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On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics
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 analogydoi:10.5445/ir/1000076609 fatcat:pzsrmsug7jbmxa3j5efm2d3wfm