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On \(A_p\) weights and the Landau equation

  • Maria Gualdani
  • Nestor Guillen
Article
  • 16 Downloads

Abstract

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of very soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for very soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrödinger operator whose coefficients are controlled by \(A_p\)-weights.

Mathematics Subject Classification

35B65 35K57 35B44 35K61 35Q20 35P15 

Notes

Acknowledgements

MPG is supported by NSF DMS-1412748 and DMS-1514761. NG is partially supported by NSF-DMS 1700307. NG would like to thank the Fields Institute for Research in Mathematical Sciences, where part of the work in this manuscript was carried out in the Fall of 2014. MPG would like to thank NCTS Mathematics Division Taipei for their kind hospitality. The authors thank Luis Silvestre for many fruitful communications, as well as the anonymous referee for many useful remarks that helped improve this paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsGeorge Washington UniversityWashingtonUSA
  2. 2.Royal Institute of Technology (KTH)StockholmSweden
  3. 3.Department of MathematicsUniversity of MassachusettsAmherstUSA

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