线性玻尔兹曼碰撞算符：I.用离散内能变量和多组分混合物模拟多原子分子
Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures
摘要：
玻尔兹曼方程的线性化碰撞算子可以以自然的方式写成正乘法算子、碰撞频率和积分算子的和。单原子积分算子的紧性是一个经典的结果，而混合物的相应结果是最近才得到的。在这项工作中，多原子单物种，其中的多原子性是由一个离散的内部能量变量建模的操作程序的紧性进行了研究。以碰撞算子的概率公式为出发点，在碰撞核的某些假设下，证明了积分算子是Hilbert-Schmidt积分算子与近似Hilbert-Schmidt积分算子的和，从而得到了紧性.线性化碰撞算子的自伴性如下。此外，边界-包括-碰撞频率的硬球模型。然后，它遵循的线性碰撞算子是一个弗雷德霍姆算子。结果可以推广到混合物。为了简洁起见，仅考虑单原子物质的混合物的情况。
英文摘要：
The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In this work the compactness of the operator for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable, is studied. 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 approximately Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere model. Then it follows that the linearized collision operator is a Fredholm operator. The results can be extended to mixtures. For brevity, only the case of mixtures for monatomic species is accounted for.
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