Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations

Abstract

We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form ${\partial }_{t}u+H(x,t,Du,D^{2}u)=0$ in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $\epsilon F(x,t,D^{2}u)$. Assuming that $u\in C_x^{\alpha }$, $u_0\in C^{\eta }$, $H\in C_x^{\beta }$ and having power growth $\gamma$ in the gradient entry, we establish a convergence rate of order ${\epsilon }^{\min \left\{\frac{\eta }{2},\frac{\beta +\gamma (\alpha -1)}{\beta +\gamma (\alpha -1)+2-\alpha }\right\}}$. Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on $H$. The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with Hölder coefficients.