On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion

TL;DR

This paper proves the existence of classical solutions with certain regularity for the Hamilton-Jacobi equation involving critical fractional diffusion, using new Hölder estimates for related advection-diffusion equations.

Contribution

It introduces a novel Hölder estimate technique to establish $C^{1,eta}$ regularity for solutions of Hamilton-Jacobi equations with critical fractional diffusion.

Findings

Existence of classical $C^{1,eta}$ solutions for the Hamilton-Jacobi equation with fractional diffusion.

Development of a new Hölder estimate for advection-diffusion equations with bounded vector fields.

Extension of regularity results to non-convex Hamiltonians.

Abstract

We prove that the Hamilton Jacobi equation for an arbitrary Hamiltonian $H$ (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical $C^{1,\alpha}$ solutions. The proof is achieved using a new H\"older estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free.