Nonlinear asymptotic stability in $L^{\mathrm{\infty }}$ for Lipschitz solutions to scalar conservation laws

In this note, we show nonlinear stability in $L^{\mathrm{\infty }}$ for Lipschitz solutions to genuinely nonlinear, multi-dimensional scalar conservation laws. As an application, we are able to compute explicit algebraic decay rates of the $L^{\mathrm{\infty }}$ norm of perturbations of global-in-time Lipschitz solutions, including perturbations of planar rarefaction waves. Our analysis uses the De Giorgi method applied to the kinetic formulation and is an extension of the method introduced recently by Silvestre in [Comm. Pure Appl. Math., 72(6):1321-1348, 2019].