Regularity for parabolic integro-differential equations with very irregular kernels

Abstract

We prove Hölder regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof which avoids the use of a convex envelop as well as give a new covering argument which is better suited to the fractional order setting. Our main result involves a class of kernels which may contain a singular measure, may vanish at some points, and are not required to be symmetric. This new generality of integro-differential operators opens the door to further applications of the theory, including some regularization estimates for the Boltzmann equation.

References in corpus (5)

• Regularity theory for fully nonlinear integro-differential equations
• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• Regularity results for nonlocal parabolic equations
• Regularity for fully non linear equations with non local drift
• Intrinsic scaling properties for nonlocal operators II

Cited by in corpus (14)

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• Intrinsic scaling properties for nonlocal operators II
• Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts
• Global in Time Classical Solutions to the 3D Quasi-geostrophic System for Large Initial Data
• Fractional Fokker-Planck Equation with General Confinement Force
• An $\alpha$-stable limit theorem under sublinear expectation
• Min-max formulas for nonlocal elliptic operators on Euclidean space
• From the free boundary condition for Hele-Shaw to a fractional parabolic equation
• Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion
• Dini estimates for nonlocal fully nonlinear elliptic equations