: Šárka Nečasová (Praha)

MSC:

35Q35	PDEs in connection with fluid mechanics
35D30	Weak solutions to PDEs

Keywords:

nonlinear Boussinesq system; weak-renormalized solution; viscosity