Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln
Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln
Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln
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Juliette Hell<br />
Freie <strong>Universität</strong> Berlin<br />
Dynamics at Infinity<br />
DMV Tagung 2011 - <strong>Köln</strong>, 19. - 22. September<br />
We interpret blow-up phenomena as heteroclinic orbits connecting a bounded invariant set to infinity. The<br />
question addressed is: Which parts of the bounded dynamics give birth to exploding trajectories? Via<br />
Poincaré compactification, infinity is made accessible to topological methods of detection of heteroclinics,<br />
such as Conley index theory. Furthermore the artificial dynamics in the sphere at infinity provides information<br />
on the shape of the blow-up. This talk emphasizes applications of compactification methods to partial<br />
differential equations whose leading or<strong>der</strong> is a homogenous polynomial of degree d. In particular we will<br />
focus on slowly non-dissipative reaction-diffusion equations (joined work with Nitsan Ben-Gal).<br />
Literatur<br />
Nitsan Ben-Gal. Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs Ph.D. Thesis,<br />
Brown University, 2010,<br />
Juliette Hell. Conley Index at Infinity Doktorarbeit, Freie <strong>Universität</strong> Berlin, 2010<br />
Juliette Hell. Conley Index at Infinity. Preprint, arXiv:1103.5335v1<br />
Sebastian Herr<br />
<strong>Universität</strong> Bonn<br />
Nonlinear dispersive equations in critical spaces<br />
I will review harmonic analysis methods for solving the initial value problem associated to nonlinear<br />
dispersive evolution equations. For certain nonlinear Schrödinger equations and other interesting models<br />
such as the KP-II equation, the Zakharov system or pseudo-relativistic Hartree equations I am going<br />
to report on recent progress concerning the well-posedness problem in (approximately) scale-invariant<br />
Sobolev spaces.<br />
Vu Hoang<br />
GRK 1294/<strong>Institut</strong> für Analysis, Karlsruher <strong>Institut</strong> für Technologie (KIT)<br />
Analysis of semi-infinite periodic structures<br />
Traditionally, the Floquet transform is used to analyze periodic structures which are infinitely extended<br />
in all space directions. Since there is the possibility of building optical devices from photonic crystal<br />
materials, the mathematical un<strong>der</strong>standing of structures arising from the truncation of an infinite crystal<br />
is also of great importance. A typical model situation is a half-space of photonic crystal or a semi-infinite<br />
periodic waveguide; here, the un<strong>der</strong>lying domain is not invariant with respect to integer translations, so it<br />
is not clear how to apply Floquet-Bloch theory. In this talk, we give an overview of recent work leading<br />
to a new radiation condition for semi-infinite periodic structures and also to representation theorems for<br />
solutions of periodic problems in half-spaces.<br />
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