$L^2$-type contraction for systems of conservation laws

L2-contraction for Shock Waves of Scalar Viscous Conservation Laws

We consider the L-contraction up to a shift for viscous shocks of scalar viscous conservation laws with strictly convex fluxes in one space dimension. In the case of a flux which is a small perturbation of the quadratic burgers flux, we show that any viscous shock induces a contraction in L, up to a shift. That is, the L norm of the difference of any solution of the viscous conservation law, wi...

L-type contraction for systems of conservation laws

The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in L1. However it is not a contraction in Lp for any p > 1. Leger showed in [18] that for a convex flux, it is however a contraction in L2 up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy m...

L2-contraction of Large Planar Shock Waves for Multi-dimensional Scalar Viscous Conservation Laws

We consider a L-contraction (a L-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift. The shift function, depending both on the time and space variables, solves a viscous HamiltonJacobi equation with source terms. We consider a suitably small L-perturbation around a viscous planar shock wave with arbitrarily large stren...

L2 formulation of multidimensional scalar conservation laws

Numerical Integrations of Systems of Conservation Laws of Mixed Type

The systems of conservation laws have been used to model dynamical phase transitions in, for example, the propagating phase boundaries in solids and the van der Waals uid. When integrating such mixed hyperbolic-elliptic systems the Lax-Friedrichs scheme is known to give the correct solutions selected by a viscosity-capillarity criterion except a spike at the phase boundary which does not go awa...