1:00 pm Friday, September 9, 2016
Analysis: The solution of the Gevrey smoothing conjecture for the non-cutoff homogenous Boltzmann equation for Maxwellian molecules by
Dirk Hundertmark [mail] (Karlsruhe Institute of Technology) in RLM 10.176
It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the non-cutoff homogenous Boltzmann equation with initial datum in , i.e., finite mass, energy and entropy, should immediately become Gevrey regular. So far, the best available results show that the solution becomes regular for positive times. Gevrey regularity is also known for weak solutions of the linearised Boltzmann equation, where one studies solutions close to a Maxwellian distribution, or under additional decay assumptions on the solutions. The main problem for establishing Gevrey regularity is that, in order to use the coercivity results on the non-cutoff Boltzmann collision kernel, one has to bound a non-linear and non-local commutator of the Boltzmann kernel with certain sub-Gaussian weights in Fourier space and this control has only been established for polynomial weights, which then only yields --smoothing. We prove, under the sole assumption that the initial datum is in , that the weak solution of the homogenous Boltzmann becomes Gevrey regular for strictly positive times. The main ingredient in the proof is a new way of estimating the non-local and non-linear commutator. Joint work with Jean-Marie Barbaroux, Tobias Ried, and Semjon Wugalter. Submitted by
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