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On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics

Ried, Tobias

Abstract (englisch):

This thesis deals with some nonlinear and nonlocal effective equations arising in kinetic theory and nonlinear optics.

First, it is shown that the homogeneous non-cutoff Boltzmann equation for Maxwellian molecules enjoys strong smoothing properties:
In the case of power-law type particle interactions, we prove the Gevrey smoothing conjecture. For Debye-Yukawa type interactions, an analogous smoothing effect is shown.
In both cases, the smoothing is exactly what one would expect from an analogy to certain heat equations of the form $\partial_t u = f(-\Delta)u$, with a suitable function $f$, which grows at infinity, depending on the interaction potential.
The results presented work in arbitrary dimensions, including also the one-dimensional Kac-Boltzmann equation.

In the second part we study the entropy decay of certain solutions of the Kac master equation, a probabilistic model of a gas of interacting particles. It is shown that for initial conditions corresponding to $N$ particles in a thermal equilibrium and $M\leq N$ particles out of equilibrium, the entropy relative to the thermal state decays exponentially to a fraction of the initial relative entropy, with a rate that is essentially independent of the number of particles. ... mehr

Volltext §
DOI: 10.5445/IR/1000076609
Veröffentlicht am 16.11.2017
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Hochschulschrift
Publikationsjahr 2017
Sprache Englisch
Identifikator urn:nbn:de:swb:90-766090
KITopen-ID: 1000076609
Verlag Karlsruher Institut für Technologie (KIT)
Umfang XV, 210 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Analysis (IANA)
Prüfungsdatum 18.10.2017
Schlagwörter Boltzmann equation, Kac equation, Gevrey Smoothing, Smoothing properties of weak solutions, Entropy decay, Thermostated Systems, Correlation inequalities, Solitary waves, Dispersion management, DM solitons, Saturating nonlinearities
Referent/Betreuer Hundertmark, D.
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