Global Existence Results for the Anisotropic Boussinesq System in Dimension Two

Details Global Existence Results for the Anisotropic Boussinesq System in Dimension Two Global Existence Results for the Anisotropic Boussinesq System in Dimension Two

2008-09-29

arxiv.org/abs/0809.4984v1

Mathematics

Analysis of PDEs

Scientific paper

We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.

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