We investigate the smoothing effect of the spatially inhomogeneous Boltzmann equation without an angular cut-off, under the Navier-Stokes scaling. For Maxwellian molecules or hard potentials with singular angular kernels, we demonstrate that the solutions become analytic at positive times when the angular singularities are sufficiently strong and lie within the optimal Gevrey class when the singularities are mild. The analysis is based on carefully selected vector fields with time-dependent coefficients and quantitative estimates of directional derivatives, which reveal the behavior of the kinetic-fluid transition.
Qi AN
,
Xin HU
,
Weixi LI
. REGULARIZATION EFFECT OF THE BOLTZMANN EQUATION UNDER NAVIER-STOKES TYPE SCALING[J]. Acta mathematica scientia, Series B, 2025
, 45(6)
: 2607
-2628
.
DOI: 10.1007/s10473-025-0613-9
[1] Alexandre R, Hérau F, Li W X. Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff. J Math Pures Appl,2019, 126(9): 1-71
[2] Alexandre R, Morimoto Y, Ukai S, et al. Regularizing effect and local existence for the non-cutoff Boltzmann equation. Arch Ration Mech Anal, 2010, 198(1): 39-123
[3] Alexandre R, Morimoto Y, Ukai S, et al. The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J Funct Anal, 2012, 262(3): 915-1010
[4] Chen H, Hu X, Li W X, Zhan J. The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off. Sci China Math, 2022, 65(3): 443-470
[5] Chen H, Li W X, Xu C J. Gevrey hypoellipticity for a class of kinetic equations. Comm Partial Differential Equations, 2011, 36(4): 693-728
[6] Chen J L, Li W X, Xu C J. Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials. Ann Inst H Poincaré C Anal Non Linéaire,2025, 42(4): 933-970
[7] Desvillettes L. About the regularizing properties of the non-cut-off Kac equation. Comm Math Phys, 1995, 168(2): 417-440
[8] Duan R, Li W X, Liu L. Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation. Adv Math, 2022, 395: Paper 108159
[9] Duan R, Liu S, Sakamoto S, Strain R M. Global mild solutions of the Landau and non-cutoff Boltzmann equations. Comm Pure Appl Math, 2021, 74(5): 932-1020
[10] Gressman P T, Strain R M. Global classical solutions of the Boltzmann equation without angular cut-off. J Amer Math Soc, 2011, 24(3): 771-847
[11] Imbert C, Silvestre L. The weak Harnack inequality for the Boltzmann equation without cut-off. J Eur Math Soc (JEMS), 2020, 22(2): 507-592
[12] Jiang N, Xu C J, Zhao H. Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions. Indiana Univ Math J, 2018, 67(5): 1817-1855
[13] Lions P L. On Boltzmann and Landau equations. Philos Trans Roy Soc London Ser A, 1994, 346(1679): 191-204
[14] Lions P L, Masmoudi N. From the Boltzmann equations to the equations of incompressible fluid mechanics. I. Arch Ration Mech Anal, 2001, 158(3): 173-193
[15] Lions P L, Masmoudi N. From the Boltzmann equations to the equations of incompressible fluid mechanics. II. Arch Ration Mech Anal, 2001, 158(3): 195-211
[16] Saint-Raymond L.Hydrodynamic Limits of the Boltzmann Equation. Berlin: Springer-Verlag, 2009
