Pointwise Greenʹs Function Bounds for Multidimensional Scalar Viscous Shock Fronts

We determine detailed pointwise bounds for the Greenʹs function of the linearized operator about a multidimensional scalar viscous shock front. These extend the pointwise semigroup methods introduced by Howard and Zumbrun in the one-dimensional case to multidimensions, sharpening Lp estimates obtained by Goodman and Miller using a weighted norm approach. Moreover, our results apply to shocks of arbitrary strength, as previous results did not. As described in a companion paper, the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear Lp-asymptotic behavior of the front under small perturbation. The analysis of the multidimensional case involves several new features not found in the one-dimensional case, concerned with the geometry of propagating signals.