Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case

We study the speed of convergence in $L^\infty$ norm of the vanishing viscosity process for Hamilton-Jacobi equations with uniformly or strictly convex Hamiltonian terms with superquadratic behavior. Our analysis boosts previous findings on the rate of convergence for this procedure in $L^p$ norms, showing rates in sup-norm of order $\mathcal{O}(\eps^\beta)$, $\beta\in(1/2,1)$, or $\mathcal{O}(\eps|\log\eps|)$ with respect to the vanishing viscosity parameter $\eps$, depending on the regularity of the initial datum of the problem and convexity properties of the Hamiltonian. Our proofs are based on integral methods and avoid the use of techniques based on stochastic control or the maximum principle.