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

DMS奖摘要奖#：0204225PI：陈贵强西北大学研究所：应用数学项目经理：凯瑟琳·马夫里普利斯题目：守恒定律和流体动力学及相关偏微分方程中的非线性问题研究人员继续研究守恒定律和流体动力学中的非线性问题以及相关的非线性偏微分方程及其应用，并分析和开发有效的非线性方法。这项研究的目标是双重的：(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、各种本构关系的可压缩流体、奇异极限问题、多相问题和Riemann问题，以获得新的物理见解，指导有效的非线性方法的制定，并找到提出非线性守恒律的正确函数空间，并发展稳定和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、激波捕捉技术、能量方法和相关势能技术，以形成新的、更有效的非线性方法，并解决守恒定律和流体动力学中各种更重要的非线性问题。本研究计划中的非线性问题和相关偏微分方程组涉及到气体动力学、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、生物物理和材料科学等领域。该奖项将支持对这些非线性问题和相关偏微分方程解的可解性、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。这项研究将有助于加深对非线性现象的理解，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