Stability of shocks among inviscid limits of Navier-Stokes equation

We study the stability of a 1D shock for the p-system (the 1D isentropic Euler equation in the Lagrangian space). We show that such a shock is stable with respect to initial values perturbations in the energy space, in the class of inviscid limits of Navier-stokes equation.

The result is based on the theory of a-contraction with shifts for viscous shocks of Navier-Stokes equation. The method allows to show the uniform stability with respect to the viscosity. Stability results on the inviscid model are then inherited at the inviscid limit, thanks to the fact that large perturbations, independent of the viscosity, can be considered at the Navier-Stokes level. These stability results hold in the class of wild perturbations of inviscid limits, without any regularity restriction (non even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.

This is a joint work with Moon-Jin Kang.