Decay and Continuity of the Boltzmann Equation in Bounded Domains
Decay and Continuity of the Boltzmann Equation in Bounded Domains
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DOI：
10.1007/s00205-009-0285-y
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发表时间：
2008-01
影响因子：
2.5
通讯作者：
Yan Guo
中科院分区：
文献类型：
--
作者：
Yan Guo
边界效应在动力学方程中自然发生，边界效应对由玻尔兹曼方程控制的稀气体动力学至关重要。我们开发了一个数学理论来研究四种基本类型的边界条件下玻尔兹曼解的时间衰减和连续性：流入，反弹反射，镜面反射和漫反射。我们建立了指数衰减的L ∞范数的硬势的一般类光滑域附近的绝对麦克斯韦。此外，在凸域中，我们也建立连续性，这些玻尔兹曼解决方案远离放牧集的边界。我们的贡献是基于一个新的L2衰变理论和它的相互作用与微妙的L ∞衰变分析的线性Boltzmann方程在存在许多重复的相互作用的边界。
Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in theL∞norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a newL2decay theory and its interplay with delicateL∞decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.