686 research outputs found
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
Metastability of solitary roll wave solutions of the St. Venant equations with viscosity
We study by a combination of numerical and analytical Evans function
techniques the stability of solitary wave solutions of the St. Venant equations
for viscous shallow-water flow down an incline, and related models. Our main
result is to exhibit examples of metastable solitary waves for the St. Venant
equations, with stable point spectrum indicating coherence of the wave profile
but unstable essential spectrum indicating oscillatory convective instabilities
shed in its wake. We propose a mechanism based on ``dynamic spectrum'' of the
wave profile, by which a wave train of solitary pulses can stabilize each other
by de-amplification of convective instabilities as they pass through successive
waves. We present numerical time evolution studies supporting these
conclusions, which bear also on the possibility of stable periodic solutions
close to the homoclinic. For the closely related viscous Jin-Xin model, by
contrast, for which the essential spectrum is stable, we show using the
stability index of Gardner--Zumbrun that solitary wave pulses are always
exponentially unstable, possessing point spectra with positive real part.Comment: 42 pages, 9 figure
Nonlinear stability of viscous roll waves
Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic
conservation laws, we show that spectral stability implies nonlinear stability
for spatially periodic viscous roll wave solutions of the one-dimensional St.
Venant equations for shallow water flow down an inclined ramp. The main new
issues to be overcome are incomplete parabolicity and the nonconservative form
of the equations, which leads to undifferentiated quadratic source terms that
cannot be handled using the estimates of the conservative case. The first is
resolved by treating the equations in the more favorable Lagrangian
coordinates, for which one can obtain large-amplitude nonlinear damping
estimates similar to those carried out by Mascia--Zumbrun in the related shock
wave case, assuming only symmetrizability of the hyperbolic part. The second is
resolved by the observation that, similarly as in the relaxation and detonation
cases, sources occurring in nonconservative components experience greater than
expected decay, comparable to that experienced by a differentiated source.Comment: Revision includes new appendix containing full proof of nonlinear
damping estimate. Minor mathematical typos fixed throughout, and more
complete connection to Whitham averaged system added. 42 page
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
- …