Kinetic Techniques for Hyperbolic and Multiscale Problems

点击翻译按钮获取中文摘要

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.