Abstract
In this paper, we investigate the regularizing effect of a non-local operator on first order Hamilton-Jacobi equations. We prove that there exists a unique solution that is $C^2$ in space and $C^1$ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a $W^{1,\infty}$ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of $C^\infty$ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in $L^\infty$ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.