Some a priori estimates for the homogeneous Landau equation with soft potentials

Radjesvarane  Alexandre; Jie Liao; Chunjin  Lin

doi:10.3934/krm.2015.8.617

Article Contents

2015, Volume 8, Issue 4: 617-650. Doi: 10.3934/krm.2015.8.617

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Some a priori estimates for the homogeneous Landau equation with soft potentials

1.
Arts et Metiers Paris Tech, Paris 75013
2.
School of Science, East China University of Science and Technology, Shanghai, 200237
3.
Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098

Received: May 2014

Revised: June 2015

Published: December 2015

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Abstract

This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in $L^2$ space hold true for moderately soft potential cases $ \gamma \in[-2, 0) $ without any smallness assumption on the initial data. For very soft potential cases $ \gamma \in[-3, -2) $, which cover in particular the Coulomb case $\gamma=-3$, we get local in time estimates of weak solutions in $L^{2}$.
In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $ \gamma \in[-3, -2) $, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality.

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Mathematics Subject Classification: Primary: 35B45, 35D30; Secondary: 82C40.

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