Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels

Jean Dolbeault (1) , Peter Markowich (2) , Dietmar Oelz (3) , Christian Schmeiser (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Faculty of Mathematics [Vienna]
3 Institute for Analysis and Scientific Computing [Wien]
4 RICAM - Johann Radon Institute for Computational and Applied Mathematics

Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proven for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The result are drift-diffusion equations with nonlinear diffusion. The most notable examples are of porous medium or fast diffusion type, with exponent ranging from 0 to 5/3, in dimension 3.

Mots clés: Kinetic equation; Macroscopic limit; Diffusion limit; Boltzmann equation; Equilibrium distribution function; Gibbs state; Porous medium equation; Fast diffusion equation; Relaxation time approximation; Compensated compactness

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser. Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels. Archive for Rational Mechanics and Analysis, 2007, 186 (1), pp.133-158. ⟨hal-00005892⟩