Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic-parabolic systems

Toan Nguyen, Kevin Zumbrun

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems including the Navier-Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundary conditions, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ ≥ 1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive ("shock-type") layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.

Original languageEnglish (US)
Pages (from-to)547-598
Number of pages52
JournalJournal des Mathematiques Pures et Appliquees
Volume92
Issue number6
DOIs
StatePublished - Dec 1 2009

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Parabolic Systems
Hyperbolic Systems
Boundary Layer
Boundary layers
Nonlinear Stability
Strictly
Evans Function
Spectral Stability
Time and motion study
Ideal Gas
Gas dynamics
Gas Dynamics
Linear Stability
Magnetohydrodynamics
Asymptotic stability
Green's function
Stability Condition
Asymptotic Stability
Navier Stokes equations
Shock

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems including the Navier-Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundary conditions, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ ≥ 1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive ({"}shock-type{"}) layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.",
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Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic-parabolic systems. / Nguyen, Toan; Zumbrun, Kevin.

In: Journal des Mathematiques Pures et Appliquees, Vol. 92, No. 6, 01.12.2009, p. 547-598.

Research output: Contribution to journalArticle

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