Compactness properties and local existence of weak solutions to the Landau equation
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- by Hyung Ju Hwang and Jin Woo Jang
- Proc. Amer. Math. Soc. 148 (2020), 5141-5157
- DOI: https://doi.org/10.1090/proc/15173
- Published electronically: September 24, 2020
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Abstract:
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.References
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Bibliographic 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: hjhwang@postech.ac.kr
- 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: jangjinw@iam.uni-bonn.de
- 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: https://doi.org/10.1090/proc/15173
- MathSciNet review: 4163828