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Nguyen, Toan

Stability of multidimensional viscous shocks for symmetric systems with variable multiplicities. (English) Zbl 1197.35045

Duke Math. J. 150, No. 3, 577-614 (2009).
The author establishes 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 more than one. The obtained result 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, the author drops the technical assumption on the structure of the so-called glancing set that was used in the previous investigations. The key idea for the improvements is to use a new simple argument for obtaining a bound for the \(L^1\to L^p\) resolvent in the low-frequency regimes by employing the recent construction of the degenerate Kreiss symmetrizers by Gues, Metivier, 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 previously been established entirely by nonlinear energy estimates.
Reviewer: Michael I. Gil’ (Beer-Sheva)

Cited in 4 Documents

MSC:

35B35 Stability in context of PDEs
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L67 Shocks and singularities for hyperbolic equations
76W05 Magnetohydrodynamics and electrohydrodynamics

Keywords:

hyperbolic-parabolic systems; glancing set; low-frequency resolvent bound
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\textit{T. Nguyen}, Duke Math. J. 150, No. 3, 577--614 (2009; Zbl 1197.35045)
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References:

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