Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data

2019/09/12 by Henderson, Christopher, Snelson, Stanley, Tarfulea, Andrei · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.05914

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.

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• Uniqueness for the Homogeneous Landau-Coulomb Equation in L^3/2
• On Hydrodynamic Implosions and the Landau-Coulomb Equation
• The M-Operator and Uniqueness of Nonlinear Kinetic Equations

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