[Paper Review] Viscous Boundary Value Problems for Symmetric Systems with Variable Multiplicities

Olivier Guès, Guy Métivier+2 more|arXiv (Cornell University)|Jul 3, 2006

Advanced Mathematical Physics Problems16 references4 citations

TL;DR

This paper establishes a framework for analyzing viscous boundary layers and shock waves in multidimensional symmetric hyperbolic systems with variable characteristic multiplicities, extending symmetrizer construction and stability theory beyond the block structure condition. It proves a generalized Zumbrun–Serre–Rousset theorem for variable multiplicities and identifies novel viscous coupling effects that can induce instability, with applications to magnetohydrodynamics (MHD).

ABSTRACT

Extending investigations of Métivier&Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to ``real'' or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the Zumbrun--Serre--Rousset theorem, valid for variable multiplicities, characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system. The new effects of viscosity are seen to be surprisingly subtle; in particular, viscous coupling of crossing hyperbolic modes may induce a destabilizing effect. We illustrate the theory with applications to magnetohydrodynamics.

Motivation & Objective

• To extend spectral stability analysis for viscous boundary layers and shocks beyond systems satisfying the block structure condition.
• To develop a generalized symmetrizer construction in the low-frequency regime for symmetric systems with variable characteristic multiplicities.
• To characterize the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system, valid for variable multiplicities.
• To analyze viscous coupling effects that may destabilize crossing hyperbolic modes.
• To apply the theory to magnetohydrodynamics (MHD), identifying failure of decoupling and instability conditions in MHD shocks.

Proposed method

• Constructs symmetrizers for the low-frequency viscous linearized system using a generalized block structure condition.
• Introduces a reduced viscous system to analyze the $H \to 0$ limit and prove the generalized Zumbrun–Serre–Rousset theorem.
• Applies block reduction techniques to decouple modes and analyze the structure of the symmetrizer in the presence of variable multiplicities.
• Uses Fourier-Laplace analysis and Evans function theory to study spectral stability in the low- and high-frequency regimes.
• Analyzes the viscous coupling matrix $B^\sharp$ in MHD to detect destabilizing effects from non-diagonal entries.
• Applies the theory to MHD by deriving the viscous system, eigenstructure, and jump conditions, and checks decoupling and stability conditions.

Experimental results

Research questions

• RQ1 Can symmetrizer construction be extended to viscous systems with variable characteristic multiplicities, beyond the block structure condition?
• RQ2 What is the role of viscous coupling in destabilizing crossing hyperbolic modes, and how does it affect stability?
• RQ3 How does the limiting hyperbolic system emerge as a nonsingular limit of the reduced viscous system in the low-frequency regime?
• RQ4 Under what conditions does the decoupling condition fail in MHD boundary layers, and what are the consequences?
• RQ5 Do slow Lax shocks in MHD violate the decoupling condition, and does this imply instability?

Key findings

• The decoupling condition fails in MHD at modes where $\xi \cdot v = 0$ and $\nu \neq \rho\mu$, indicating viscous coupling between modes.
• Viscous coupling can induce destabilizing effects, particularly when $B^\sharp_{2,-2} \neq 0$, which occurs when $\mu - \nu/\rho \neq 0$.
• For slow Lax shocks in MHD, the decoupling condition is never satisfied, implying potential instability due to mode coupling.
• The generalized Zumbrun–Serre–Rousset theorem holds for variable multiplicities, characterizing the hyperbolic limit as a nonsingular limit of the reduced viscous system.
• In the $H \to 0$ limit, the system reduces to isentropic Euler equations, and the eigenvalue $\lambda_0$ is totally nonglancing if $u_3 \notin \{-c, 0, c\}$.
• For fast Lax shocks, the assumptions of the main stability theorem are satisfied, indicating potential for stability under the generalized framework.