Abstract
This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with H\older continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker-Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.
Highlights
We conclude the introduction by mentioning that the well-posedness result for the toy nonlinear model can be improved in two directions
Given an open set Q ⊂ R × Rd × Rd and β > 0, we say that a function g : Q → R
We investigate here how to recover the fact that a given function g lies in C2+α only knowing that its free transport and its velocity second order derivatives lie in Cα
Summary
— The left hand side can be understood as a Hölder regularity of (kinetic) order 2 + α, according the specific definition of Hölder spaces Cβ, β 0, given (Definition 2.2). We compare this result to the classical. The known a priori estimates that are preserved in time for this equation are L1(Td × Rd) and C1μ f C2μ, where μ denotes the Gaussian (2π)−d/2e−|v|2/2 They are not sufficient to derive uniqueness or bootstrap higher regularity. The Schauder estimate from Theorem 1.1, together with the Hölder regularity from [GIMV19] (see Theorem 4.3), allows us to prove global well-posedness of Eq (1.4) in Sobolev spaces.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
