Global solutions to a nonlinear Fokker-Planck equation

Xingang Zhang; Zhe Liu; Ling Ding; Bo Tang; Xingang Zhang; Zhe Liu; Ling Ding; Bo Tang

doi:10.3934/math.2023822

AIMS Mathematics

2023, Volume 8, Issue 7: 16115-16126. doi: 10.3934/math.2023822

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Global solutions to a nonlinear Fokker-Planck equation

1.
School of Computer Science and Technology, Nanyang Normal University, Nanyang, Henan 473061, China
2.
Nanyang Normal University, Nanyang, Henan 473061, China
3.
School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China
4.
Hubei Key Laboratory of Power System Design and Test for Electrical Vehicle, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China

Received: 22 October 2022 Revised: 21 January 2023 Accepted: 17 March 2023 Published: 05 May 2023
MSC : 35A01, 35Q84

In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space $ H^2_{x}L^2_v $ by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, J. Stat. Phys., 173 (2018), 222–241.), the regularity assumption on the initial data is much weaker.
- nonlinear Fokker-Planck equation,
- global existence,
- energy method,
- Cauchy problem,
- Priori estimates
Citation: Xingang Zhang, Zhe Liu, Ling Ding, Bo Tang. Global solutions to a nonlinear Fokker-Planck equation[J]. AIMS Mathematics, 2023, 8(7): 16115-16126. doi: 10.3934/math.2023822

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Abstract

In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space $ H^2_{x}L^2_v $ by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, J. Stat. Phys., 173 (2018), 222–241.), the regularity assumption on the initial data is much weaker.

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