Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries

Authors: Hyung Ju Hwang, Juhi Jang and Juan J. L. Velázquez
Journal: Quart. Appl. Math.
MSC (2010): Primary 35Q84, 35K65, 35A20; Secondary 35Q70, 35R60, 35R06, 60H15, 60H30, 47D07
DOI: https://doi.org/10.1090/qam/1507
Published electronically: June 19, 2018
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Abstract: In this paper we compute asymptotics of solutions of the kinetic Fokker-Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if $ r < r_c$. The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point $ (x,v)=(0,0)$. In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper (Nonuniqueness for the kinetic-Fokker-Planck equation with inelastic boundary conditions) to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.

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Hyung Ju Hwang
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang, GyungBuk 790-784, Republic of Korea
Email: hjhwang@postech.ac.kr

Juhi Jang
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089 – and – Korea Institute for Advanced Study, Seoul, Korea
Email: juhijang@usc.edu

Juan J. L. Velázquez
Affiliation: Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: velazquez@iam.uni-bonn.de

DOI: https://doi.org/10.1090/qam/1507
Keywords: Fokker--Planck equation, nonuniqueness of solutions, measure-valued solutions, inelastic boundary condition, singular set
Received by editor(s): March 4, 2018
Published electronically: June 19, 2018
Additional Notes: The first author was partly supported by the Basic Science Research Program (NRF-2017R1E1A1A03070105) through the National Research Foundation of Korea.
The second author was supported in part by NSF grants DMS-1608492 and DMS-1608494.
The authors acknowledge support through the CRC 1060: The Mathematics of Emergent Effects at the University of Bonn, that is funded through the German Science Foundation (DFG)
Article copyright: © Copyright 2018 Brown University

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