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Nonuniqueness for the Kinetic Fokker–Planck Equation with Inelastic Boundary Conditions

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Abstract

We describe the structure of solutions of the kinetic Fokker–Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient r describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (J Math Kyoto Univ 2:227–235 1963) of the properties of the stochastic process associated to the Fokker–Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if r < rc and unique if \({r_{c} < r \leqq 1.}\) In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the Fokker–Planck equation. In the proof of the results of this paper we use several asymptotic formulas and computations in the companion paper (Hwang in Q Appl Math 2018).

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Acknowledgements

The authors would like to thank the Hausdorff Center for Mathematical Sciences of the University of Bonn and the Pohang Mathematics Institute, where parts of this work were done. H.J.H. is partly supported by the Basic Science Research Program (NRF-2017R1E1A1A03070105) through the National Research Foundation of Korea. J.J. is 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”, which is funded through the German Science Foundation (DFG). We thank Seongwon Lee for helping with the figures in the paper.

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Authors and Affiliations

  1. Department of Mathematics, Pohang University of Science and Technology, Pohang, GyungBuk, 790-784, Republic of Korea

    Hyung Ju Hwang

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Juhi Jang

  3. Korea Institute for Advanced Study, Seoul, Korea

    Juhi Jang

  4. Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Juan J. L. Velázquez

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  1. Hyung Ju Hwang
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  2. Juhi Jang
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  3. Juan J. L. Velázquez
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Correspondence to Hyung Ju Hwang.

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Communicated by P. Rabinowitz

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Hwang, H.J., Jang, J. & Velázquez, J.J.L. Nonuniqueness for the Kinetic Fokker–Planck Equation with Inelastic Boundary Conditions. Arch Rational Mech Anal 231, 1309–1400 (2019). https://doi.org/10.1007/s00205-018-1299-0

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