Contractivity and asymptotics in wasserstein metrics for viscous nonlinear scalar conservation laws

In this work, recent results concerning the long time asymptotics of one-dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-dependent asymptotic profiles for a large class of convection-diffusion problems with fully general nonlinearities and with degenerate diffusion.