空间非均匀线性Boltzmann方程在硬势情形下解的最优指数收敛速率

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1853-1863.

空间非均匀线性Boltzmann方程在硬势情形下解的最优指数收敛速率

孙宝燕*()

^{烟台大学数学与信息科学学院山东烟台 264005}

收稿日期:2021-07-22 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 孙宝燕 E-mail:bysun@ytu.edu.cn
基金资助:
烟台大学博士科研启动基金(2219008)

Optimal Exponential Decay for the Linear Inhomogeneous Boltzmann Equation with Hard Potentials

Baoyan Sun*()

^{School of Mathematics and Information Sciences, Yantai University, Shandong Yantai 264005}

Received:2021-07-22 Online:2021-12-26 Published:2021-12-02
Contact: Baoyan Sun E-mail:bysun@ytu.edu.cn
Supported by:
the Scientific Research Foundation of Yantai University(2219008)

摘要/Abstract

摘要：

该文研究空间非均匀线性Boltzmann方程在周期区域上非角截断硬势情形下解的渐近行为.在带有多项式权的$L_{v}^{1} L_{x}^{2}\left(\langle v\rangle^{k}\right)$空间中得到了解的最优指数收敛速率.将从谱理论和半群理论的角度来对该方程进行分析，主要方法是借助于在Hilbert空间中的强制性估计、算子分解以及空间拉大理论等.

关键词: 线性Boltzmann方程, 硬势, 多项式权, 谱隙, 指数衰减

Abstract:

In this paper, we consider the asymptotic behavior of solutions to the linear spatially inhomogeneous Boltzmann equation for hard potentials in the torus. We obtain an optimal rate of exponential convergence towards equilibrium in a Lebesgue space with polynomial weight $L_{v}^{1} L_{x}^{2}\left(\langle v\rangle^{k}\right)$. This model is analyzed from a spectral point of view and from the point of view of semigroups. Our strategy is taking advantage of the spectral gap estimate in the Hilbert space with inverse Gaussian weight, the factorization argument and the enlargement method.

Key words: Linear Boltzmann equation, Hard potentials, Polynomial weight, Spectral gap, Exponential decay

中图分类号:

O175

孙宝燕. 空间非均匀线性Boltzmann方程在硬势情形下解的最优指数收敛速率[J]. 数学物理学报, 2021, 41(6): 1853-1863.

Baoyan Sun. Optimal Exponential Decay for the Linear Inhomogeneous Boltzmann Equation with Hard Potentials[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1853-1863.

参考文献

1	Bisi M , Ca?izo J A , Lods B . Entropy dissipation estimates for the linear Boltzmann operator. J Funct Anal, 2015, 269 (4): 1028- 1069
2	Ca?izo J A , Einav A , Lods B . On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials. J Math Anal Appl, 2018, 462 (1): 801- 839
3	Lods B , Mokhtar-Kharroubi M . Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in $L^{1}$-spaces. Math Meth Appl Sci, 2017, 40 (18): 6527- 6555
4	Lods B , Mouhot C , Toscani G . Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models. Kinet Relat Models, 2008, 1 (2): 223- 248
5	Bisi M , Ca?izo J A , Lods B . Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath. SIAM J Math Anal, 2011, 43 (6): 2640- 2674
6	Ca?izo J A , Lods B . Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath. Nonlinearity, 2016, 29 (5): 1687- 1715
7	Mouhot C , Neumann L . Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity, 2006, 19 (4): 969- 998
8	Gualdani M P, Mischler S, Mouhot C. Factorization of non-symmetric operators and exponential H-theorem. 2010, arXiv: 1006.5523
9	Mischler S , Mouhot C . Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation. Arch Ration Mech Anal, 2016, 221 (2): 677- 723
10	Tristani I . Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off. J Stat Phys, 2014, 157 (3): 474- 496
11	Hérau F , Tonon D , Tristani I . Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off. Commun Math Phys, 2020, 377 (1): 697- 771
12	Carrapatoso K . Exponential convergence to equilibrium for the homogenenous Landau equation with hard potentials. Bull Sci Math, 2015, 139 (7): 777- 805
13	Carrapatoso K , Tristani I , Wu K C . Cauchy problem and exponential stability for the inhomogeneous Landau equation. Arch Ration Mech Anal, 2016, 221 (1): 363- 418
14	Li F C , Sun B Y . Optimal exponential decay for the linearized ellipsoidal BGK model in weighted Sobolev spaces. J Stat Phys, 2020, 181 (2): 690- 714
15	Li F C , Wu K C . Semigroup decay of the linearized Boltzmann equation in a torus. J Differential Equations, 2016, 260 (3): 2729- 2749
16	Wu K C . Pointwise behavior of the linearized Boltzmann equation on a torus. SIAM J Math Anal, 2014, 46 (1): 639- 656
17	Ca?izo J A , Cao C Q , Evans J , Yolda? H . Hypocoercivity of linear kinetic equations via Harris's theorem. Kinet Relat Models, 2020, 13 (1): 97- 128
18	Alonso A , Morimoto Y , Sun W R , Yang T . Non-cutoff Boltzmann equation with polynomial decay perturbations. Rev Mat Iberoam, 2021, 37 (1): 189- 292
19	Desvillettes L , Mouhot C . Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Arch Ration Mech Anal, 2009, 193 (2): 227- 253
20	Mouhot C , Strain R M . Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J Math Pures Appl, 2007, 87 (5): 515- 535
21	Mouhot C . Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun Partial Differ Equ, 2006, 31 (9): 1321- 1348
22	Yang T , Yu H J . Spectrum analysis of some kinetic equations. Arch Ration Mech Anal, 2016, 222 (2): 731- 768
23	Sun B Y . Exponential convergence for the linear homogeneous Boltzmann equation for hard potentials. Appl Math Comput, 2018, 339, 727- 737

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