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 frac ... mehr

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 frac ... mehr

Zugehörige Institution(en) am KIT |
Institut für Analysis (IANA) |

Publikationstyp |
Hochschulschrift |

Jahr |
2017 |

Sprache |
Englisch |

Identifikator |
URN: urn:nbn:de:swb:90-766090 KITopen ID: 1000076609 |

Verlag |
Karlsruhe |

Umfang |
XV, 210 S. |

Abschlussart |
Dissertation |

Fakultät |
Fakultät für Mathematik (MATH) |

Institut |
Institut für Analysis (IANA) |

Prüfungsdatum |
18.10.2017 |

Referent/Betreuer |
Prof. D. Hundertmark |

Schlagworte |
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 |

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