Solutions of Nonlinear Hyperbolic and Mixed Type Partial Differential Equations

当飞机的飞行速度接近音速(所谓的跨音速状态)时，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；这两者被一个表面隔开，在那里，气流完全是音速的(即具有音速)。音速表面附着在翼型上，当音速表面与翼型相交时，流动的特性，如阻力和升力，沿翼型变化很大。这会导致翼型产生严重的拉力。这一现象的数学描述是由混合型偏微分方程(PDE)给出的，这是本项目的主要主题。类似的方程在流体和量子力学、广义相对论、生物科学和等离子体物理中也有其他应用。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，尽管最近对混合型偏微分方程组进行了广泛的研究，但关于其解的行为的许多基本数学问题仍然没有解决。在这个项目中，首席调查员将研究更简单、经常使用的系统和初始数据类别，这些系统和类别在理解波相互作用、解的渐近行为及其稳定性方面起着至关重要的作用。该项目还将作为研究生和本科生的培训基地，他们将为这项研究做出贡献。我们将研究几个PDE模型系统，它们的解涉及所谓的奇异激波，其中至少一个状态变量以加权狄拉克增量函数的形式发展出一个极端集中。这些都可以作为积木，以获得更广泛的全球知识、洞察力和视角，以便及时了解在一个空间层面上守恒律系统的大解决方案的存在。几个空间维度的微分方程式也将被考虑。球对称解的简单情况是本项目的主要主题之一。这个项目将使用动力系统、几何、调和和傅立叶分析的各种工具。这项研究的最终目标是找到提供可压缩流体流动模型信息的构建块，并可以在定义良好的构建方案中使用，以近似任何初始数据的解。

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.