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Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules. (English) Zbl 1382.35177
It has long been suspected that the non-cutoff Boltzmann operator with a singular cross section kernel has similar coercivity properties to the fractional Laplacian operator. This conjecture was proven in [Arch. Ration. Mech. Anal. 152, No. 4, 327–355 (2000; Zbl 0968.76076)] by R. Alexandre et al. Thanks to this discovery, it was then conjectured that the fully nonlinear homogeneous Boltzmann equation enjoys regularity properties similar to those of the heat equation with a fractional Laplacian. In particular, it is expected that the weak solutions of the fully nonlinear non-cutoff homogeneous Boltzmann equation with initial datum in the class of functions with finite mass, energy and entropy become instantaneously Gevrey regular for strictly positive times. The authors prove this conjecture for Maxwellian molecules. The result covers both the weak and strong singularity regimes for the fractional Laplacian. It applies to the Boltzmann equation in the space of any dimension starting dimension one. The physical case appears in dimension three when the fractional Laplacian has weak singularity.

MSC:
35Q20 Boltzmann equations
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
35R11 Fractional partial differential equations
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