Mathematical Sciences: Mathematical Modelling in Fluid Dynamics

本课题的研究方向是应用 数学建模问题的分析与计算 流体动力学 正在研究的具体问题来自 磁流体动力学以及不可压缩的动力学 流体. 一个普遍和统一的变分方法将是 被用来解决各种问题， 静磁场的存在性、规律性和稳定性 平衡和稳定的涡流具有一个空间对称性。 这些问题涉及到非线性椭圆特征值 从基本的物理原理出发， 具有许多非线性约束的优化问题。 新 求解这类变分问题的迭代方法 将被开发，其收敛性能将是 分析性地检查。 类似的方法将适用于改善数值求解器 用非线性投影法求解控制演化方程 法 调查还涉及某些非线性 流体中的波动，特别是密度的内波， 旋流中的分层流体和惯性波。 目标 的研究是为了提高流体的理论方面 动力学通过研究相关的非线性偏微分 方程 ***

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. ***