Prototype Systems of Multidimensional Conservation Laws

这个项目研究守恒律系的二维黎曼问题，包括气体动力学的欧拉方程。多维守恒定律的理论还处于初级阶段，而对黎曼问题的分析是指出多维系统中出现的奇点类型的第一步。最近的数值模拟已经提供了在马赫茎形成点上意外奇异行为的证据。这些研究需要分析证实，并将从简单的原型例子中获得洞察力。在自相似方法中，系统改变类型--远离原点的双曲型(“超音速流”)和靠近原点的混合双曲型-椭圆型(“亚音速区”)。对亚声速解的分析引出了具有混合边界条件(Dirichlet和斜导数)的混合型偏微分方程解的自由边界问题。以前的工作在激波反射点附近局部地解决了这类自由边界问题，但只针对一个简单的相互作用(规则反射)，并且只针对简化的系统(其中椭圆型方程与双曲型系统解耦)。目前的项目将为更复杂的问题开发新的技术，当边界条件不是一致倾斜的，当亚音速系统中的方程没有解耦时。最后，对发散稀疏面问题的一个原型实例的研究将有助于揭示奇异的马赫杆形成点。守恒定律模拟了空气动力学和连续介质力学中的基本问题。更好的理解在现实世界的应用中是有影响的。例如，构成天气和气候预测以及核反应堆安全分析基础的大规模数值模拟是基于守恒定律形式的偏微分方程组。理论上的进步，特别是在奇异行为分析方面的进步，将会使数字代码变得更有效、更可靠。在开展这项研究时，研究人员将在成功的基础上向初级研究人员，包括在数学中代表性不足的群体成员，介绍守恒定律理论，并扩大他们的职业视野。

This project studies two-dimensional Riemann problems for systems of conservation laws, including the Euler equations of gas dynamics. The theory of multi-dimensional conservation laws is in its infancy, and this analysis of Riemann problems is a first step that will indicate the types of singularities that arise in multi-dimensional systems. Recent numerical simulations have presented evidence of unexpectedly singular behavior at the formation points of Mach stems. These studies call for analytical confirmation, and insight will be gained from simple prototype examples. In the self-similar approach, the systems change type -- hyperbolic far from the origin ("supersonic flow") and of mixed hyperbolic-elliptic type near the origin ("subsonic region"). Analysis of the subsonic solution gives rise to free boundary problems in mixed-type partial differential equations with mixed boundary conditions (Dirichlet and oblique derivative). Earlier work solved such free boundary problems locally near shock reflection points, but only for one simple interaction (regular reflection) and only for simplified systems (where the elliptic equation decoupled from the hyperbolic system). The current project will develop new techniques for the more complicated problems that arise when the boundary condition is not uniformly oblique, and when the equations in the subsonic system do not decouple. Finally, study of a prototype example of diverging rarefactions problems will shed light on singular Mach stem formation points. Conservation laws model fundamental problems in aerodynamics and continuum mechanics. Better understanding has consequences in real world applications. For example, the large-scale numerical simulations that form the basis of weather and climate prediction and of nuclear reactor safety analysis are based on partial differential equations in the form of conservation laws. Theoretical advances, particularly in the analysis of singular behavior, will find their way into making numerical codes more efficient and more reliable. In carrying out this research, the investigators will build on success in introducing beginning researchers, including members of groups underrepresented in mathematics, to the theory of conservation laws, and in expanding their career horizons.