The fundamental solution of a nonlinear kinetic Fokker-Planck equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911548511879168 |
|---|---|
| author | Brigati, Giovanni Carlier, Guillaume Dolbeault, Jean |
| author_facet | Brigati, Giovanni Carlier, Guillaume Dolbeault, Jean |
| contents | This paper is devoted to a fundamental solution of a nonlinear kinetic equation involving a porous medium or fast diffusion operator acting on velocities. Such a nonlinearity has interesting scaling properties, which result in a self-similar behaviour of the fundamental solution. Here fundamental solution means a Dirac distribution initial datum which moreover governs the large time asymptotics of a large class of solutions. Using a self-similar change of variables, the equation becomes a nonlinear kinetic Fokker-Planck equation with harmonic confinement and the intermediate asymptotics regime is transformed into a stability property of a special stationary solution, which attracts the solutions for large times. In the homogeneous case (pure nonlinear diffusion), the problem is reduced to a classical nonlinear diffusion equation with Barenblatt-Pattle self-similar profiles. Unexpectedly, this beautiful structure is preserved at kinetic level, with remarkable consequences for relative entropy estimates, detailed intermediate asymptotics and nonlinear diffusion limits in adapted functional spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26650 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The fundamental solution of a nonlinear kinetic Fokker-Planck equation Brigati, Giovanni Carlier, Guillaume Dolbeault, Jean Analysis of PDEs 35B40, 35A08, 35K65, 35Q84, 82C40 This paper is devoted to a fundamental solution of a nonlinear kinetic equation involving a porous medium or fast diffusion operator acting on velocities. Such a nonlinearity has interesting scaling properties, which result in a self-similar behaviour of the fundamental solution. Here fundamental solution means a Dirac distribution initial datum which moreover governs the large time asymptotics of a large class of solutions. Using a self-similar change of variables, the equation becomes a nonlinear kinetic Fokker-Planck equation with harmonic confinement and the intermediate asymptotics regime is transformed into a stability property of a special stationary solution, which attracts the solutions for large times. In the homogeneous case (pure nonlinear diffusion), the problem is reduced to a classical nonlinear diffusion equation with Barenblatt-Pattle self-similar profiles. Unexpectedly, this beautiful structure is preserved at kinetic level, with remarkable consequences for relative entropy estimates, detailed intermediate asymptotics and nonlinear diffusion limits in adapted functional spaces. |
| title | The fundamental solution of a nonlinear kinetic Fokker-Planck equation |
| topic | Analysis of PDEs 35B40, 35A08, 35K65, 35Q84, 82C40 |
| url | https://arxiv.org/abs/2603.26650 |