Local solutions of the Landau equation with rough, slowly decaying initial data

Henderson, Christopher; Snelson, Stanley; Tarfulea, Andrei

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 6, pp. 1345-1377.

Nous considérons le problème de Cauchy pour l'équation de Landau non-homogène en espace avec potentiels mous dans le cas de grandes (i.e. non perturbatrices) données initiales. Nour construisons une solution pour toute donnée initiale bornée et mesurable à décroissance polynomiale uniforme dans la variable de vitesse, et qui satisfait une hypothèse technique de borne inférieure (il est toujours permis d'avoir des régions de vide). Pour être unique dans cette famille générale, nous devons supposer que les données initiales sont Hölder continues. Nos hypothèses sont beaucoup plus faibles, en termes de régularité et de décroissance, que les résultats de la littérature sur le caractère bien-posé dans le cas de grandes données. Nous dérivons également un critère de continuation pour nos solutions qui est, pour les potentiels très mous, une amélioration par rapport à l'état de l'art.

We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results in the literature. We also derive a continuation criterion for our solutions that is, for the case of very soft potentials, an improvement over the previous state of the art.

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