We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the
This content will become publicly available on April 1, 2025
Hölder regularity of the Boltzmann equation past an obstacle
Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].
more » « less- Award ID(s):
- 2047681
- NSF-PAR ID:
- 10538947
- Publisher / Repository:
- John Wiley & Sons
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 77
- Issue:
- 4
- ISSN:
- 0010-3640
- Page Range / eLocation ID:
- 2331 to 2386
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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