CAREER: Large and Multi-Dimensional Solutions of Conservation Laws

理解非线性偏微分方程（PDE）的性质是现代纯数学和应用数学中的一个基本挑战。该研究项目的目标是获得新的洞察到进化PDE的三个具体领域。 (A)一维守恒律系统的大解：大数据和非线性效应的组合效应构成了一个基本的挑战，同时在应用中具有明显的重要性。在现有工作的基础上，该项目旨在进一步给出爆破解的例子，了解其稳定性，并确定保证大解全局存在的假设。(B)多维守恒律系统特殊系统解的存在性和定性性质。多维方程表现出非常丰富的行为，目前还没有普遍的存在性结果的整体解决方案。最近的爆破例子表明，一维系统的小变分理论不能推广到几个空间维。这项研究旨在采取“自下而上”的方法，从具体案例中获得见解。系统将被选择提供简化的，但不平凡的，强结构约束的例子。我们的目标是创建一个可应用于更通用系统的方法工具箱。分析和数值工具将被用来深入了解解决方案的结构。(C)流体流动的Navier-Stokes方程在广泛的应用中提供了一个重要的基本模型。该项目的目标是研究三个问题：真空（空化）的形成，包括温度依赖性的传输系数（粘度和导热性）的影响，以及大振幅的多维流动。各种项目还将解决在计算大型或多维解决方案以及包含真空的流动时面临的重大数值挑战。 这些项目旨在更好地理解非线性机制如何与广泛用于物理模型的方程中的一维或多维效应相互作用，这些模型包括材料特性、流体流动和气象学。 虽然这些问题是独立的理论兴趣，他们也是在科学和工程应用中的明显重要性。其中一些项目需要开发和测试高性能计算机代码，这些代码在日常模拟中具有重要应用。分析技术，建模和数值计算的综合方法将增强对流体流动及其应用的基本理解。

Understanding the properties of nonlinear Partial Differential Equations (PDE) is a fundamental challenge in modern pure and applied mathematics. The goal of the research projects is to gain new insightinto three specific areas of evolutionary PDE. (A) Large solutions of one-dimensional systems of conservation laws:The combined effects of large data and nonlinear effects pose a basic challenge as well as being of obvious importance in applications. Building on existing work this project aims at giving further examples of blowup solutions, understanding their stability properties, and identifying assumptions guaranteeing global existence of large solutions.(B) Existence and qualitative properties of solutions for particular systems of multi-dimensional systems of conservation laws. Multi-dimensional equations display an exceedingly richvariety of behaviors and there is currently no general existence result available for global solutions. Recent examples of blowup demonstrate that the small variation theory for one-dimensionalsystems cannot be generalized to several space dimensions. The research aims at a "bottom-up" approach where insight is obtained from specific cases. The systems will be chosen to provide simplified, but non-trivial, examples with strong structural constraints. The goal is to create a toolbox of methods that can be applied to more general systems. Both analytical and numerical tools will be employed to gain insight into the structure of the solutions.(C) The Navier-Stokes equations for fluid flow provide a basicmodel of importance in a wide range of applications. The goal of this project is to investigate three issues: formation of vacuums (cavitation), the effect of including temperature dependence in the transport coefficients (viscosities and heat conductivity), and multi-dimensional flows with large amplitudes. The various projects will also address the significant numerical challenges one faces in computing large or multi-dimensional solutions, and flows containing vacuums. Vice versa, exact solutionswill be used to benchmark various computational codes.The projects aims at a better understanding of how nonlinear mechanisms interact with one- or multi-dimensional effects in equations that are extensively used in physical models, ranging from properties of materials and to fluid flow and meteorology. While these issues are of independent theoretical interest they are also of obvious importance in applications to Science and Engineering. Several of the projects require development and testing of high performance computer codes with important applications in everyday simulations. A combined approach of analytical techniques, modeling, and numerical calculations will enhance basic understanding of fluid flow and its applications.