Preprint No.
A-00-10
Stefan Liebscher
Stable, oscillatory viscous profiles of weak, non-Lax shocks in
systems of stiff balance laws
Abstract:
This thesis is devoted to a phenomenon in hyperbolic balance laws,
which is similar in spirit to the Turing instability.
The combination of two individually stabilising effects can lead
to quite rich dynamical behaviour, like instabilities,
oscillations, or pattern formation.
Our problem is composed of two ingredients.
As a first part we have a strictly hyperbolic
conservation law which has rarefaction waves and shocks with
monotone viscous profiles as elementary solutions.
The second part is a source term which, alone, would describe a simple,
stable kinetic behaviour: all trajectories end by converging
monotonically to an equilibrium.
The balance law, constructed of these two parts, however,
can support viscous shock profiles with oscillatory tails.
They emerge from a Hopf-like bifurcation point that belongs
to a curve of equilibria of the associated travelling-wave system.
The linearised flow at the rest points along this curve possesses a pair
of conjugate complex eigenvalues which crosses the imaginary axis
at the Hopf-point.
The nature of the oscillatory shocks as well as their
stability properties are the subjects of this thesis.
Our main result establishes
convective stability of oscillatory viscous profiles to
weak shocks with extreme speed: if the speed of the wave
exceeds any characteristic speed, then the profile is linearly stable in
a suitable exponentially weighted space.
For intermediate speeds, the profiles are absolutely unstable.
Methods of two areas, hyperbolic conservation laws and dynamical systems,
are combined in this work. This makes it necessary to give a brief
review of basic facts and methods of both fields.
Keywords: Hyperbolic balance laws, viscous profiles,
stable oscillatory shocks
Mathematics Subject Classification (MSC91): 35L67, 34C23, 34C37
Language: ENG
Available: Pr-A-00-10.ps (2.5MB),
Pr-A-00-10.ps.gz (0.7MB)
Contact: Stefan Liebscher, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (liebsch@math.fu-berlin.de)
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