Fluid dynamic limits of kinetic equations. I. Formal derivations

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.

[1]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[2]  T. Nishida.  Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation , 1978 .

[3]  Takaaki Nishida,et al.  On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation , 1979 .

[4]  R. Caflisch.  The fluid dynamic limit of the nonlinear boltzmann equation , 1980 .

[5]  Thomas C. Sideris,et al.  Formation of singularities in three-dimensional compressible fluids , 1985 .

[6]  Y. Sone,et al.  STEADY GAS FLOWS PAST BODIES AT SMALL KNUDSEN NUMBERS : BOLTZMANN AND HYDRODYNAMIC SYSTEMS(Mathematical Analysis of Fluid and Plasma Dynamics I) , 1987 .

[7]  Carlo Cercignani.  The Boltzmann Equation , 1988 .

[8]  Pierre-Louis Lions,et al.  Regularity of the moments of the solution of a Transport Equation , 1988 .

[9]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[10]  STEADY GAS FLOWS PAST BODIES AT SMALL KNUDSEN NUMBERS : BOLTZMANN AND HYDRODYNAMIC SYSTEMS(Mathematical Analysis of Fluid and Plasma Dynamics I) , 1988 .

[11]  Raffaele Esposito,et al.  Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation , 1989 .

[12]  P. Lions,et al.  On the Cauchy problem for Boltzmann equations: global existence and weak stability , 1989 .

[13]  Claude Bardos,et al.  Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles , 1989 .

[14]  C. D. Levermore,et al.  Density variations in weakly compressible flows , 1992 .

[15]  François Golse,et al.  Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation , 1993 .