REGULARIZATION EFFECT OF THE BOLTZMANN EQUATION UNDER NAVIER-STOKES TYPE SCALING

doi:10.1007/s10473-025-0613-9

Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (6): 2607-2628.doi: 10.1007/s10473-025-0613-9

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REGULARIZATION EFFECT OF THE BOLTZMANN EQUATION UNDER NAVIER-STOKES TYPE SCALING

Qi AN¹, Xin HU², Weixi LI^3,*

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. Wuhan Institute for Math & AI, Wuhan University, Wuhan 430072, China;
3. School of Mathematics and Statistics & Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

Received:2025-04-01 Revised:2025-06-14 Online:2025-11-25 Published:2025-11-14
Contact: ^*Weixi LI, E-mail: wei-xi.li@whu.edu.cn
About author:Qi AN, E-mail: anqiann@whu.edu.cn; Xin HU, E-mail: hux@whu.edu.cn
Supported by:
Natural Science Foundation of China (12325108, 12131017, 12221001) and the Natural Science Foundation of Hubei Province (2019CFA007).

Abstract

Abstract: 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.

Key words: scaled Boltzmann equation, non cut-off, spatially inhomogeneous, smoothing effect

CLC Number:

35B65

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.

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References

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REGULARIZATION EFFECT OF THE BOLTZMANN EQUATION UNDER NAVIER-STOKES TYPE SCALING

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