Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation wi

We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function f I L1(RN) yields a control of vf in Sobolev norms as soon as f is locally bounded below. Under this additional assumption of lower bound, our result is an improvement of a recent estimate given by P.-L. Lions, and is optimal in a certain sense.

[1]  Cédric Villani,et al.  On the spatially homogeneous landau equation for hard potentials part ii : h-theorem and applications , 2000 .

[2]  Cédric Villani,et al.  On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations , 1998 .

[3]  Laurent Desvillettes,et al.  A proof of the smoothing properties of the positive part of Boltzmann's kernel , 1998 .

[4]  Radjesvarane Alexandre.  Sur le taux de dissipation d'entropie sans troncature angulaire , 1998 .

[5]  Thierry Goudon.  Sur l'équation de boltzmann homogène et sa relation avec l'équation de landau-fokker-planck: influence des collisions rasantes , 1997 .

[6]  L. Desvillettes.  Regularization for the non-cutoff 2D radially symmetric boltzmann equation with a velocity dependent cross section , 1996 .

[7]  Laurent Desvillettes,et al.  About the regularizing properties of the non-cut-off Kac equation , 1995 .

[8]  Pierre-Louis Lions,et al.  On Boltzmann and Landau equations , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[9]  C. Cercignani.  A remarkable estimate for the solutions of the Boltzmann equation , 1992 .

[10]  G. Toscani.  New a priori estimates for the spatially homogeneous Boltzmann equation , 1992 .

[11]  Eric A. Carlen,et al.  Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation , 1992 .

[12]  A. A. Arsen’ev,et al.  ON THE CONNECTION BETWEEN A SOLUTION OF THE BOLTZMANN EQUATION AND A SOLUTION OF THE LANDAU-FOKKER-PLANCK EQUATION , 1991 .

[13]  Leif Arkeryd,et al.  Intermolecular forces of infinite range and the Boltzmann equation , 1981 .

[14]  Radjesvarane Alexandre.  Sur l'oprateur de Boltzmann non linaire 3D sans troncature angulaire , 1998 .

[15]  L. Desvillettes.  Regularization properties of the 2-dimensional non radially symmetric non cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules , 1997 .

[16]  Bernt Wennberg,et al.  A maxwellian lower bound for solutions to the Boltzmann equation , 1997 .

[17]  P. Lions,et al.  Compactness in Boltzmann’s equation via Fourier integral operators and applications. III , 1994 .

[18]  B. Wennberg.  Regularity in the Boltzmann equation and the Radon transform , 1994 .

[19]  H. McKean.  Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas , 1966 .

[20]  T. Carleman,et al.  Problèmes mathématiques dans la théorie cinétique des gaz , 1957 .

[21]  G. B..  The Dynamical Theory of Gases , 1916, Nature.