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 strength, while the BV-norm or the L∞-norm of the perturbation can be large. Quite different from the previous results, 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-norm, then the L-contraction holds true for the viscous shock wave up to a spatially inhomogeneous shift function. Moreover, as the time t tends to infinity, the L-contraction holds true up to a (spatially homogenous) timedependent shift function. In particular, if we choose some special initial perturbations, then we can prove a L-convergence of the solutions towards the associated shock profile up to a time-dependent shift.