2004/01/30 by Guy Métivier, Métivier, Guy, Kevin Zumbrun +1 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #math.AP
paper · pdf · doi:10.48550/arxiv.math/0401444
arxiv created 2004/01/30 · openalex publication_date 2004/01/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. Our methods apply also to the viscous case.