Problems in Fluid Mechanics and Related Nonlinear Partial Differential Equations and Applications

该奖项将支持流体力学和相关偏微分方程的数学问题及其应用的研究，以及有效数值方法的分析和发展。本研究计划的目标有三个：研究具有各种本构关系的可压缩流体、渐近问题、多相问题、跨声速流动等重要非线性问题，并开发稳定高效的数值算法；研究非线性偏微分方程的耗散机制，特别是可压缩粘性气体流动的耗散机制；并为这些问题的研究分析和发展新的理论技术。流体动力学的问题出现在许多科学和工程领域，例如气体动力学、水力学、弹性、塑性、燃烧、磁流体动力学、多相流、相变、半导体中的电荷输运、离子的生物输运、蚀刻和沉积过程。物理过程通常由某些非线性偏微分方程组控制。该奖项将支持研究这些非线性系统的可解性及其解的定性行为，以及有效的理论和数值方法的分析和发展。这将有助于更好地理解这些流动现象及其支配规律。

The award will support research on mathematical problems in fluid mechanics and related partial differential equations and their applications, along with the analysis and development of efficient numerical methods. The objective of this research program is threefold: investigate important nonlinear problems including compressible fluids with various constitutive relations, asymptotic problems, multiphase problems, transonic flows, and develop stable and efficient numerical algorithms; study dissipation mechanisms for nonlinear partial differential equations, especially for compressible viscous gas flow; and analyze and develop new theoretical techniques for the study of these problems.Problems in fluid dynamics arise in many areas of science and engineering,for example in gas dynamics, hydraulics, elasticity, plasticity, combustion, magnetohydrodynamics, multiphase flow, phase transition, charge transport in semiconductors, biological transport of ions, and etching and deposition processes. The physical processes are typically governed by certain nonlinear systems of partial differential equations. The award will support research onthe solvability of these nonlinear systems and the qualitative behavior of their solutions, as well as on the analysis and development of efficient theoretical and numerical methods. This will lead to a better understanding ofthese flow phenomena and their governing laws.