Hydrodynamic limits: some improvements of the relative entropy method

Abstract The present paper is devoted to the study of the incompressible Euler limit of the Boltzmann equation via the relative entropy method. It extends the convergence result for well-prepared initial data obtained by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80]. It explains especially how to take into account the acoustic waves and relaxation layer, and thus to obtain convergence results under weak assumptions on the initial data. The study presented here requires in return some nonuniform control on the large tails of the distribution, which is satisfied for instance by the classical solutions close to a Maxwellian built by Guo [Y. Guo, The Vlasov–Poisson–Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002) 1104–1135].

[1]  Laure Saint-Raymond,et al.  Hydrodynamic Limits of the Boltzmann Equation , 2009 .

[2]  Radyadour Kh. Zeytounian,et al.  Fluid Dynamic Limits of the Boltzmann Equation , 2002 .

[3]  Harold Grad,et al.  Asymptotic Theory of the Boltzmann Equation , 1963 .

[4]  Jean Leray,et al.  Séminaire sur les équations aux dérivées partielles , 1977 .

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

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

[7]  Stéphane Mischler Ceremade.  KINETIC EQUATIONS WITH MAXWELL BOUNDARY CONDITION , 2004 .

[8]  W. Craig,et al.  Strong solutions of the Boltzmann equation in one spatial dimension , 2006 .

[9]  Horng-Tzer Yau,et al.  Relative entropy and hydrodynamics of Ginzburg-Landau models , 1991 .

[10]  François Golse,et al.  Kinetic equations and asympotic theory , 2000 .

[11]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[12]  Yan Guo,et al.  The Boltzmann equation in the whole space , 2004 .

[13]  N. Bellomo,et al.  ON THE CAUCHY PROBLEM FOR THE BOLTZMANN EQUATION , 1995 .

[14]  Stéphane Mischler,et al.  Kinetic equations with Maxwell boundary conditions , 2008, 0812.2389.

[15]  B. Desjardins,et al.  Low Mach number limit of viscous compressible flows in the whole space , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[17]  N. Masmoudi.  Ekman layers of rotating fluids: The case of general initial data , 2000 .

[18]  Yan Guo,et al.  The Vlasov‐Poisson‐Boltzmann system near Maxwellians , 2002 .

[19]  Laure Saint-Raymond.  Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles , 2002 .

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

[21]  C. Cercignani.  Global Weak Solutions of the Boltzmann Equation , 2005 .

[22]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[23]  F. Golse,et al.  Fluid dynamic limits of kinetic equations. I. Formal derivations , 1991 .

[24]  Denis Serre,et al.  Handbook of mathematical fluid dynamics , 2002 .

[25]  Pierre-Louis Lions,et al.  Une approche locale de la limite incompressible , 1999 .

[26]  D. Hilbert,et al.  Begründung der kinetischen Gastheorie , 1912 .

[27]  S. Schochet.  Fast Singular Limits of Hyperbolic PDEs , 1994 .

[28]  Emmanuel Grenier,et al.  On the nonlinear instability of Euler and Prandtl equations , 2000 .

[29]  P. Lions.  Conditions at infinity for boltzmann's equation. , 1994 .

[30]  Laure Saint-Raymond,et al.  The Incompressible Navier-Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials , 2008, 0808.0039.

[31]  L. Saint-Raymond,et al.  Chapter 5 - On the influence of the Earth's Rotation on Geophysical Flows , 2007 .

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

[33]  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 .

[34]  B. Desjardins,et al.  Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations , 2006 .

[35]  R. Caflisch.  Communications in Mathematical Physics © by Springer-Verlag 1980 The Boltzmann Equation with a Soft Potential II. Nonlinear, Spatially-Periodic , 2022 .

[36]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases , 1954 .

[37]  L. Saint-Raymond,et al.  Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit , 2003 .