论文标题
多组分多原子气体的线性化玻尔兹曼碰撞算子的紧凑特性
Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas
论文作者
论文摘要
线性化的玻尔兹曼碰撞运算符在许多玻尔兹曼方程的研究中至关重要，其主要特性至关重要。分解为正乘算子的总和，碰撞频率和积分操作员是微不足道的。对于单变性单物种的积分算子的紧凑性是一个经典的结果，而最近获得了单变性混合物和多原子单物种的相应结果。这项工作涉及多原子物种多组分混合物的操作员的紧凑性，其中多原子性是由离散的内部能量变量建模的。通过将碰撞算子作为起点的概率表述，可以通过证明整体操作员是希尔伯特·史密特（Hilbert-Schmidt）积分操作员和操作员的总和，在碰撞核上的某些假设下，这是Hilbert-Schmidt积分运算符的统一限制。这些假设本质上是对毕业生对单一单一物种的假设的概括。线性化碰撞算子的自相关性随之而来。此外，在像模型这样的硬球体中获得了碰撞频率的界限 - 包括 - 碰撞频率。然后是线性化碰撞操作员是弗雷德姆操作员，并且还获得了其域。
The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for monatomic mixtures and polyatomic single species are more recently obtained. This work concerns the compactness of the operator for a multicomponent mixture of polyatomic species, where the polyatomicity is modeled by a discrete internal energy variable. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. The assumptions are essentially generalizations of the Grad's assumptions for monatomic single species. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model. Then it follows that the linearized collision operator is a Fredholm operator, and its domain is also obtained.
