Regularity for parabolic integro-differential equations with very irregular kernels

By Russell W. Schwab, Luis Silvestre. 2014

101 citations recorded by the source corpus.

We prove H\"older 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d92badfe80913be66e6c69c7a8323e1840dda43

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Nonlocal Diffusion and Applications (2015, background): “Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here, the fully nonlinear or singular/degenerate equations are not taken into account, only very few applications are di”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “Popular models that are being investigated are integro-differential operators with irregular or rough kernels, as in [213], where the form is”
Nonlocal elliptic equations in bounded domains: a survey (2015, unknown): “• Interior regularity of solutions: Bass–Levin [6], Kassmann [33], Caffarelli–Silvestre [13, 14, 15], Barles–Chasseigne–Imbert [4], Kassmann–Mimica [34, 35], Schwab–Silvestre [53], and Serra [54].”
Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential (2014, unknown): “After the submission of this article, the authors learned of related work of Silvestre [2016] on the Boltzmann equation, covering the spatially inhomogeneous case.”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, methodology): “The assumptions in [56] are sufficiently general to be applicable to the space homogeneous Boltzmann equation.”