Mathematical Sciences: Nonlinear Partial Differential Equations and Fluid Dynamics and Mechanics

这是一个研究控制可压缩流体流动的方程的激波和其他不连续解的项目。例如，关于耗散机制在两个或多个激波的相互作用中所起的作用以及这些激波的长期稳定性，仍有许多未解之谜。首席研究员将使用分析和渐近工具来回答这些问题和相关问题。除了通常的流体动力学方程外，首席研究员还将研究磁流体动力学方程系统、气体动力学理论和浅水波浪理论。在流体动力学的某些问题中，摩擦力等耗散力可能很小，但其作用不可忽视。例如，气体的粘度（内摩擦）非常小，但它会影响穿过气体的物体的大规模行为。首席研究员将研究这种小的耗散机制如何影响气体中冲击波的形成和传播。冲击波有点像音爆，它是高速流动的一个重要特征。

This is a project to study shock wave and other discontinuous solutions of the equations that govern the flow of compressible fluids. For example, there are still many unanswered questions regarding the role that dissipative mechanisms play in the interaction of two or more shock waves and the long-time stability of such waves. The principal investigator will use analytic and asymptotic tools to answer these questions and related ones. In addition to the usual equations of fluid dynamics the principal investigator will study systems of equations in magneto-hydrodynamics, the kinetic theory of gases, and the theory of shallow water waves. Dissipative forces like friction can be small in certain problems in fluid dynamics, but their effect cannot be neglected. For example, the viscosity (internal friction) of a gas is very small and yet it affects the large-scale behavior of objects that move through the gas. The principal investigator will study how such small dissipative mechanisms can influence the formation and propagation of shock waves in gases. A shock wave is something like a sonic boom, and it is an important feature of flows at high speeds.