Mathematical Sciences: Mathematical Modelling in Fluid Dynamics

该项目的研究方向是分析和计算在流体动力学数学建模问题中的应用。所研究的具体问题是从磁流体力学和不可压缩流体动力学中得出的。本文将采用一种通用的、统一的变分方法来解决关于静磁平衡和具有一个空间对称的稳定涡旋流动的存在性、规律性和稳定性的各种问题。这些问题涉及非线性椭圆特征值问题，由基本物理原理表述为具有许多非线性约束的优化问题。新的迭代方法来解决这类变分问题将发展，其收敛性质将分析研究。类似的方法将被用于改进非线性投影技术来求解控制演化方程的数值解。研究还涉及了流体中的某些非线性波动，特别是密度分层流体中的内波和旋流中的惯性波。研究的目的是通过研究相关的非线性偏微分方程来提高流体动力学的理论方面。＊＊＊

Research on this project is directed toward the application of analysis and computation to mathematical modeling problems in fluid dynamics. Specific problems under study are drawn from magnetohydrodynamics as well as the dynamics of incompressible fluids. A general and unified variational approach will be employed to address a variety of questions concerning the existence, regularity and stability of both magnetostatic equilibria and steady vortex flows having one spatial symmetry. These questions involve nonlinear elliptic eigenvalue problems formulated from basic physical principles as optimization problems with many nonlinear constraints. New iterative methods for solving this class of variational problems will be developed, and their convergence properties will be examined analytically. Similar methods will be adapted to improve numerical solvers for the governing evolution equations by a nonlinear projection technique. The investigations also address certain nonlinear wave motions in fluids, especially internal waves in density- stratified fluids and inertial waves in swirling flows. The goal of the research is to enhance the theoretical aspects of fluid dynamics by studying the relevant nonlinear partial differential equations. ***