Stability of multidimensional viscous shocks for symmetric systems with variable multiplicities

Abstract

We establish long-time stability of multidimensional 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 Zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity, yielding the first such stability result for (fast) MHD shocks. At the same time, we are able to drop a technical assumption on the 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 an $L^1\to L^p$ resolvent bound in low-frequency regimes by employing the recent construction of degenerate Kreiss symmetrizers by Guès, Métivier, Williams, and Zumbrun. Thus, at the low-frequency resolvent bound level, our analysis gives an alternative to the earlier pointwise Green-function approach of Zumbrun. High-frequency solution operator bounds have been previously established entirely by nonlinear energy estimates