# 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

 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 Referent/Betreuer Prof. D. Hundertmark 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
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
KITopen Landing Page