Mathematical Sciences: Nonlinear Partial Differential Equations in Continuum Mechanics

9600137 Xin PI提议研究连续介质力学和其他自然科学分支中产生的一类非线性偏微分方程，包括可压缩和不可压缩、无粘和粘性流体的Euler和Navier-Stokes方程、导电可压缩粘性流体的电磁场动力学方程、弹性方程、动力学理论中的非线性Boltzmann型方程。以及燃烧、多相流等方程。主要目的是了解激波、膨胀波、孤子、扩散波、涡片和边界层等基本非线性波的动力学和相互作用，这些波在一般流动的研究中起着重要的作用。特别地，建议分析小尺度耗散或松弛过程对大尺度物理量及其运动的影响。其他问题如强解和弱解的整体存在性、基本波型的非线性稳定性、弱熵解的唯一性、解的层行为、不连续流计算的数值方法、动力学模型的流体动力极限等也将被探讨。计划继续目前的努力，扩大现有的技术，并开发新的工具来研究其中的一些问题。其中一个主要思想是通过考虑潜在不连续流的更多内部结构来改进用于研究光滑流动或非松弛系统的传统方法，并从理论上和数值上研究这些问题。由物理平衡定律引起的偏微分方程的非线性演化支配着自然界中许多最广泛遇到的现象。这些非线性系统的解决方案仍然是波传播研究的中心主题，也是理解基本物理现象的基础，这些现象包括可压缩和不可压缩流体、声波和电磁散射，以及非线性复合材料、反应气体混合物、光纤、磁流体力学和广义相对论中的波传播。本课题拟研究这些非线性系统的一些基本问题，这些问题直接受到图像处理、航空、复合材料、声纳、雷达、医学成像、地球物理勘探、天气预报、油藏模拟等许多领域的重要应用的启发。例如，研究设计混合型偏微分方程尖锐锋面的高分辨率多维数值计算方法，对油藏模拟、复合材料设计、成像处理以及水和空气中油等污染物的传播具有重要意义；对边界层理论的研究是理解湍流、设计超音速飞机等的关键因素之一。因此，本项目的完成对上述领域的技术改进和新发展具有重要意义。*＊*

9600137 Xin The PI proposes to study a class of nonlinear partial differential equations arising from Continuum Mechanics and other branches of natural sciences, which includes the Euler and Navier-Stokes equations for both compressible and incompressible, inviscid and viscous fluids, the equations of electro-magneto field dynamics for electrically conducting compressible viscous fluids, the equations of elasticity, nonlinear Boltzmann type equations in kinetic theory, and equations for combustion, multi-phase flows, etc. The main goal is to gain understanding of the dynamics and interactions of basic nonlinear waves such as shock waves, expansion waves, solitons, diffusive waves, vortex sheets, and boundary layers, which play important roles in the study of the general flows. In particular, it is proposed to analyze the effects of the small scale dissipation or relaxation processes on large scale physical quantities and their motions. Other questions such as global existence of both strong and weak solutions, nonlinear stability of basic wave patterns, uniqueness of weak entropy solutions, layer behavior of solutions, numerical methods for the computation of discontinuous flows, and fluid-dynamic limit for kinetic models are also to be explored. It is planned to continue the current effort to expand the existing techniques and to develop new tools for studying some of these problems. One of the main idea is to refine the conventional methods designed to study smooth flows or unrelaxed systems by taking into account more internal structures of the underlying discontinuous flows, and study these problems both theoretically and numerically. %%% Nonlinear evolution of partial differential equations arising from physical balance laws govern many of the most widely encountered phenomena in nature. Solutions to these nonlinear systems remain a central topic in the study of wave propagations and are fundamental to the understanding of basic physical phenomena ranging from compressible and incompressible fluids, acoustic waves and electromagnetic scattering to wave propagations in nonlinear composite materials, in reactive gas mixtures, in fiber optics, as well as in magnetohydrodynamics and general relativity. In this project, It is proposed to study some of the basic questions for these nonlinear systems, which are motivated directly by important applications to many fields such as image processing, aeronautics, composite material, sonar, radar, medical imaging, geophysical exploration, weather prediction, oil reservoir simulations etc.. For example, the proposed study of design of high resolution multi-dimensional numerical methods for calculating sharp fronts for mixed type partial differential equations is extremely important for the oil reservoir simulations, design of composite materials, imaging processing, and propagation of pollutants such as oil in water and air; the proposed study of boundary layer theory is one of the key ingredients in the understanding of turbulence, and the design of supersonic aircrafts, etc.. Thus the accomplishment of this project should be of considerable importance in the improvement and new development of technologies in the above mentioned fields. ***