Well-posedness of the spatially homogeneous Landau equation for soft potentials

Nicolas Fournier (1) , Hélène Guérin (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 IRMAR - Institut de Recherche Mathématique de Rennes

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka.

Mots clés: quadratic cost

Mots clés: Fokker-Planck-Landau equation; soft potentials; plasma physics; uniqueness; Wasserstein distance; quadratic cost.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Nicolas Fournier, Hélène Guérin. Well-posedness of the spatially homogeneous Landau equation for soft potentials. Journal of Functional Analysis, 2009, 256 (8), pp.2542-2560. ⟨10.1016/j.jfa.2008.11.008⟩. ⟨hal-00289384⟩