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 23

1	Bisi M , Cañizo J A , Lods B . Entropy dissipation estimates for the linear Boltzmann operator. J Funct Anal, 2015, 269 (4): 1028- 1069 doi: 10.1016/j.jfa.2015.05.002
2	Cañizo 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 doi: 10.1016/j.jmaa.2017.12.052
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 doi: 10.1002/mma.4473
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 doi: 10.3934/krm.2008.1.223
5	Bisi M , Cañizo 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 doi: 10.1137/110837437
6	Cañizo J A , Lods B . Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath. Nonlinearity, 2016, 29 (5): 1687- 1715 doi: 10.1088/0951-7715/29/5/1687
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 doi: 10.1088/0951-7715/19/4/011
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 doi: 10.1007/s00205-016-0972-4
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 doi: 10.1007/s10955-014-1066-z
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 doi: 10.1007/s00220-020-03682-8
12	Carrapatoso K . Exponential convergence to equilibrium for the homogenenous Landau equation with hard potentials. Bull Sci Math, 2015, 139 (7): 777- 805 doi: 10.1016/j.bulsci.2014.12.002
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 doi: 10.1007/s00205-015-0963-x
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 doi: 10.1007/s10955-020-02595-z
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 doi: 10.1016/j.jde.2015.10.012
16	Wu K C . Pointwise behavior of the linearized Boltzmann equation on a torus. SIAM J Math Anal, 2014, 46 (1): 639- 656 doi: 10.1137/13090482X
17	Cañizo J A , Cao C Q , Evans J , Yoldaş H . Hypocoercivity of linear kinetic equations via Harris's theorem. Kinet Relat Models, 2020, 13 (1): 97- 128 doi: 10.3934/krm.2020004
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 doi: 10.1007/s00205-009-0233-x
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 doi: 10.1016/j.matpur.2007.03.003
21	Mouhot C . Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun Partial Differ Equ, 2006, 31 (9): 1321- 1348 doi: 10.1080/03605300600635004
22	Yang T , Yu H J . Spectrum analysis of some kinetic equations. Arch Ration Mech Anal, 2016, 222 (2): 731- 768 doi: 10.1007/s00205-016-1010-2
23	Sun B Y . Exponential convergence for the linear homogeneous Boltzmann equation for hard potentials. Appl Math Comput, 2018, 339, 727- 737

[1]	章春国,付煜之,刘宇标. 二维Mindlin-Timoshenko板系统的稳定性与最优性[J]. 数学物理学报, 2021, 41(5): 1465-1491.
[2]	叶耀军, 胡月. 一类高阶波动方程的整体解及指数衰减估计[J]. 数学物理学报, 2018, 38(1): 110-121.
[3]	章春国, 张海燕, 谷尚武. 一类具有局部记忆阻尼的弱耦合系统的能量衰减估计[J]. 数学物理学报, 2015, 35(1): 194-209.
[4]	晋守博, 陈攀峰, 孙善辉. 耗散线性Boltzmann方程的解的正则性[J]. 数学物理学报, 2011, 31(6): 1662-1668.
[5]	黄玲娣, 陈泽乾. 加权解析Lipschitz空间的等价范数[J]. 数学物理学报, 2005, 25(5): 663-672.
[6]	吕建生. 格点巢分形上的渗流模型[J]. 数学物理学报, 2000, 20(4): 521-527.