This project developed novel methods to analyze stochastic versions of fundemental equations arrising in geophysical scale fluid dynamics. We derived a well-possedness theory (existence-uniqueness-continuous dependence on data) for several systems: the Primitive equations of the oceans and atmosphere and for the Euler and Zakharov-Kuznestov equations. We also addressed certain parameter estimation problems for these and related fluids systems. Detailed descriptions of these projects (divided into 5 subcategories) are below. Extensive conference participation and organization were carried out to promote the disemination of these ideas. Two Ph.d student were mentored at Indiana University during the period of the fellowship. [1] Stochastic Partial Differential Equations In Climate Modeling Description: The Primitive equations are a basic model in geophysical fluid dynamics which serve as the analytical core of advanced numerical climate models. We have developed a more of less complete mathematical theory of the Stochastic Primitive Equations. In the course of this work we developed mathematical tools that have proven use for the study of other nonlinear stochastic partial differential equations (see the projects concerning the Euler and Zakharov-Kuznestov equa- tions below). Current work in progress seeks to develop the numerical analysis of these equations. The project was motivated initially by interactions with a climate scientists [2] Parameter Estimation for Nonlinear Stochastic Partial Differential Equations Description: We have developed methods for parameter estimation for nonlinear SPDEs. Previous results in this area only applied to linear partial differential equations and we had to develop new methods both for the derivation of estimators and for their rigorous analysis. Although our initial results were focused on the 2D Stochastic Navier-Stokes Equations with additive noise, the methods developed are more general and are currently being applied to other classes of equations. [3] Invariant Measures for Dissipative Dynamical Systems Description: We have study a class of invariant measures associated to time averaged observations of a very generic class of dissipative dynamical systems. The work extends to a general context a framework used in the study of turbulent fluids. We consider applications for systems with memory which have a more hyperbolic character. [4] Inviscid Fluids and the Stochastic Euler’s Equations Description: We develop the basic local (and in 2D global) well-posedness theory for a stochastic version of the Euler equations. The work identifies situations where noise can be shown to extend the possible existence time of solutions. [5] Stochastic Partial Differential Equations in with Disspersive Structure. Description: We study of stochastic version of the Zakharov-Kuznestov equations an important model in plasma physics which has some features in common with the KdV equations.
