Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD

Abstract

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$. For $p=0$, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.

32pp

References in corpus (5)

• Stable manifolds for an orbitally unstable NLS
• Nonlinear stability of time-periodic viscous shocks
• Center stable manifolds for quasilinear parabolic pde and conditional stability of nonclassical viscous shock waves
• Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions
• Conditional stability of unstable viscous shocks

Cited by in corpus (6)

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• Majda and ZND models for detonation: nonlinear stability vs. formation of singularities
• Ion-acoustic shock in a collisional plasma