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

Abstract

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. Résumé 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.

On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics

“…The standard scaling technique and the Hölder estimate up to the initial time given by Lemma 4.7 (ii) can improve the integrability with respect to t in the energy estimate so that Grönwall's inequality becomes admissible, see (4.23) below for the precise expression. This kind of phenomena was also noticed in [17] (see the remarks in §1.4.2). Besides, the global energy estimate of the equation (4.1) is not available when the spatial domain is unbounded, since there is no decay of the solution as |x| → ∞.…”

“…We develop it with rough initial data and without smallness or lower bound assumptions as in Theorem 1.1 and Remark 1.2. An analogue of the local well-posedness for the Landau equation with low-regularity and non-perturbative initial data was established in [17], which is a follow-up study of their previous work [16]. When the drift-diffusion coefficient ρ β f in (1.1) is proportional to the local mass of the solution, that is when β = 1, the equations (1.1), (1.2) have the same quadratic homogeneity as the Landau equation, but global bounds and conservation laws for (1.1), (1.2) are simpler.…”

“…A subtle point of this lower bound result lies in the possibilities of the degeneracy of solutions as t → 0 + or t → ∞, which leads to two delicate issues. First, with the same difficulty as mentioned in [17], in order to prove the uniqueness of the Cauchy problem (1.1), the nondegeneracy of diffusion up to the initial time is required so that the apriori estimates can be still applicable. The uniqueness is then achieved by using the scaling argument and Grönwall's lemma, since the Hölder estimate around the initial time implies that the integrand in the inequality of Grönwall's type is improved to be integrable with respect to the time variable, see the proof of Theorem 4.11 below for more details.…”

Local well-posedness of the Boltzmann equation with polynomially decaying initial data

“…Indeed, this has already been accomplished for the closely-related Landau equation in [24] in the analogous parameter regime. An upshot of such a continuation criterion, were it established, is the ability to construct classical solutions from rough initial initial data as accomplished for the Landau equation [25]. These will be the subject of a future work.…”

“…It would be interesting to extend our existence result to the two remaining cases, γ ∈ (−3, max(−3/2 − 2s, −3)] and γ ≥ 0; we leave this for future work. Another open issue is decreasing the required regularity of the initial data, as in our recent work [30] on the closely related Landau equation, which required only polynomial decay in v and Hölder continuity for f in to establish existence and uniqueness (and if the assumption of Hölder continuity is dropped, we can establish existence but not uniqueness).…”