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Seminar 28.4.2016: Sinisa Slijepcevic

PostPosted: Mon Apr 18, 2016 1:40 pm
by quittner
Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 28.4.2016 at 14:00 Lecture room M-223

Siniša Slijepčević (University of Zagreb):
Scalar semilinear parabolic equations on unbounded domains,
phase transitions and metric entropy of twist maps

The dynamics of semilinear parabolic equations on bounded domains is one of
the, in the words of J. Hale, "success stories" of the theory of dynamical
systems in infinite dimensions. Several tools, such as the zero (or lap)
number, the Poincaré-Bendixson theorem, etc., however, do not seem to
extend to unbounded domains (except in very special cases).

We claim that this is not the case, if one considers typical, rather than
absolute properties of solutions and asymptotics, where "typical" is with
respect to any translationally invariant measure on an appropriate phase
space of functions. We thus define a zero-number function (on the space of
measures), and prove an ergodic Poincaré-Bendixson theorem for
semilinear parabolic equations on unbounded domains, describing "typical"
asymptotics on unbounded domains.

We then use these tools to address two closely related problems. The first
one is to mathematically rigorously describe phase transitions observed by
physicists in such and related equations. The second one is a possible
approach to address the metric entropy conjecture for twist maps in
Hamiltonian dynamics. (The stationary solutions of a given semilinear
parabolic differential equation with gradient structure correspond to orbits
of an area-preserving twist map).

Any further progress in extending tools and results from bounded to
unbounded domains would potentially contribute to better understanding of
both of these problems.