TY - JOUR

T1 - Local solutions of the Landau equation with rough, slowly decaying initial data

AU - Henderson, Christopher

AU - Snelson, Stanley

AU - Tarfulea, Andrei

N1 - Funding Information:
AT was partially supported by NSF grant DMS-1816643 . SS was partially supported by a Ralph E. Powe Award from ORAU . CH was partially supported by NSF grant DMS-2003110 .
Publisher Copyright:
© 2020 L'Association Publications de l'Institut Henri Poincaré

PY - 2020/11/1

Y1 - 2020/11/1

N2 - 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.

AB - 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.

KW - Classical solutions

KW - Kinetic equations

KW - Landau equation

KW - Large data well-posedness

KW - Vacuum regions

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U2 - 10.1016/j.anihpc.2020.04.004

DO - 10.1016/j.anihpc.2020.04.004

M3 - Article

AN - SCOPUS:85085612471

VL - 37

SP - 1345

EP - 1377

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 6

ER -