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 $C^{1,α}$ 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.

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