Kinetic Techniques for Hyperbolic and Multiscale Problems

近年来，动力学方程理论与双曲系统弱解理论之间的思想交流取得了丰硕的成果。这种交流的核心在于从气体动力学理论或统计物理的微观模型推导连续统理论的问题。在这种情况下，无论是在动力学方程的框架中，还是在非线性输运的背景下，输运性质都起着至关重要的作用，因为它在多凸弹性动力学或非线性麦克斯韦模型的背景下出现微分约束。s方程。本提案的目标是对以下主题进行分析和建模工作：(i)两个守恒定律系统中的输运和振荡，（ii）碰撞动力学模型及其流体动力学极限，（iii）微分约束对多凸弹性动力学方程的影响，（iv）稀聚合物动力学理论的数学方面，以及(v)均质问题动力学技术的发展。守恒定律的双曲系统表达了连续体物理的基本定律，因此在科学建模中处于中心地位。随着微观建模和介观方程的相关推导在今天变得司空见惯，动力学建模正变得越来越明显。S工程应用。理解均匀化问题和多尺度相互作用问题中的耦合是设计不解决问题的所有微观信息的高效计算算法的必要组成部分。从小尺度过渡到大尺度的数学分析对高性能计算算法的设计具有重要意义，特别是在处理发生尺度相互作用的问题时。这些问题是现代材料科学的核心，在化学和材料工程中有许多应用。

In recent years there has seen a very fruitful exchange between ideas in the theory of kinetic equations and the theory of weak solutions for hyperbolic systems. At the core of this exchange lies the issue of deriving continuum theories from microscopic models of kinetic theory of gases or statistical physics. In this context transport properties play a crucial role whether in a framework of kinetic equations, or in a context of nonlinear transport as it appears with differential constraints in the context of polyconvex elastodynamics or nonlinear models for Maxwell?s equations. This proposal has the objectives to perform analytical and modeling work on the topics: (i) transport and oscillations in systems of two conservation laws, (ii) collisional kinetic models and their hydrodynamic limits, (iii) effect of differential constraints on the equations of polyconvex elastodynamics, (iv) mathematical aspects of kinetic theory of dilute polymers, and (v) development of kinetic techniques for homogenization problems. Hyperbolic systems of conservation laws express the basic laws of continuum physics and as such are central in modeling in the sciences. Kinetic modeling is becoming all the more pronounced, as microscopic modeling and the associated derivation of mesoscopic equations is commonplace in today?s engineering applications. Understanding of homogenization issues and the couplings in problems where multiple scales interact are necessary ingredients for the design of e cient computational algorithms that proceed without resolving all microscopic information of the problem. The mathematical analysis of issues related to the passage from small to coarser scales has significant implications on the design of high-performance computing algorithms, particularly when treating problems where scale interactions occur. Such problems are at the center of modern material science with several applications in chemical and materials engineering.