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

点击翻译按钮获取中文摘要

This is a proposal 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 investigators will use analytic and asymptotic tools to answer these questions and related ones. In addition to the usual equations of fluid dynamics the principal investigators 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 investigators 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.