Nonlinear Problems in Conservation Laws and Fluid Dynamics and Related Partial Differential Equations

项目编号：0204225项目负责人：陈桂强单位：美国西北大学项目：应用数学项目经理：Catherine mavriplis标题：守恒定律和流体动力学的非线性问题及相关偏微分方程的非线性问题及其应用的研究，以及有效的非线性方法的分析和发展。这个研究项目的目标是双重的：(1)研究重要的非线性问题，如多维跨音速激波和自由边界问题、真空问题、渐近稳定性问题、具有各种本构关系的可压缩流体、奇异极限问题、多相问题、黎曼问题等，以获得新的物理见解，指导有效的非线性方法的制定；寻找正确的函数空间来提出非线性守恒律，发展稳定快速收敛的数值方法；(2)分析和发展非线性方法，包括自由边界方法、动力学方法、几何测量方法、弱收敛方法、激波捕获技术、能量方法和相关势能技术，以制定新的、更有效的非线性方法，解决守恒定律和流体动力学中各种更重要的非线性问题。本研究项目涉及气体动力学、水力学、燃烧、磁流体力学、半导体、弹性、多相流、相变、动力学理论、生物物理学和材料科学等领域的非线性问题和相关偏微分方程。该奖项将支持对这些非线性问题和相关偏微分方程的可解性、其解的定性行为和相关应用的研究，以及应用分析和数值分析中非线性方法的分析和发展。这一研究将导致对非线性现象的更深入的理解，并将为应用提供更有效的非线性方法和理论。日期：2002年5月1日

DMS Award AbstractAward #: 0204225PI: Chen, Gui-QiangInstitution: Northwestern University Program: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Nonlinear Problems in Conservation Laws and Fluid Dynamics and Related Partial Differential EquationsThe investigator continues studies of nonlinear problems in conservation laws and fluid dynamics and related nonlinear partial differential equations and their applications, along with the analysis and development of efficient nonlinear methods. The objective of this research program is twofold: (1) to investigate important nonlinear problems such as multidimensional transonic shocks and free boundary problems, vacuum problems, asymptotic stability problems, compressible fluids with various constitutive relations, singular limit problems, multiphase problems, and the Riemann problem to gain new physical insights, to guide the formulation of efficient nonlinear methods, and to find the correct function spaces in which to pose the nonlinear conservation laws and develop the numerical methods that converge stably and rapidly; and (2) to analyze and develop nonlinear methods including free boundary methods, kinetic methods, geometric measure methods, weak convergence methods, shock capturing techniques, energy methods, and related potential techniques to formulate new, more efficient nonlinear methods and to solve various more important nonlinear problems in conservation laws and fluid dynamics.The nonlinear problems and related partial differential equations in this research program arise in such areas as gas dynamics, hydraulics, combustion, magnetohydrodynamics, semiconductor, elasticity, multiphase flow, phase transitions, kinetic theory, biophysics, and material science. The award will support research on the solvability of these nonlinear problems and related partial differential equations, the qualitative behavior of their solutions and related applications, as well as the analysis and development of nonlinear methods in applied analysis and numerical analysis. This research will lead to a deeper understanding of nonlinear phenomena and will provide more efficient nonlinear methods and theories for applications.Date: May 1, 2002