Generalized kinetic Maxwell type models of granular gases
Generalized kinetic Maxwell type models of granular gases
复制标题
颗粒气体的广义动力学麦克斯韦型模型
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
I. Gamba
中科院分区：
文献类型：
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作者：
A. Bobylev;C. Cercignani;I. Gamba
我们考虑广义动力学颗粒气体模型给出的麦克斯韦型玻尔兹曼方程。这些类型的非线性弹性或非弹性相互作用的模型，有许多应用在物理学，动力学的粒状气体，经济等，我们提出的问题，并制定其形式的空间中的特征函数，即傅立叶变换的概率措施，从一个非常一般的角度来看，包括那些与任意多项式非线性和在任何维空间。我们发现了一类广义麦克斯韦模型，满足的特性，动态缩放或自相似的解决方案，通常被称为{em均匀冷却状态}的存在性和渐近性。特别感兴趣的是一个概念，解释为一个运营商的推广通常的Lipschitz条件，允许描述的行为的解决方案，相应的初始值问题。特别是，我们提出，在最一般的情况下，存在的自相似的解决方案和研究，在概率措施的意义上，收敛的动态缩放的解决方案与柯西问题的自相似的解决方案，随着时间的推移到无穷大。此外，我们表明，这些自相似的解决方案的性质导致非经典的平衡稳定状态显示功率尾巴。这些结果适用于与玻尔兹曼方程（弹性和非弹性相互作用）相关的不同具体问题，并表明所有物理相关的解决方案的属性直接遵循本演示文稿中开发的一般理论。
We consider generalizations of kinetic granular gas models given by Boltzmann equations of Maxwell type. These type of models for non-linear elastic or inelastic interactions, have many applications in physics, dynamics of granular gases, economy, etc. We present the problem and develop its form in the space of characteristic functions, i.e. Fourier transforms of probability measures, from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. We find a whole class of generalized Maxwell models that satisfy properties that characterize the existence and asymptotic of dynamically scaled or self-similar solutions, often referred as {em homogeneous cooling states}. Of particular interest is a concept interpreted as an operator generalization of usual Lipschitz conditions which allows to describe the behavior of solutions to the corresponding initial value problem. In particular, we present, in the most general case, existence of self similar solutions and study, in the sense of probability measures, the convergence of dynamically scaled solutions associated with the Cauchy problem to those self-similar solutions, as time goes to infinity. In addition we show that the properties of these self-similar solutions lead to non classical equilibrium stable states exhibiting power tails. These results apply to different specific problems related to the Boltzmann equation (with elastic and inelastic interactions) and show that all physically relevant properties of solutions follow directly from the general theory developed in this presentation.