Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation

Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials

“…Recall for Boltzmann equations we only have Gevrey regularity of index (1 + 2s)/2s in both spatial and velocity variables (cf. [19,30,47]), which seems not optimal in view of the counterpart for Kolmogorov type operators with fractional diffusion in velocity.…”

“…Furthermore, the C ∞smoothing effect was proven by [3,5], and a higher order Gevrey regularity, inspired by the behaviors of Kolmogorov operators, was proven by Lerner-Morimoto-Pravda-Starov-Xu [47] and [19] for the1D and 3D Boltzmann equations, respectively. Recently the existence and uniqueness of some mild weak solutions was established by Duan-Liu-Sakamoto-Strain [31] and the Gevrey regularity were proven by [30]. For the perturbation setting with polynomial decay, the classical solutions were constructed by Alonso-Morimoto-Sun-Yang [10]; see also the independent works of He-Jiang [36] and Hérau-Tonon-Tristani [41].…”

“…Moreover the following estimate . Furthermore we may follow the presentation in [30] with necessary modifications, to conclude the global Gevrey smoothing effect of such low-regularity solutions in the sense that f (t, •, •) ∈ G 3/2 for any t > 0, that is, the quantitative estimate (5.1) is satisfied globally in time.…”

“…Observe the proof in [30] relies on the same trilinear and coercivity estimates for Boltzmann collision operator as that in Proposition 2.2, the pseudo-differential calculus for the symbol…”

“…which corresponds to s = 1 in the case of Landau equations. Thus the estimate (5.1) will follow without any additional difficulty by applying the same strategy as in [30], and we refer to interested reads to [16] for the detailed derivation of (5.1).…”