Conditional stability of unstable viscous shocks

Kevin Zumbrun

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arXiv · 0811.1193

Conditional stability of unstable viscous shocks

Kevin Zumbrun

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Abstract

Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^2$ perturbatoins, 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 Howard, Mascia, and Zumbrun.

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Kevin Zumbrun. 2008-11-07. Conditional stability of unstable viscous shocks. https://arxiv.org/abs/0811.1193

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