$L^2$-contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws

Abstract

We consider a $L^2$-contraction of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift. The shift function depends on the time and space variables. It solves a parabolic equation with inhomogeneous coefficients reflecting the perturbation. We consider a suitably small $L^2$-perturbation around a viscous planar shock wave of arbitrarily large strength. However, we do not impose any condition on the anti-derivative variables of the perturbation around shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in the $L^2$ norm, then the $L^2$-contraction holds true for the viscous shock wave up to a shift function which may depend on the temporal and spatial variables. Moreover, as the time $t$ tends to infinity, the $L^2$-contraction holds true up to a time-dependent shift function. In particular, if we choose some special initial perturbation, then we can prove a $L^2$ convergence of the solutions towards the associated shock profile up to a time-dependent shift.