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

TL;DR

This paper develops new regularity and a priori estimates for fractional Fokker-Planck and Hamilton-Jacobi equations with unbounded coefficients, using advanced analytical techniques to establish optimal regularity results.

Contribution

It introduces novel integrability and regularity estimates for fractional PDEs with unbounded ingredients, extending classical results to the fractional setting.

Findings

Established integrability estimates under fractional Aronson-Serrin conditions.

Proved new integral, sup-norm, and Hölder estimates for fractional Hamilton-Jacobi equations.

Derived optimal L^q-regularity results for fractional Hamilton-Jacobi equations.

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.