Some Mathematical Problems in Fluid Dynamics

ZIANE DMS-1109562这个项目解决了与解的存在性、正则性、定性特征和大时间行为有关的各种数学问题，以及某些偏微分方程组对物理参数的渐近依赖性，例如海洋的原始方程和纳维尔-斯托克斯方程。特别是，研究人员研究了海洋的无粘性原始方程，并开发了正则化方案，从收敛的角度进行了分析。该项目的第二个目标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摄动问题，以及随机边界问题。这个项目解决了与重要的流体流动物理模型有关的各种数学问题，例如纳维尔-斯托克斯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测的基本方程。根据物理基础和收集的实验数据，公认的是，大气和海洋的湍流涉及广泛的空间和时间尺度，这使得包括海洋和大气中流动的大规模环流的气候模型难以分析研究，而且计算研究的费用过高。该项目力求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。它提供了充分的近似解，并严格地证明了这些近似解的有效性。该项目还研究了在存在随机扰动的情况下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.