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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Compactness properties and local existence of weak solutions to the Landau equation
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by Hyung Ju Hwang and Jin Woo Jang PDF
Proc. Amer. Math. Soc. 148 (2020), 5141-5157 Request permission


We consider the Landau equation nearby the Maxwellian equilibrium. Based on the assumptions on the boundedness of mass, energy, and entropy in the sense of Silvestre [J. Diffential Equations 262 (2017), no. 3, 3034–3055], we enjoy the locally uniform ellipticity of the linearized Landau operator to derive local-in-time $L^\infty _{x,v}$ uniform bounds. Then we establish a compactness theorem for the sequence of solutions using the $L^\infty _{x,v}$ bounds and the standard velocity averaging lemma. Finally, we pass to the limit and prove the local existence of a weak solution to the Cauchy problem. The highlight of this work is in the low-regularity setting where we only assume that the initial condition $f_0$ is bounded in $L^\infty _{x,v}$, whose size determines the maximal time-interval of the existence of the weak solution.
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Additional Information
  • Hyung Ju Hwang
  • Affiliation: Department of Mathematics, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
  • MR Author ID: 672369
  • Email:
  • Jin Woo Jang
  • Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea
  • Address at time of publication: Institute for Applied Mathematics, University of Bonn, 53115 Bonn, Germany
  • MR Author ID: 1297316
  • ORCID: 0000-0002-3846-1983
  • Email:
  • Received by editor(s): February 19, 2019
  • Received by editor(s) in revised form: January 27, 2020
  • Published electronically: September 24, 2020
  • Additional Notes: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF- 2017R1E1A1A03070105 and NRF-2019R1A5A1028324.
    The second author was supported by the Korean IBS project IBS-R003-D1.
    The second author is the corresponding author.
  • Communicated by: Ryan Hynd
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 5141-5157
  • MSC (2010): Primary 35Q84, 35Q20, 82C40, 35B45, 34C29, 35B65
  • DOI:
  • MathSciNet review: 4163828