Article

# The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity

07/2008;
Source: arXiv

ABSTRACT We are concerned with the so-called Boussinesq equations with partial viscosity. These equations consist of the ordinary incompressible Navier-Stokes equations with a forcing term which is transported {\it with no dissipation} by the velocity field. Such equations are simplified models for geophysics (in which case the forcing term is proportional either to the temperature, or to the salinity or to the density). In the present paper, we show that the standard theorems for incompressible Navier-Stokes equations may be extended to Boussinesq system despite the fact that there is no dissipation or decay at large time for the forcing term. More precisely, we state the global existence of finite energy weak solutions in any dimension, and global well-posedness in dimension $N\geq3$ for small data. In the two-dimensional case, the finite energy global solutions are shown to be unique for any data in $L^2(\R^2).$

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22 Feb 2013

### Keywords

dimension $N\geq3$

dissipation

dissipation}

equations

finite energy global solutions

finite energy weak solutions

forcing term

global existence

global well-posedness

incompressible Navier-Stokes equations

large time

ordinary incompressible Navier-Stokes equations

partial viscosity

present paper

salinity

so-called Boussinesq equations

standard theorems

two-dimensional case

velocity field