Mixing and Mixtures in Geo- and Biophysical Flows: A Focus on Mathematical Theory and Numerical Methods

On contraction of large perturbation of shock waves, and inviscid limit problems

Moon-Jin Kang

University of Texas at Austin

This talk will start with the relative entropy method to handle the contrac- tion of possibly large perturbations around viscous shock waves of conservation laws. In the case of viscous scalar conservation law in one space dimension, we obtain L2-contraction for any large perturbations of shocks up to a Lipschitz shift depending on time. Such a time-dependent Lipschitz shift should be con- structed from dynamics of the perturbation. In the case of multidimensional scalar conservation law, the perturbations of planar shocks are L2-contractive up to a more complicated shift depending on both time and space variable, which solves a parabolic equation with inhomogeneous coefficient and force terms re- flecting the perturbation. As a consequence, the L2-contraction property im- plies the inviscid limit towards inviscid shock waves. At the end, we handle the contraction properties of admissible discontinuities of the hyperbolic system of conservation laws equipped with a strictly convex entropy.