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.