Well-posedness for the spatially inhomogeneous Fermi-Dirac-Fokker-Planck equation
Abstract
This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor f(1−f) appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint 0≤ f≤ 1. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation (BV), allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.
Related
- Weakly inhomogeneous solutions to a nonlinear Vlasov--Fokker--Planck equation in a domain
- Long-time asymptotics of nonlinear Fokker-Planck equations with inhomogeneous diffusion and natural boundary conditions
- Well-posedness and Quantitative Stability for Degenerate Fokker--Planck Equations with Rough Coefficients
- Dynamical Tensor Train Approximation for Wigner--Fokker--Planck Equation with Applications to Open Quantum Systems
- Fluctuating Hydrodynamics of Quantum Fluids from the Nonlinear Fokker-Planck Equation
BibTeX
@misc{toshpulatov2026wellposedness,
title = {{Well-posedness for the spatially inhomogeneous Fermi-Dirac-Fokker-Planck equation}},
author = {Gayrat Toshpulatov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16700},
eprint = {2610.16700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-posedness for the spatially inhomogeneous Fermi-Dirac-Fokker-Planck equation
AU - Gayrat Toshpulatov
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16700
UR - https://arxiv.org/abs/2610.16700
AB - This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor f(1−f) appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint 0≤ f≤ 1. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation (BV), allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.
ER -