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Existence and Stability of Noncharacteristic Boundary Layers for the Compressible Navier–Stokes and Viscous MHD Equations

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Abstract

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 the companion work (Gues et al. in J Differ Equ, 244, 309–387 (2008)), 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). Whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with “residual” boundary conditions. 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.

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Author information

Authors and Affiliations

  1. LATP, Université de Provence, Marseille, France

    Olivier Guès

  2. MAB, Université de Bordeaux I, Talence, France

    Guy Métivier

  3. Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA

    Mark Williams

  4. Department of Mathematics, Indiana University, Bloomington, IN, USA

    Kevin Zumbrun

Authors
  1. Olivier Guès
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  2. Guy Métivier
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  3. Mark Williams
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  4. Kevin Zumbrun
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Corresponding author

Correspondence to Mark Williams.

Additional information

Communicated by C. M. Dafermos

Research of O. Gues was partially supported by European network HYKE, HPRN-CT-2002-00282. Research of G. Métivier was partially supported by European network HYKE, HPRN-CT-2002-00282. Research of M. Williams was partially supported by NSF grants number DMS-0070684 and DMS-0401252. K. Zumbrun thanks the Universities of Bordeaux I and Provence for their hospitality during visits in which this work was partially carried out. Research of K. Zumbrun was partially supported by NSF grants number DMS-0070765 and DMS-0300487.

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Guès, O., Métivier, G., Williams, M. et al. Existence and Stability of Noncharacteristic Boundary Layers for the Compressible Navier–Stokes and Viscous MHD Equations. Arch Rational Mech Anal 197, 1–87 (2010). https://doi.org/10.1007/s00205-009-0277-y

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