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

Research paper by Luis Silvestre

Indexed on: 08 Sep '10Published on: 08 Sep '10Published in: Mathematics - Analysis of PDEs

#### 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.