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

Sun Baoyan,

School of Mathematics and Information Sciences, Yantai University, Shandong Yantai 264005

 基金资助: 烟台大学博士科研启动基金.  2219008

 Fund supported: the Scientific Research Foundation of Yantai University.  2219008

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.

Keywords： Linear Boltzmann equation ; Hard potentials ; Polynomial weight ; Spectral gap ; Exponential decay

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

## 1 引言

$$$\partial_{t}f + v \cdot \nabla_{x} f = Q(f, \mu),$$$

$$$f(0, x, v) = f_{in}(x, v)\geq0,$$$

1. 对温和奇性情形, 即$s\in(0, \frac{1}{2})$ : $f_{in}\in L_{v}^{1}L_{x}^{2}({\langle v \rangle}^{k})$, $k\geq2$,

2. 对强奇性情形, 即$s\in[\frac{1}{2}, 1)$ : $f_{in}\in L_{v}^{1}L_{x}^{2}({\langle v \rangle}^{k})$, $k\geq4$,

(2) 对于温和奇性情形, 我们考虑空间$L_{v}^{1}L_{x}^{2}({\langle v \rangle}^{k})$, $k\geq2$; 对于强奇性情形, 我们可以考虑空间$L_{v}^{1}L_{x}^{2}({\langle v \rangle}^{k})$, $k\geq4$. 我们可以看到当处理强奇性情形时, 选取的空间变小了. 对于权指标$k$的限制并不是技术原因, 而是由方程本身来决定的. 通过作拟微分变换, 我们可以发现$\rm Boltzmann$碰撞算子和分数阶的$\rm Laplace$算子类似. 实际上, 对于非角截断的$\rm Boltzmann$碰撞算子的性质和具有阶$s+\frac{1}{2}$的分数阶$\rm Laplace$算子类似, 而对于指标$k$的限制恰好能够支持导数.

(3) $\rm Cañizo$等人[17]借助概率论中$\rm Harris$定理可以得到在非角截断硬势情形下方程(1.1)–(1.2) 的亚强制性估计, 从而证明解在空间$L_{x, v}^{1}({\langle v \rangle}^{2})$中具有指数衰减性质.

(4) $\rm Alonso$等人[18]研究了在非角截断硬势情形下经典的$\rm Boltzmann$方程解的适定性. 值得注意地, 他们利用文献[8] 中的空间拉大理论证明线性化$\rm Boltzmann$算子${\mathcal L}(h): = Q(h, \mu)+Q(\mu, h)-v \cdot \nabla_{x}h $$\rm Hilbert 空间 L_{x, v}^{2}({\langle v \rangle}^{k})$$ (k >\frac{9}{2}+\frac{\gamma}{2}+2s)$中具有谱隙估计. 因此, 算子${\mathcal L}$所生成的半群$S_{{\mathcal L}}(t)$在空间$L_{x, v}^{2}({\langle v \rangle}^{k})$中满足指数衰减. 不同于之前工作, 我们在本文中首次在大空间$L_{v}^{1}L_{x}^{2}({\langle v \rangle}^{k})$中证明该结论.

## 2 定理1.1的证明

$$$\partial_{t}h = {\cal L}(h): = \overline{{\cal L}}(h)-v \cdot \nabla_{x} h,$$$

$\lambda$是算子${\cal L} $$\rm Hilbert 空间 L_{x, v}^{2}(\mu^{-\frac{1}{2}}) 中的谱隙. 第二步: 算子 {\cal B} 具有耗散性. 引理2.2 若选取参数 \epsilon>0$$ \delta>0$充分小, 可以得到算子${\cal B}+\lambda$在空间$L_{v}^{1}L_{x}^{p}({\langle v \rangle}^{k})$中耗散, 这里$p\in[1, +\infty]$, $k\geq2 $$s\in\big(0, \frac{1}{2}\big) 时或者 k\geq4$$ s\in\big[\frac{1}{2}, 1\big)$时, $\lambda$由引理2.1给出. 即

令$h$是下面线性方程的解

$$$\left\{\begin{array}{ll} \partial_{t}h = {\cal B}(h) = \overline{{\cal L}}_\epsilon(h)+{\cal B}^c_{\epsilon, \delta}(h)-\nu_\epsilon(v)\, h-v \cdot \nabla_{x}h, \\ h(0, x, v) = h_{0}(x, v). \end{array}\right.$$$

$$$I_1 \leq C_k\, k_{\epsilon}\, \|h\|_{L_{v}^{1}L_{x}^{p}({\langle v \rangle}^{k+\gamma})},$$$

$\begin{eqnarray} I_2&\leq& \Big(C_k+C_k\, O(\delta)+C_k\, K_\epsilon\, O(\delta)\Big)\, \|h\|_{L_{v}^{1}L_{x}^{p}({\langle v \rangle}^{k+\gamma})}\\ &&+K_{\epsilon}\, \int_{{\mathbb R}^{3}\times{\mathbb R}^{3}}\chi_{\delta^{-1}}\, |v-v_{*}|^\gamma \, \|h\|_{L_{x}^{p}}\, \mu(v_{*})\, {\langle v \rangle}^{k}\, {\rm d}v_{*}\, {\rm d}v. \end{eqnarray}$

$$$\frac{\rm d}{{\rm d}t}\|h\|_{L_{v}^{1}L_{x}^{p}({\langle v \rangle}^{k})}\leq -\lambda\, \|h\|_{L_{v}^{1}L_{x}^{p}({\langle v \rangle}^{k+\gamma})}.$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Bisi M , Cañizo J A , Lods B .

Entropy dissipation estimates for the linear Boltzmann operator

J Funct Anal, 2015, 269 (4): 1028- 1069

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

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

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

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

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

Mouhot C , Neumann L .

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

Nonlinearity, 2006, 19 (4): 969- 998

Gualdani M P, Mischler S, Mouhot C. Factorization of non-symmetric operators and exponential H-theorem. 2010, arXiv: 1006.5523

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

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

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

Carrapatoso K .

Exponential convergence to equilibrium for the homogenenous Landau equation with hard potentials

Bull Sci Math, 2015, 139 (7): 777- 805

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

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

Li F C , Wu K C .

Semigroup decay of the linearized Boltzmann equation in a torus

J Differential Equations, 2016, 260 (3): 2729- 2749

Wu K C .

Pointwise behavior of the linearized Boltzmann equation on a torus

SIAM J Math Anal, 2014, 46 (1): 639- 656

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

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

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

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

Mouhot C .

Explicit coercivity estimates for the linearized Boltzmann and Landau operators

Commun Partial Differ Equ, 2006, 31 (9): 1321- 1348

Yang T , Yu H J .

Spectrum analysis of some kinetic equations

Arch Ration Mech Anal, 2016, 222 (2): 731- 768

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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