Uniform stability estimates for constant-coefficient symmetric hyperbolic boundary value problems

Abstract

Answering a question left open in \cite{MZ2}, we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong $L^2$ well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly $L^2$ well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the constant-coefficient case; at least, it does not translate in an obvious way to the variable-coefficient case. The result in the hyperbolic case is derived from a more general principle that can be applied, for example, to parabolic or partially parabolic problems like the Navier-Stokes or viscous MHD equations linearized about a constant state or even a viscous shock.