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Solutions of Nonlinear Hyperbolic and Mixed Type Partial Differential Equations

非线性双曲混合型偏微分方程的解

基本信息

批准号：
1714912
负责人：
Charis Tsikkou
金额：
$ 14.1万
依托单位：
West Virginia University Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714912&HistoricalAwards=false
关键词：
Solutions Nonlinear Hyperbolic Mixed Type

项目摘要

When an airplane's flight speed is close to the speed of sound (the so-called transonic regime), the airflow around the plane has very unusual properties: in some areas the flow is supersonic, and in other areas the flow is subsonic; these two are separated by a surface where the flow is exactly sonic (i.e., has the speed of sound). The sonic surface is attached to the airfoil, and characteristics of the flow, such as drag and lift, change dramatically along the airfoil when the sonic surface is crossed. This causes a serious strain experienced by the airfoil. Mathematical description of this phenomenon is given by partial differential equations (PDE) of mixed type that are the main subject of this project. Other applications of similar equations are found in fluid and quantum mechanics, general relativity, bio-sciences, and plasma physics. Although these equations are widely used and serve as a foundation of some engineering tasks, for example, of the computer-aided design in aerodynamics, among others, and despite recent extensive studies of mixed type PDE, many fundamental mathematical questions concerning the behavior of their solutions are still unresolved. In this project, the Principal Investigator will study simpler, frequently used systems and classes of initial data that play a paramount role in understanding the wave interaction, asymptotic behavior of solutions and their stability. This project will also serve as a training ground for graduate and undergraduate students who will contribute to this research. Several model systems of PDE will be studied whose solutions involve the so-called singular shocks, where at least one state variable develops an extreme concentration in the form of a weighted Dirac delta function. These can be used as building blocks for gaining a broader knowledge, insight and perspective on global in time existence of large solutions to systems of conservation laws in one spatial dimension. Differential equations in several spatial dimensions will also be considered. Simpler cases of solutions with spherical symmetry are among the principal topics of this project. Diverse tools from dynamical systems, geometry, harmonic and Fourier analysis will be used in this project. The ultimate goal of this research is to find building blocks that provide information on models of compressible fluid flow and can be used in a well-defined construction scheme to approximate solutions for any initial data.

当飞机的飞行速度接近音速时（所谓的跨音速区域），飞机周围的气流具有非常不寻常的特性：在某些区域，气流是超音速的，而在其他区域，气流是亚音速的;这两者被一个表面分开，在这个表面上，气流完全是音速的（即，具有声音的速度）。音速面附着在翼型上，当音速面穿过翼型时，气流的特性，如阻力和升力，沿翼型沿着发生显著变化。这导致翼型件经受严重的应变。这一现象的数学描述是由混合型偏微分方程（PDE），这是本项目的主要课题。类似方程的其他应用在流体和量子力学、广义相对论、生物科学和等离子体物理学中。尽管这些方程被广泛使用并作为一些工程任务的基础，例如空气动力学中的计算机辅助设计等，并且尽管最近对混合型偏微分方程进行了广泛的研究，但许多有关其行为的基本数学问题解仍然没有得到解决。在这个项目中，主要研究者将研究更简单，经常使用的系统和初始数据类，在理解波的相互作用，解的渐近行为及其稳定性方面发挥着至关重要的作用。该项目也将作为研究生和本科生谁将有助于这项研究的培训基地。几个模型系统的偏微分方程将研究其解决方案涉及所谓的奇异冲击，其中至少有一个状态变量开发一个极端浓度的形式加权狄拉克δ函数。这些可以被用来作为积木获得更广泛的知识，洞察力和全球的时间存在的大型解决方案，在一个空间维度的守恒定律系统的观点。在几个空间维度的微分方程也将被考虑。球对称解的简单情况是这个项目的主要课题之一。在这个项目中将使用动力系统，几何，谐波和傅立叶分析的各种工具。本研究的最终目标是找到构建块，提供可压缩流体流动模型的信息，并可用于一个定义明确的建设计划，以近似任何初始数据的解决方案。