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

Abstract : It has long been suspected that the non-cutoff Boltzmann operator has similar coerciv-ity 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 fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in L 1 2 (R d) ∩ L log L(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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Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules. Archives of rational mechanics and analysis, Springer-Verlag, 2017, 225 (2), pp. 601-661. ⟨10.1007/s00205-017-1101-8⟩. ⟨hal-01253130⟩

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