On the Rate of Relaxation for the Landau Kinetic Equation and Related Models
On the Rate of Relaxation for the Landau Kinetic Equation and Related Models
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朗道动力学方程及相关模型的弛豫率研究
DOI：
10.1007/s10955-017-1814-y
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发表时间：
2017
影响因子：
1.6
通讯作者：
Zhang, Chenglong
中科院分区：
文献类型：
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作者：
Bobylev, Alexander;Gamba, Irene M.;Zhang, Chenglong
通过考虑线性Landau型方程径向解的相对简单情形，研究了朗道动力学方程及相关模型向平衡态的弛豫速率.众所周知的困难是演化算子没有谱隙，即它的谱不与零分离。因此，我们并不期望在大的时间值上有纯指数松弛。我们的工作的主要目标之一是数值识别的大时间渐近松弛到平衡。我们回顾应变和郭的工作（拱大鼠机械肛门187：287-339 2008，通讯部分差异Equ 31：17-429 2006），谁严格表明，预期的法律放松是与一些。在这份手稿中，我们找到了一个启发式的方法，执行渐近的方法，找到这个“三分之二定律”，然后研究这个问题的数值。更具体地说，线性朗道方程近似的一组常微分方程的基础上扩展广义Laguerre多项式。我们详细分析了相应的二次型和这些常微分方程的解决方案。结果表明，该解决方案有两个不同的渐近阶段的大值的λ和最大阶的多项式sN：第一个集中在中间渐近同意的“三分之二定律”的适度大值的λ，然后第二个绝对的，纯指数渐近非常大，预期的线性常微分方程。我们相信，在有限维近似中出现中间渐近性对于函数空间中的不同类型的方程（一些偏微分方程，软势的玻尔兹曼方程等）来说一定是一种普遍的行为。并且我们的方法可以应用于相关问题。
We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purely exponential relaxation for large values of time. One of the main goals of our work is to numerically identify the large time asymptotics for the relaxation to equilibrium. We recall the work of Strain and Guo (Arch Rat Mech Anal 187:287–339 2008, Commun Partial Differ Equ 31:17–429 2006), who rigorously show that the expected law of relaxation iswith some. In this manuscript, we find an heuristic way, performed by asymptotic methods, that finds this “law of two thirds”, and then study this question numerically. More specifically, the linear Landau equation is approximated by a set of ODEs based on expansions in generalized Laguerre polynomials. We analyze the corresponding quadratic form and the solution of these ODEs in detail. It is shown that the solution has two different asymptotic stages for large values of timetand maximal order of polynomialsN: the first one focus on intermediate asymptotics which agrees with the “law of two thirds” for moderately large values of timetand then the second one on absolute, purely exponential asymptotics for very larget, as expected for linear ODEs. We believe that appearance of intermediate asymptotics in finite dimensional approximations must be a generic behavior for different classes of equations in functional spaces (some PDEs, Boltzmann equations for soft potentials, etc.) and that our methods can be applied to related problems.
非线性朗道-福克-普朗克动力学方程的确定性和随机方法及其在等离子体物理中的应用
影响因子：
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作者：
I. Potapenko;A. V. Bobylev;E. Mossberg
通讯作者：
E. Mossberg