Asymptotic behavior of multidimensional scalar viscous shock fronts

Abstract

Making use of detailed pointwise Green's function bounds obtained in a previous work for the linearized equations about the wave, we give a straightforward derivation of the (nonlinear) Lp -asymptotic behavior of a scalar (planar) viscous shock front under perturbations in L1 ∩ L∞ with first moment in the normal direction to the front, in all dimensions d ≥ 2. For dimension d ≥ 3, we establish sharp Lp decay rates by a much simpler argument using only Lp information on the Green's function, for perturbations merely in L1 ∩ L∞. These results simplify and greatly extend previous results of Goodman-Miller and Goodman, respectively, which were obtained under assumptions of weak shock strength and artificial (identity) viscosity, and, in the case of asymptotic behavior, exponential decay of perturbations in the direction normal to the shock front. For perturbations localized as (1 + lx1l)-1 in the normal direction, but not possessing a first moment, we give a refined picture of the linearized Lp -asymptotic behavior different from the near-field approximation of Goodman and Miller.