| Date: Friday, January 21, 2011
Title: Global Well-Posedness for the 2D Boussinesq System Without Heat Diffusion
With Anisotropic Viscosity
Abstract: In this talk I will discuss global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in ocean dynamics. This work improves the global well-posedness results established recently by R.
Danchin and M. Paicu for the Boussinesq system with anisotropic viscosity and zero diffusion. In proving the uniqueness result, we have used an alternative approach by writing the transported temperature (density) as $\theta = \Delta\xi$ and adapting the techniques of V. Yudovich for the 2D incompressible Euler equations. This new idea allows us to establish uniqueness results with fewer assumptions on the initial data for the transported quantity $\theta$. Furthermore, this new technique allows us to establish uniqueness results without having to resort to the paraproduct calculus of J. Bony.
If time permits I will also discuss the global regularity of an inviscid $\alpha$-regularization for the two-dimensional inviscid, non-diffusive Boussinesq system of equations, which we call the Boussinesq-Voigt equations and a Voigt-$\alpha$ regularization for the inviscid 3D Boussinesq equations with diffusion.
Speaker: Evelyn Lunasin
Institution: University of Michigan
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