Some Mathematical Problems in Fluid Dynamics

本项目研究与某些偏微分方程系统（如海洋的原始方程和Navier-Stokes方程）解的存在性、规律性、定性特征和大时间行为以及对物理参数的渐近依赖有关的各种数学问题。特别是，研究者研究了海洋的无粘原始方程，并发展了从收敛的观点分析的正则化方案。该项目的第二个目的是研究原始方程的随机扰动。这包括研究带有乘性噪声的随机原始方程的奇异摄动问题，以及随机边界问题。该项目解决了与流体流动的重要物理模型有关的各种数学问题，如Navier-Stokes方程和海洋的原始方程。这些方程是地球物理流体动力学和气候预报的基本方程。基于物理基础和收集的实验数据，大气和海洋湍流涉及广泛的空间和时间尺度，这使得包括海洋和大气大尺度环流在内的气候模型很难进行分析研究，而计算研究则非常昂贵。该项目旨在为这些模型提供一个严格的数学框架，并确定其适用范围的可能限制。它提供了足够的近似解，并严格地建立了这些近似的有效性。该项目还在随机扰动存在的情况下研究这些重要的地球物理模型。

ZianeDMS-1109562 This project addresses various mathematical questions relating to the existence, regularity, qualitative features, and large-time behavior of solutions and to the asymptotic dependence on physical parameters of certain systems of partial differential equations, such as the primitive equations of the ocean and the Navier-Stokes equations. In particular, the investigator studies the inviscid primitive equations of the ocean and develops regularizing schemes that are analyzed from the point of view of convergence. A secondary aim of the project is to investigate stochastic perturbations of the primitive equations. This includes the study of singular perturbation problems of the stochastic primitive equation with a multiplicative noise, as well as stochastic boundary problems. This project addresses various mathematical questions relating to important physical models of fluid flows, such as the Navier-Stokes equations and the primitive equations of the ocean. These equations are the fundamental equations of geophysical fluid dynamics and climate prediction. It is well established, based on physical grounds and collected experimental data, that atmospheric and oceanic turbulent flows involve a broad spectrum of spatial and time scales, which makes climate models that include large-scale circulations of flows in the ocean and atmosphere too difficult to study analytically and extraordinarily expensive to study computationally. The project seeks to provide a rigorous mathematical framework for these models and to determine possible limits on the range of their applicability. It provides adequate approximate solutions and establishes rigorously the validity of these approximations. The project also studies these important geophysical models in the presence of random perturbations.