A Weak Formulation of the Boltzmann Equation Based on the Fourier Transform

Abstract In this article we present an alternative formulation of the spatially homogeneous Boltzmann equation. Rewriting the weak form of the equation with shifted test functions and using Fourier techniques, it turns out that the transformed problem contains only a three-fold integral. Explicit formulas for the transformed collision kernel are presented in the case of VHS models for hard and soft potentials. For isotropic Maxwellian molecules, a classical result by Bobylev is recovered, too.

[1]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[2]  Sergej Rjasanow,et al.  Mapping Properties of the Boltzmann Collision Operator , 2005 .

[3]  S. Rjasanow,et al.  Difference scheme for the Boltzmann equation based on the Fast Fourier Transform , 1997 .

[4]  Lorenzo Pareschi,et al.  Fast methods for the Boltzmann collision integral , 2004 .

[5]  Bernt Wennberg,et al.  Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff , 2005 .

[6]  Lorenzo Pareschi,et al.  A Fourier spectral method for homogeneous boltzmann equations , 1996 .

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

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

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

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

[11]  Sergej Rjasanow,et al.  Numerical solution of the Boltzmann equation on the uniform grid , 2002, Computing.

[12]  Cl'ement Mouhot,et al.  Regularity Theory for the Spatially Homogeneous Boltzmann Equation with Cut-Off , 2004, math/0607539.

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

[14]  Radjesvarane Alexandre,et al.  LITTLEWOOD–PALEY THEORY AND REGULARITY ISSUES IN BOLTZMANN HOMOGENEOUS EQUATIONS I: NON-CUTOFF CASE AND MAXWELLIAN MOLECULES , 2005 .

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

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

[17]  S. Rjasanow,et al.  Fast deterministic method of solving the Boltzmann equation for hard spheres , 1999 .

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

[19]  Lorenzo Pareschi,et al.  Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit , 2010, Numerische Mathematik.