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

摘要

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.