Strong Shock Waves in Cosmology, General Relativity, and Classical Fluids

点击翻译按钮获取中文摘要

This project investigates fundamental open problems in the theory of nonlinear partial differential equations concerning large amplitude or highly oscillatory solutions that generate shock waves. The first part of the project addresses both mathematical and cosmological issues related to shock wave propagation in the Einstein equations of general relativity. The second part of the project addresses unsolved problems regarding oscillatory and resonant interactions of strong waves in classical fluids, particularly the Euler equations for a compressible fluid.Shock waves are steep fronts that develop and propagate in fluids. Mathematical models of such fluid flow involve highly nonlinear differential equations that have solutions with very singular structure, which is difficult to analyze. For large amplitude or highly oscillatory solutions that generate shock waves, there is no general mathematical theory that applies, and many fundamental problems remain unsolved. Such problems are the topics of this research project, which develops the mathematical theory of gravitational shock waves and of shock waves in compressible fluids. The work has potential application to cosmological models of the universe and to modeling of transonic flow for aircraft design.