Transport equations with nonlocal diffusion and applications to 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 $$s\in (1/2,1)$$ s ∈ ( 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 Hölder 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 Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

A priori Lipschitz estimates for nonlinear equations with mixed local and nonlocal diffusion via the adjoint-Bernstein method

“…With the aid of Proposition 5.3 we first prove the following sup-norm estimate for solutions to (4). This slightly extends [16,Proposition 3.7] and [28,Theorem 2.3] to problems with mixed diffusion.…”

“…Lipschitz), so that an estimate on ∂ t u readily follows by the maximum principle. We finally emphasize that a regularity estimate starting from a continuous initial datum and suitable weak solutions can be obtained working at the level of difference quotients, as already done first in [16] for Lipschitz regularity and then in [17,28] for Hölder regularization properties.…”

“…x,t as in [28], together with the embeddings of (trace) fractional Sobolev spaces. Maximal regularity for the heat equation with mixed diffusion in Lebesgue spaces holds by Lemma 3.3.…”

“…and use the estimate in Corollary 5.9 of [28] combined with the Young's inequality to conclude the gradient bound. We emphasize that an assumption like f ∈ L q (0, T ; W 1,q (T N )) lies in between those assumed for the use of the weak Bernstein method and the ones to run the Ishii-Lions argument.…”

“…The drawback of this approach is the requirement on γ (that must be of subquadratic growth) and on the initial data, which needs to be more regular respect to Theorem 2.1. The same idea works even for problems driven by the sole fractional Laplacian using interpolation estimates and the L ∞ bounds in Theorem 2.3 of [28], provided that γ < 2s, s ∈ (1/2, 1). Still, the same approach of this manuscript can be refined to obtain maximal regularity estimates via new Hölder and L p bounds for mixed diffusion problems, see [17].…”