Local conditional regularity for the Landau equation with Coulomb potential

Immanuel Ben Porat

doi:10.3934/krm.2022010

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2022, Volume 15, Issue 5: 775-791. Doi: 10.3934/krm.2022010

This issue Previous Article Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium Next Article A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis

Local conditional regularity for the Landau equation with Coulomb potential

Immanuel Ben Porat

CMLS, Ecole polytechnique, 91128 Palaiseau cedex, Paris, France

Received: June 2021

Revised: February 2022

Early access: March 2022

Published: October 2022

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Abstract

This paper studies the regularity of Villani solutions of the space homogeneous Landau equation with Coulomb interaction in dimension 3. Specifically, we prove that any such solution belonging to the Lebesgue space $ L_{t}^{\infty}L_{v}^{q} $ with $ q>3 $ in an open cylinder $ (0,S)\times B $, where $ B $ is an open ball of $ \mathbb{R}^{3} $, must have Hölder continuous second order derivatives in the velocity variables, and first order derivative in the time variable locally in any compact subset of that cylinder.

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Mathematics Subject Classification: Primary: 35A09.

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