Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

The incompressible Euler limit of the Boltzmann equation with accommodation boundary condition

Pages: 159 – 190

DOI: https://dx.doi.org/10.4310/CMS.2012.v10.n1.a9

Authors

Claude Bardos (Laboratoire J.-L. Lions, Université Paris-Diderot, Paris, France)

François Golse (Centre de Mathématiques L. Schwartz, Ecole Polytechnique, Palaiseau, France)

Lionel Paillard (Centre de Mathématiques L. Schwartz, Ecole Polytechnique, Palaiseau, France)

Abstract

The convergence of solutions of the Navier-Stokes Equations, set in a domain with boundary, to solutions of the Euler Equations in the large Reynolds number limit is a challenging open problem in both 2 and 3 space dimensions. In particular it is distinct from the question of existence in the large of a smooth solution of the initial-boundary value problem for the Euler Equations. The present paper proposes three results in that direction. First, if the solutions of the Navier-Stokes Equations satisfy a slip boundary condition with vanishing slip coefficient in the large Reynolds number limit, we show by an energy method that they converge to the classical solution of the Euler Equations on its time interval of existence. Next we show that the incompressible Navier- Stokes limit of the Boltzmann Equation with Maxwell’s accommodation condition at the boundary is governed by the Navier-Stokes Equations with slip boundary condition, and we express the slip coefficient at the fluid level in terms of the accommodation parameter at the kinetic level. This second result is formal, in the style of [Bardos-Golse-Levermore, J. Stat. Phys., 63, 323–344, 1991]. Finally, we establish the incompressible Euler limit of the Boltzmann Equation set in a domain with boundary with Maxwell’s accommodation condition assuming that the accommodation parameter is small enough in terms of the Knudsen number. Our proof uses the relative entropy method following closely [L. Saint-Raymond, Arch. Ration. Mech. Anal., 166, 47–80, 2003] in the case of the 3-torus, except for the boundary terms, which require special treatment.

Keywords

Navier-Stokes equations, Euler equations, Boltzmann equation, fluid dynamic limit, inviscid limit, slip coefficient, Maxwell’s accommodation boundary condition, accommodation parameter, relative entropy method, dissipative solutions of the Euler equations

2010 Mathematics Subject Classification

35Q30, 76Bxx, 76D05, 82B40

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Published 14 October 2011