Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities

TL;DR

This paper proves the long-time stability of multi-dimensional viscous shocks in symmetric hyperbolic-parabolic systems with variable multiplicities, including MHD equations, by introducing new resolvent bounds and removing previous technical assumptions.

Contribution

It extends stability results to systems with variable multiplicities and includes MHD shocks, using novel resolvent bounds and simplifying previous technical conditions.

Findings

Established stability of multi-dimensional viscous shocks with variable multiplicities.

Introduced a new $L^1\to L^p$ resolvent bound approach for low-frequency regimes.

Removed technical assumptions on the glancing set structure.

Abstract

We establish long-time stability of multi-dimensional viscous shocks of a general class of symmetric hyperbolic--parabolic systems with variable multiplicities, notably including the equations of compressible magnetohydrodynamics (MHD) in dimensions $d\ge 2$. This extends the existing result established by K. Zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity, yielding the first such a stability result for (fast) MHD shocks. At the same time, we are able to drop a technical assumption on structure of the so--called glancing set that was necessarily used in previous analyses. The key idea to the improvements is to introduce a new simple argument for obtaining a $L^1\to L^p$ resolvent bound in low--frequency regimes by employing the recent construction of degenerate Kreiss' symmetrizers by O. Gu\`es, G. M\'etivier, M. Williams, and K. Zumbrun. Thus, at the low-frequency resolvent bound level, our analysis gives an alternative to the earlier pointwise Green's function approach of K. Zumbrun. High--frequency solution operator bounds have been previously established entirely by nonlinear energy estimates.