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

Abstrakt:

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.