Well-posedness of the spatially homogeneous Landau equation for soft potentials |
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Authors: | Nicolas Fournier |
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Institution: | a LAMA UMR 8050, Faculté de Sciences et Technologies, Université Paris Est, 61, Avenue du Général de Gaulle, 94010 Créteil Cedex, France b IRMAR UMR 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France |
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Abstract: | 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 moderately soft potentials, we assume additionally 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 H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46 (1) (1978-1979) 67-105]. |
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Keywords: | Fokker-Planck-Landau equation Soft potentials Plasma physics Uniqueness Wasserstein distance Quadratic cost |
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