Nonlinear Waves in One-Dimensional and Multi-Dimensional Conservation Laws

本研究计划的重点是研究一维和多维守恒律系统的解。尽管对受多维守恒定律支配的问题进行了广泛的实验和数值探索，但没有理论可以描述作为这些问题的解决方案而出现的非线性波的特性。PI和他的同事Keyfitz最近的研究结果表明，对于一大类二维问题，包括可压缩流的标准方程，对它们的自相似解的分析会导致许多结果被整个一类问题所共享。它们包括一个存在唯一性定理，适用于流动为超音速的区域，以及对流动从超音速变为亚音速的边界可能出现的奇点的描述。为了完成分析，需要一个跨音速激波位置的自由边界问题理论。这方面的初步结果令人鼓舞，PI计划继续朝这个方向进行研究。研究工具包括必须求解一类新型退化椭圆方程的自由边界问题的分析，以及奇点的渐近分析。完成后，该项目的结果将有助于解决多维波相互作用和激波反射问题领域的开放问题，其中包括斜面倾斜激波反射的存在性定理，二维初等波的分岔准则，以及用于捕获解中可能奇点的一般存在理论的正确函数空间。在一维守恒定律中，PI最近发现了一些新的现象，需要进一步的探索和理解。它们包括经典弱自相似黎曼解的不存在性和有界振幅、高频、振荡解取代经典黎曼解的存在性。利用补偿紧性方法证明了这些解在测度值意义上满足守恒律系统。了解不存在的原因，以及振荡的物理意义，是这个研究项目的目标之一。守恒定律是描述技术核心过程的数学方程，如高速流动、超音速射流，以及在环境工程和油藏模拟中出现的多孔介质流动。了解守恒律解的结构对于成功模拟这些现象至关重要。主要的困难在于守恒定律的解承认“激波”，它对应于流动性质的突然变化。激波影响高速流动的稳定性，在石油开采模拟中具有重要意义。此外，根据PI最近的发现，多维守恒定律承认解决方案中的“奇点”，这些解决方案在理论上尚未被理解，并且难以在数值上解决。它们的存在影响了模拟的结果。这个项目提出了一种理解这些奇点的原始方法。更一般地说，在这个项目中，PI提出了一种有组织的方法来发展一种通用理论，该理论将描述多维守恒定律（如描述高速流动的方程）的解的结构。由于研究一维守恒定律的方法不能推广到一个以上的空间维度，这一研究领域是广阔的。然而，由于几乎所有的技术过程都是多维的，因此发展这样一种理论势在必行。除了本建议的研究方面外，研究人员还成功地将涉及非线性守恒定律的工作方面应用于研究生教育，并建议在类似方向上继续进行这项工作。

The focus of this research proposal is on the study of solutions ofone-dimensional and multi-dimensional systems of conservation laws.In spite of an extensive experimental and numerical exploration of problemsthat are governed by MULTI-DIMENSIONAL conservation laws, there is no theorythat would describe properties of nonlinear waves that arise as solutions ofthese problems. Recent results by the PI and co-worker Keyfitz indicate thatfor a large class of two-dimensional problems, including standard equations ofcompressible flow, an analysis of their self-similar solutions leads to manyresults that are shared by the entire class of problems. They include an existence and uniqueness theorem that holds in the region where the flow is supersonic, and a description of possible singularities that may arise at the boundary where the flow changes from supersonic to subsonic. To complete the analysis, a theory of free-boundary problems for the positions of transonic shocks is needed. Preliminary results in this vein are encouraging, and the PI, plans to continue research in this direction. Tools of study include analysis of free-boundary problems for which a degenerate elliptic equation of novel type has to be solved, and asymptotic analysis of singularities. When completed, the results of this project will contribute to solving open problems in the field of multi-dimensional wave interactions and shock reflection problems, which include an existence theorem for oblique shock reflection by a ramp, bifurcation criteria for two-dimensional elementary waves, and the correct function space for a general existence theory that will capture possible singularities in the solution. In ONE-DIMENSIONAL conservation laws several novel phenomena, recently discovered by the PI, call for further exploration and understanding. They include nonexistence of classical weak self-similar Riemann solutions and the presence of bounded amplitude, high frequency, oscillatory solutions replacing the classical ones. Using compensated compactness methods these solutions were shown to satisfy the system of conservation laws in a measure-valued sense. Understanding what causes nonexistence, and what is the physical meaning of the oscillations, is one of the goals of this research project.Conservation laws are mathematical equations that describe processes central totechnology, such as high-speed flows, supersonic jets, as well as flows throughporous media which arise in environmental engineering and reservoir simulation.Understanding the structure of solutions of conservation laws is crucial for asuccessful simulation of these phenomena. The main difficulty lies in the factthat solutions to conservation laws admit ``shock waves'', which correspond tothe sudden, abrupt changes in the flow properties. Shock waves influence thestability of high-speed flows, and are crucial in the simulation of oil recovery. In addition, based on the recent PI's findings, multi-dimensional conservation laws admit ``singularities'' in the solutions that have not yet been theoretically understood and which are difficult to resolve numerically. Their presence influences the outcome of the simulations. This project proposes an original approach towards understanding these singularities. Even more generally, in this project the PI proposes an organized approach towards the development of a general theory that would describe the structure of solutions of multi-dimensional conservation laws such as the equations that describe high-speed flows. Because the methods used to study one-dimensional conservation laws cannot be generalized to more than one space dimension, this research area is wide open. However, since almost all processes in technology are multi-dimensional, the development of such a theory is an imperative. In addition to the research aspects of this proposal, aspects of the work involving nonlinear conservation laws have been used successfully by the investigator in graduate student education, and continuation of this effort in similar directions is proposed.