TY - JOUR
T1 - Stability of multidimensional viscous shocks for symmetric systems with variable multiplicities
AU - Nguyen, Toan
PY - 2009/12
Y1 - 2009/12
N2 - 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 ≥ 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 L1 → Lp 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.
AB - 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 ≥ 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 L1 → Lp 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.
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U2 - 10.1215/00127094-2009-060
DO - 10.1215/00127094-2009-060
M3 - Article
AN - SCOPUS:77953521608
SN - 0012-7094
VL - 150
SP - 577
EP - 614
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 3
ER -