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

Abstract

We consider a L-2-contraction (a L-2-type stability) of large viscous shock waves for the multidimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in L-2-norm, then the L-2-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BY-norm or the L-infinity-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time t tends to infinity, the L-2-contraction holds true up to a (spatially homogeneous) time-dependent shift function. In particular, if we choose some special initial perturbations, then L-2-convergence of the solutions towards the associated shock profile can be proved up to a time-dependent shift.