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 sin (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-regularity for fractional Hamilton–Jacobi equations.
Highlights
In this paper, we analyze the regularity properties of transport equations of Fokker–Planck-type and Hamilton–Jacobi equations with fractional diffusion driven by a fractional power of the Laplacian, (−)s, with subcritical order s ∈ ( 1).In particular, we address well-posedness, parabolic Bessel regularity, and integrability estimates for solutions to fractional Fokker–Planck equations of the form−∂t ρ(x, t) + (− )sρ(x, t) + div(b(x, t) ρ(x, t)) = 0 in Qτ := Td × (0, τ ), ρ(x, τ ) = ρτ (x) in Td, (1)where the nonlocal diffusion operator is defined on the flat torus Td ≡ Rd \Zd [83], under “rough” integrability conditions on the velocity field, mainly when either b ∈ L
These estimates are fundamental to study the regularity properties for PDEs arising in Mean Field Games, cf [37,41,81,82], see [23] for the time-fractional framework
Appendix A collects some properties for advection equations with fractional diffusion
Summary
We analyze the regularity properties of transport equations of Fokker–. This bound is obtained by duality, following [37,38,77], via a maximal regularity estimate for nonlocal equations with divergence-type terms of the form σ bρ σ + ρτ p These estimates are fundamental to study the regularity properties for PDEs arising in Mean Field Games, cf [37,41,81,82], see [23] for the time-fractional framework. Integral and sup-norm estimates for the parabolic problem have been already addressed in the paper [38] for the viscous case s = 1, and we recover those results by letting s → 1. Appendix A collects some properties for advection equations with fractional diffusion
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