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.
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This paper is dedicated to Joel Lebowitz on his 60th-birthday.
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Bardos, C., Golse, F. & Levermore, D. Fluid dynamic limits of kinetic equations. I. Formal derivations. J Stat Phys 63, 323–344 (1991). https://doi.org/10.1007/BF01026608
- Boltzmann equation
- Chapman-Enskog expansion
- incompressible Navier stokes equation
- renormalized and weak solutions