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Well-posedness of the spatially homogeneous Landau equation for soft potentials by Nicolas Fournier; Hélène Guérin is a Mathematics article available to read on EtoBox.
What is Well-posedness of the spatially homogeneous Landau equation for soft potentials about?
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 wellposedness 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].
Who reads Well-posedness of the spatially homogeneous Landau equation for soft potentials?
It is typically read by researchers, students, and practitioners in Mathematics.
- Author
- Nicolas Fournier; Hélène Guérin
- Publisher
- Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-1236)
- Published
- 2009
- Language
- EN
- Field
- Mathematics (Physical Sciences)