Asymptotic Behavior of Multidimensional Scalar Relaxation Shocks

TL;DR

This paper analyzes the stability and decay rates of multidimensional scalar relaxation shocks, providing pointwise Green function bounds and nonlinear asymptotic behavior under spectral stability assumptions.

Contribution

It establishes pointwise Green function bounds and nonlinear decay rates for multidimensional relaxation shocks with scalar equilibrium models, extending stability analysis.

Findings

Linearized stability established for multidimensional relaxation shocks.

Sharp decay rates for perturbed weak shocks in general relaxation systems.

Pointwise bounds for the Green function in multidimensional settings.

Abstract

We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear $L^{2}$ asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously symmetrizable relaxation systems, under small $L^{1}\cap H^{[d/2]+3}$ perturbations with first moment in the normal direction to the front.