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 ... mehrtion of the initial relative entropy, with a rate that is essentially independent of the number of particles.
Finally, we investigate the existence of dispersion management solitons. Using variational techniques, we prove that there is a threshold for the existence of minimisers of a nonlocal variational problem, even with saturating nonlinearities, related to the dispersion management equation.