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The Schauder estimate in kinetic theory with application to a toy nonlinear model

This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with Hölder 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.