Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials

Fournier, Nicolas; Heydecker, Daniel

doi:10.1016/j.anihpc.2021.02.004

Fournier, Nicolas ¹ ; Heydecker, Daniel ²

¹ a Sorbonne Université, LPSM-UMR 8001, Case courrier 158, 75252 Paris Cedex 05, France
² b University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, CB30WA, United Kingdom

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1961-1987.

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Résumé

We establish the well-posedness and some quantitative stability of the spatially homogeneous Landau equation for hard potentials, using some specific Monge-Kantorovich cost, assuming only that the initial condition is a probability measure with a finite moment of order p for some $p > 2$ . As a consequence, we extend previous regularity results and show that all non-degenerate measure-valued solutions to the Landau equation, with a finite initial energy, immediately admit analytic densities with finite entropy. Along the way, we prove that the Landau equation instantaneously creates Gaussian moments. We also show existence of weak solutions under the only assumption of finite initial energy.

Reçu le : 2020-09-29
Révisé le : 2021-01-25
Accepté le : 2021-02-05

DOI : 10.1016/j.anihpc.2021.02.004

Classification : 82C40, 60K35
Mots-clés : Fokker-Planck-Landau equation, Uniqueness, Wasserstein distance, Coupling, Stochastic differential equations

Affiliations des auteurs :

Fournier, Nicolas ¹ ; Heydecker, Daniel ²

¹ a Sorbonne Université, LPSM-UMR 8001, Case courrier 158, 75252 Paris Cedex 05, France
² b University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, CB30WA, United Kingdom

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     title = {Stability, well-posedness and regularity of the homogeneous {Landau} equation for hard potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1961--1987},
     publisher = {Elsevier},
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Fournier, Nicolas; Heydecker, Daniel. Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1961-1987. doi : 10.1016/j.anihpc.2021.02.004. https://www.numdam.org/articles/10.1016/j.anihpc.2021.02.004/

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