GUÈS, Olivier; METIVIER, Guy; WILLIAMS, Mark; ZUMBRUN, Kevin

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GUÈS, Olivier
Laboratoire d'Analyse, Topologie, Probabilités [LATP]

METIVIER, Guy
Institut de Mathématiques de Bordeaux [IMB]

WILLIAMS, Mark
Department of Mathematics [Chapel Hill]

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Archive for Rational Mechanics and Analysis. 2010, vol. 197, p. pp 1 -- 87

Springer Verlag

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For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier-Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in a previous work, is to show that existence and stability of arbitrary amplitude exact boundary-layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an Evans function (uniform Evans stability). Our second is to show that uniform Evans stability for small-amplitude layers is equivalent to Evans stability of the limiting constant layer, which in turn can be checked by a linear-algebraic computation. Finally, for a class of symmetric-dissipative systems including the physical examples mentioned above, we carry out energy estimates showing that constant (and thus small-amplitude) layers always satisfy uniform Evans stability. This yields existence of small-amplitude multi-dimensional boundary layers for the compressible Navier-Stokes and MHD equations. For both equations these appear to be the first such results in the compressible case.< Réduire

Mots clés en anglais

Boundary layers

small viscosity regularization

Navier-Stokes equations

fluid mechanics

magneto-hydrodynamics

Evans functions

stability

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Existence and stability of noncharacteristic boundary-layers for the compressible Navier-Stokes and viscous MHD equations

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