Calculus of Variations and Geometric Measure Theory
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A. Goffi

Transport equations with nonlocal diffusion and applications to Hamilton-Jacobi equations

created by goffi on 02 Jan 2021
modified on 16 Jul 2021

[BibTeX]

Accepted Paper

Inserted: 2 jan 2021
Last Updated: 16 jul 2021

Journal: Journal of Evolution Equations
Year: 2020
Doi: 10.1007/s00028-021-00720-3

ArXiv: 2101.00615 PDF

Abstract:

We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish integrability estimates under a fractional version of the Aronson-Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans' nonlinear adjoint method we prove new integral, sup-norm and H\"older estimates for weak and strong solutions to fractional Hamilton-Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calder\'on-Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo-Nirenberg inequalities, we deduce optimal $L^q$-regularity for fractional Hamilton-Jacobi equations.

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