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

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 C1,α solutions. The proof is achieved using a new Hölder estimate for solutions of advection–diffusion equations of order one with bounded vector fields that are not necessarily divergence free.