Math @ Duke
Publications [#9830] of Andrea L Bertozzi
- Andrea Bertozzi, Andreas Munch, Michael Shearer, and Kevin Zumbrun, Stability of compressive and undercompressive thin film travelling waves,
European J. of Appl. Math., 12(3), pp.253-291, 2001
(last updated on 2001/08/15)
Recent studies of liquid films driven by competing forces
due to surface tension gradients and gravity reveal
that undercompressive traveling waves play an important role
dynamics when the competing forces are comparable.
In this paper we provide a theoretical framework for
assessing the spectral
stability of compressive and undercompressive traveling
in thin film models.
Associated with the linear stability problem is an Evans
which vanishes precisely at eigenvalues of the linearized
The structure of an index related to the Evans function
results for stability of compressive waves.
A new formula for the index in the undercompressive case
yields results consistent with stability.
In considering stability of
undercompressive waves to transverse perturbations,
there is an apparent
inconsistency between long-wave asymptotics of the largest
and its actual behavior. We show that this paradox
is due to the unusual structure of the eigenfunctions
and we construct a revised long-wave asymptotics.
We conclude with numerical computations of the largest
comparisons with the asymptotic results,
and several open problems associated with our findings.
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