Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff

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

“…Morimoto, Ukai, Xu and Yang [30] proved the same H ∞ regularising effect using suitable test functions in the weak formulation of the problem.…”

“…We are interested in the Gevrey smoothing effect, namely that under the (physical) assumptions of finite mass, energy and entropy of the initial data, weak solutions of the homogeneous Boltzmann equation without cutoff are Gevrey functions for any strictly positive time. This question was treated in the case of the linearised Boltzmann equation in the homogeneous setting by Morimoto et al [30], where they proved that, given 0 < ν < 1, weak solutions of the linearized Boltzmann equation belong to the space G 1 ν (R 3 ) for any positive times. Still in a linearised setting, Lerner, Morimoto, Pravda-Starov and Xu [24] proved a Gelfand-Shilov smoothing effect, which includes Gevrey regularity, on radially symmetric solutions of the homogeneous non-cutoff Boltzmann equation for Maxwellian molecules.…”

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

“…Remark 1.5. This regularity is much weaker than the Gevrey regularity we proved in [4] for singular kernels of the form (3), but it is much stronger than the H ∞ smoothing shown in [11]. Moreover, it is exactly the right type of regularity one would expect for a coercive term of the form (7) from the analogy with the heat equation (11).…”

“…In [11] it has been shown that weak solutions to the Cauchy problem (1) with Debye-Yukawa type interactions enjoy an H ∞ smoothing property, i.e. starting with arbitrary initial datum f 0 ≥ 0, f 0 ∈ L 1 2 ∩ L log L, one has f (t, ·) ∈ H ∞ for any positive time t > 0.…”

“…Let T > 0 be arbitrary (but finite). By the already known H ∞ smoothing property of the homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction, see [11], for any 0 < t 0 < T one has…”