On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries. (English) Zbl 1404.35446
Summary: In this paper, we compute asymptotics of solutions of the kinetic Fokker-Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if \( r < r_c\). The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point \((x,v)=(0,0)\). In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper [“Nonuniqueness for the kinetic-Fokker-Planck equation with inelastic boundary conditions”, Preprint, arXiv:1509.03366] to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.
MSC:
| 35Q84 | Fokker-Planck equations |
| 35K65 | Degenerate parabolic equations |
| 35A20 | Analyticity in context of PDEs |
| 35Q70 | PDEs in connection with mechanics of particles and systems of particles |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35R06 | PDEs with measure |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 47D07 | Markov semigroups and applications to diffusion processes |
Keywords:
Fokker-Planck equation; nonuniqueness of solutions; measure-valued solutions; inelastic boundary condition; singular setReferences:
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