Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
May 23-26, 2011

Van Vleck Hall
Department of Mathematics
University of Wisconsin -Madison
Madison, WI 53706

About incompressible Euler limit of solutions of Navier-Stokes and Boltzmann equation in the presence of boundary effects

Prof. Claude Bardos

University of Paris VII

Abstract:  This is a report on a on going joint work with Fran├žois Golsea and Lionel Paillard.

It is well known that in any dimension (even in 2d the limit (when the Reynold number goes to $\infty$ ) is in presence of boundary a challenging open problem...

Results are simpler when the fluid satisfies a Navier boundary condition and the problem is completely open when the fluid satisfies for finite Reynold number a Dirichlet boundary condition. The only general (always valid) mathematical result being a classical theorem of Tosio Kato.

In the incompressible finite Reynold number limit the solution of the Boltzmann equation (with boundary and accomodation effect) converges to a Leray solution of the Navier Stokes equattion with a Navier Boundary condition which depends on the accomodation codfficient (Kazuo Aoki...Nader Masmoudi and Laure Saint Raymond)

On the other hand it has been observed by several researchers from Yoshio Sone to Laure Saint Raymond that with convenient scalings (infinite Reynolds number) and in the absence of boundary the Boltzmann equation leads to the incompressible Euler equation.

Hence we try to adapt to this limit, in presence of boundary with accomodation, what is known or conjectured at the level of the Navier Stokes limit.