ON THE ASYMPTOTIC ANALYSIS OF KINETIC MODELS TOWARDS THE COMPRESSIBLE EULER AND ACOUSTIC EQUATIONS

This paper deals with the analysis of the asymptotic limit for models of the mathematical kinetic theory to the nonlinearized compressible Euler equations or to the acoustic equations when the Knudsen number e tends to zero. Existence and uniqueness theorems are proven for analytic initial fluctuations on the time interval independent of the small parameter e. As e tends to zero, the solution of kinetics models converges strongly to the Maxwellian whose fluid-dynamics parameters solve the Euler and the acoustic systems. The general results are specifically applied to the analysis of the Boltzmann and BGK equations.

[1]  The Incompressible Navier–Stokes for the Nonlinear Discrete Velocity Models , 2002, nlin/0306036.

[2]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

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

[4]  Mean-Field Approximation of Quantum Systems and Classical Limit , 2002, math-ph/0205033.

[5]  Pierre-Louis Lions,et al.  From the Boltzmann Equations¶to the Equations of¶Incompressible Fluid Mechanics, I , 2001 .

[6]  A. Bellouquid.  Global Existence and Large-Time Behavior for BGK Model for a Gas with Non-constant Cross Section , 2003 .

[7]  Kiyoshi Asano,et al.  The Euler limit and initial layer of the nonlinear Boltzmann equation , 1983 .

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

[9]  Louis Nirenberg,et al.  An abstract form of the nonlinear Cauchy-Kowalewski theorem , 1972 .

[10]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[11]  Abdelghani Bellouquid,et al.  A DIFFUSIVE LIMIT FOR NONLINEAR DISCRETE VELOCITY MODELS , 2003 .

[12]  L. Bonilla,et al.  High-field limit of the Vlasov-Poisson-Fokker-Planck system: A comparison of different perturbation methods , 2000, cond-mat/0007164.

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

[14]  G. Papanicolaou,et al.  The fluid‐dynamical limit of a nonlinear model boltzmann equation , 1979 .

[15]  François Golse,et al.  Stokes‐Fourier and acoustic limits for the Boltzmann equation: Convergence proofs , 2002 .

[16]  From microscopic to macroscopic models: Asymptotic analysis of the broadwell model toward the wave equation , 2002 .

[17]  L. Boltzmann.  Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen , 1970 .

[18]  Takaaki Nishida,et al.  A note on a theorem of Nirenberg , 1977 .

[19]  Claude Bardos,et al.  The Classical Incompressible Navier-Stokes Limit of the Boltzmann Equation(Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics) , 1991 .

[20]  François Golse,et al.  THE VLASOV–POISSON SYSTEM WITH STRONG MAGNETIC FIELD IN QUASINEUTRAL REGIME , 2003 .

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

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

[23]  F. Golse,et al.  La méthode de l’entropie relative pour les limites hydrodynamiques de modèles cinétiques , 2000 .

[24]  Mirosław Lachowicz,et al.  On the initial layer and the existence theorem for the nonlinear Boltzmann equation , 1986 .

[25]  L. Saint-Raymond.  THE GYROKINETIC APPROXIMATION FOR THE VLASOV–POISSON SYSTEM , 2000 .

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

[27]  N Bellomo,et al.  Lecture Notes on Mathematical Theory of the Boltzmann Equation , 1995 .

[28]  Juan Soler,et al.  PARABOLIC LIMIT AND STABILITY OF THE VLASOV–FOKKER–PLANCK SYSTEM , 2000 .