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

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.

RésuméNous étudions la stabilité des couches limites non-caractéristiques de grande amplitude d'une classe de systèmes hyperboliques–paraboliques comprenant les équations de Navier–Stokes de la dynamique des gaz, et les équations de la magnétohydrodynamique compressible. Nous considérons les cas de conditions au bord rentrantes et sortantes. Notre résultat principal affirme que la stabilité linéaire et la stabilité nonlinéaire sont équivalentes à une condition d'annulation de la fonction d'Evans. Cette condition est facilement vérifiable numériquement, et dans certains cas analytiquement.