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Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules

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Abstract

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties to the fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties to the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in \({L^1_2(\mathbb{R}^d)\cap L {\rm log} L(\mathbb{R}^d)}\), i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

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Authors and Affiliations

  1. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Jean-Marie Barbaroux

  2. Institute for Analysis, Karlsruhe Institute of Technology (KIT), Englerstraße 2, 76131, Karlsruhe, Germany

    Dirk Hundertmark, Tobias Ried & Semjon Vugalter

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  1. Jean-Marie Barbaroux
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Barbaroux, JM., Hundertmark, D., Ried, T. et al. Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules. Arch Rational Mech Anal 225, 601–661 (2017). https://doi.org/10.1007/s00205-017-1101-8

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