Fluid dynamic limits of kinetic equations. I. Formal derivations |
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Authors: | Claude Bardos François Golse David Levermore |
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Affiliation: | (1) Département de Mathématiques, Université Paris VII, 755251 Paris Cédex 05, France;(2) Departement of Mathematics, University of Arizona, 85721 Tucson, Arizona |
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Abstract: | The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.This paper is dedicated to Joel Lebowitz on his 60th-birthday. |
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Keywords: | Boltzmann equation Chapman-Enskog expansion incompressible Navier stokes equation renormalized and weak solutions |
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