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Solutions to the Fokker-Planck equation with time-dependent force and source in ℝ3

  • Renjun Duan
  • Fucai Li
  • Jinkai Ni
  • Zhu Zhang
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Abstract

This paper concerns the global solutions of the kinetic Fokker-Planck (KFP) equation in ℝ3 driven by a given external force E=E(t,x) and an inhomogeneous source term S0=S0(t,x,v). First, when E and S0 are sufficiently small in a suitable hybrid homogeneous Besov-Sobolev framework without any time-decay assumption, we establish the global well-posedness of the Cauchy problem in the hybrid homogeneous space Lv2(Ḃ 1/22,∞(ℝ3x) ∩ḢN(ℝ3x)) with N≥ 3. We further obtain the uniform stability of the Cauchy problem via semi-group theory and a refined energy method. Second, when both E and S0 are time-periodic of the period T, we show the existence of time-periodic solutions with the same period T and their asymptotic stability as t→ ∞. Moreover, we derive the time-decay estimate of the perturbation around time-periodic solutions for initial data in Lp-norm with $ {3}/{2}< p \leq 2$. Finally, when both E and S0 are independent of t, we establish the existence of stationary solutions and their asymptotic stability. Our results provide a unified treatment of time-periodic dynamics and stationary states for the KFP equation with small force and source terms.

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Cite

BibTeXRIS
BibTeX
@misc{duan2026solutions,
  title = {{Solutions to the Fokker-Planck equation with time-dependent force and source in $\mathbb R^3$}},
  author = {Renjun Duan and Fucai Li and Jinkai Ni and Zhu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23191},
  eprint = {2610.23191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23191},
}
RIS
TY  - UNPB
TI  - Solutions to the Fokker-Planck equation with time-dependent force and source in ℝ^3
AU  - Renjun Duan
AU  - Fucai Li
AU  - Jinkai Ni
AU  - Zhu Zhang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23191
UR  - https://arxiv.org/abs/2610.23191
AB  - This paper concerns the global solutions of the kinetic Fokker-Planck (KFP) equation in ℝ^3 driven by a given external force E=E(t,x) and an inhomogeneous source term S_0=S_0(t,x,v). First, when E and S_0 are sufficiently small in a suitable hybrid homogeneous Besov-Sobolev framework without any time-decay assumption, we establish the global well-posedness of the Cauchy problem in the hybrid homogeneous space L_v^2(Ḃ^ 1/2_2,∞(ℝ^3_x) ∩Ḣ^N(ℝ^3_x)) with N≥ 3. We further obtain the uniform stability of the Cauchy problem via semi-group theory and a refined energy method. Second, when both E and S_0 are time-periodic of the period T, we show the existence of time-periodic solutions with the same period T and their asymptotic stability as t→ ∞. Moreover, we derive the time-decay estimate of the perturbation around time-periodic solutions for initial data in L^p-norm with $ {3}/{2}< p \leq 2$. Finally, when both E and S_0 are independent of t, we establish the existence of stationary solutions and their asymptotic stability. Our results provide a unified treatment of time-periodic dynamics and stationary states for the KFP equation with small force and source terms.
ER  -