On the Boltzmann equation for long‐range interactions

We study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long-range interactions, while singularities in the relative velocity variable occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates, this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions. © 2001 John Wiley & Sons, Inc.

[1]  Pierre Degond,et al.  THE FOKKER-PLANCK ASYMPTOTICS OF THE BOLTZMANN COLLISION OPERATOR IN THE COULOMB CASE , 1992 .

[2]  Cédric Villani,et al.  Conservative forms of Boltzmann's collision operator: Landau revisited , 1999 .

[3]  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.

[4]  Existence et unicité de la solution renormalisée d'un problème parabolique non linéaire assez général , 1999 .

[5]  Radjesvarane Alexandre.  On some related non homogeneous 3D Boltzmann models in the non cutoff case , 2000 .

[6]  Radjesvarane Alexandre.  Une définition des solutions renormalisées pour l'équation de Boltzmann sans troncature angulaire , 1999 .

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

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

[9]  P. Lions.  Global solutions of kinetic models and related questions , 1993 .

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

[11]  C. Villani.  Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation wi , 1999 .

[12]  Radjesvarane Alexandre,et al.  Entropy Dissipation and Long-Range Interactions , 2000 .

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

[14]  François Golse,et al.  The Acoustic Limit for the Boltzmann Equation , 2000 .

[15]  François Golse,et al.  On a model Boltzmann equation without angular cutoff , 2000 .

[16]  Cédric Villani,et al.  On the spatially homogeneous landau equation for hard potentials part i : existence, uniqueness and smoothness , 2000 .

[17]  Carlo Cercignani,et al.  Mathematical Methods in Kinetic Theory , 1970 .

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

[19]  Pierre-Louis Lions,et al.  Lp regularity of velocity averages , 1991 .

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

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

[22]  T. Goudon.  On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions , 1997 .

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

[24]  Cédric Villani,et al.  Contribution à l'étude mathématique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas , 1998 .

[25]  P. Lions,et al.  From the Boltzmann Equations¶to the Equations of¶Incompressible Fluid Mechanics, II , 2001 .

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

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

[28]  Radjesvarane Alexandre,et al.  On the Landau approximation in plasma physics , 2004 .

[29]  P. Lions,et al.  On the Fokker-Planck-Boltzmann equation , 1988 .

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

[31]  Laurent Desvillettes,et al.  On asymptotics of the Boltzmann equation when the collisions become grazing , 1992 .

[32]  B. Wennberg.  The Geometry of Binary Collisions and Generalized Radon Transforms , 1997 .

[33]  R. J. DiPerna,et al.  Global solutions of Boltzmann's equation and the entropy inequality , 1991 .

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

[35]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

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

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

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

[39]  Cédric Villani,et al.  On the Cauchy problem for Landau equation: sequential stability, global existence , 1996, Advances in Differential Equations.

[40]  D. Blanchard,et al.  Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[42]  Radjesvarane Alexandre.  Around 3D Boltzmann non linear operator without angular cutoff, a new formulation , 2000 .

[43]  P. Lions.  Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire , 1998 .