Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Compactness properties and local existence of weak solutions to the Landau equation


Authors: Hyung Ju Hwang and Jin Woo Jang
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
Published electronically: September 24, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35Q84, 35Q20, 82C40, 35B45, 34C29, 35B65

Retrieve articles in all journals with MSC (2010): 35Q84, 35Q20, 82C40, 35B45, 34C29, 35B65


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: 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

DOI: https://doi.org/10.1090/proc/15173
Keywords: Boltzmann equation, Landau equation, collisional kinetic theory, velocity averages
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
Article copyright: © Copyright 2020 American Mathematical Society