Hyperbolic boundary value problems for symmetric systems with variable multiplicities

Abstract

We extend the Kreiss-Majda theory of stability of hyperbolic initial-boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does not hold: namely, simultaneously symmetrizable systems with characteristics of variable multiplicity, satisfying at points of variable multiplicity either a ''totally nonglancing'' or a ''nonglancing and linearly splitting'' condition. At the same time, we give a simple characterization of the block structure condition as ''geometric regularity'' of characteristics, defined as analyticity of associated eigenprojections. The totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics, and likewise for our main physical examples of MHD or Maxwell equations for a crystal. Together with previous analyses of spectral stability carried out by Gardner-Kruskal and Blokhin-Trakhinin, this yields immediately a number of new results of nonlinear inviscid stability of shock waves in MHD in the cases of parallel or transverse magnetic field, and recovers the sole previous nonlinear result, obtained by Blokhin-Trakhinin by direct ''dissipative integral'' methods, of stability in the zero-magnetic field limit. We also discuss extensions to the viscous case. r 2004 Published by Elsevier Inc.

Lecture notes : Stability of Noncharacteristic Viscous Boundary Layers

“…The reader is refereed to [MéZu2] and [GMWZ6] for complete proofs of the results quoted in this section.…”

“…By symmetry, −τ is a semi-simple eigenvalue of A(p,ξ), say of multiplicity m. In [MéZu2], it is proved that the assumption on E − k implies that the multiplicity of µ k as an eigenvalue of H 0 (p,ζ) =Ȟ(p,ζ, 0) is equal to m. Denote by V k a N × m matrix the columns of which form a basis of E k , so that…”

“…For the inviscid MHD, the multiple eigenvalues are algebraically regular, but some are not geometrically regular (see [MéZu2]). …”

“…Recall from [MéZu2] the following definition. If τ is a root of multiplicity m of the polynomial ∆(p, ·, ξ), then by hyperbolicity, the Taylor expansion of ∆ at (p, τ , ξ) at the order m − 1 vanishes so that …”

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

“…In particular, (H2) may be weakened to allow the case of nonstrictly hyperbolic but constant multiplicity systems with stable viscosity matrices treated in [MZ2]; extensions to the variablemultiplicity case are discussed in [MZ2,GMWZ4]. Also, (H3) may be weakened to allow overcompressive shocks as in [Z1]; see Remark 10.1 below.…”