Seminár z kvalitatívnej teórie diferenciálnych rovníc - Siniša Slijepčević (28.4.2016)
vo štvrtok 28.4.2016 o 14:00 hod. v miestnosti M/223
Od: Pavol Quittner
Prednášajúci: Siniša Slijepčević (University of Zagreb)
Názov: Scalar semilinear parabolic equations on unbounded domains, phase transitions and metric entropy of twist maps
Termín: 28.4.2016, 14:00, M/223
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.