428 research outputs found
Conditional stability of unstable viscous shocks
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
perturbatoins, converging time-asymptotically to a translate of
the unperturbed wave. That is, for a shock with unstable eigenvalues, we
establish conditional stability on a codimension- manifold of initial data,
with sharp rates of decay in all . For , we recover the result of
unconditional stability obtained by Howard, Mascia, and Zumbrun
The refined inviscid stability condition and cellular instability of viscous shock waves
Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and
Texier and Zumbrun, we propose as a mechanism for the onset of cellular
instability of viscous shock and detonation waves in a finite-cross-section
duct the violation of the refined planar stability condition of Zumbrun--Serre,
a viscous correction of the inviscid planar stability condition of Majda. More
precisely, we show for a model problem involving flow in a rectangular duct
with artificial periodic boundary conditions that transition to
multidimensional instability through violation of the refined stability
condition of planar viscous shock waves on the whole space generically implies
for a duct of sufficiently large cross-section a cascade of Hopf bifurcations
involving more and more complicated cellular instabilities.
The refined condition is numerically calculable as described in
Benzoni-Gavage--Serre-Zumbrun
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