The Schauder estimate for kinetic integral equations

Abstract

We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. We study equations of the form $f_t+v\cdot {\nabla }_{x}f={\mathcal{L}}_{v}f+c$, where ${\mathcal{L}}_v$ is an integro-differential diffusion operator of order $2s$ acting in the $v$-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of ${\mathcal{L}}_v$, we obtain an a priori estimate for $f$ in a properly scaled Hölder space.

There was a silly typo in equation (1.3) from the first version that could cause confusion. This was our only correction in the second version

References in corpus (2)

• Schauder estimates for solutions of linear parabolic integro-differential equations
• Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts

Cited by in corpus (4)

• Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in nondivergence form
• Optimal Schauder estimates for kinetic Kolmogorov equations with time measurable coefficients
• Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations
• On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics