2005/08/22 by Olivier Guès, Olivier Gues, Gues, Olivier +6
Engineering · Mathematics · #35L50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #msc:35L50
paper · pdf · doi:10.48550/arxiv.math/0508376
15pp
arxiv created 2005/08/22 · openalex publication_date 2005/08/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Answering a question left open in \citeMZ2, 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 L2 well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly L2 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.