CAREER: Large and Multi-Dimensional Solutions of Conservation Laws

了解非线性偏微分方程(PDE)的性质是现代纯数学和应用数学的一个基本挑战。研究项目的目标是对进化PDE的三个具体领域获得新的见解。(A)一维守恒律系统的大型解：大型数据和非线性效应的综合影响是一项基本挑战，在应用中具有明显的重要性。在已有工作的基础上，本项目旨在给出爆破解的进一步例子，了解其稳定性性质，并确定保证整体存在大解的假设。(B)多维守恒律组特定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，目前还没有关于整体解的普遍存在结果。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度。这项研究的目标是“自下而上”的方法，即从具体的案例中获得洞察力。这些系统将被选择来提供具有强烈结构约束的简化但不是微不足道的例子。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。将使用分析和数值工具来了解解的结构。(C)流体流动的Navier-Stokes方程提供了一个在广泛应用中具有重要意义的基本模型。这个项目的目标是研究三个问题：真空的形成(空化)，在输运系数中考虑温度相关性的影响(粘度和导热系数)，以及大振幅的多维流动。不同的项目还将解决人们在计算大型或多维解以及包含真空的流动时面临的重大数值挑战。反之亦然，精确的解将被用来作为各种计算代码的基准。该项目旨在更好地理解非线性机制如何与物理模型中广泛使用的方程中的一维或多维效应相互作用，从材料的性质到流体流动和气象学。虽然这些问题具有独立的理论意义，但它们在科学和工程应用方面也具有明显的重要性。其中几个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、建模和数值计算相结合的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.