A fundamental challenge faced in fluid dynamics is that the basic governing equations do not account for persistent, randomly driven fluctuations that arise due to uncertainties in modeling from empirical, numerical and physical sources. In view of the wide progress made in computation and given the ubiquity of probabilistic methods in applications there is a clear need to better understand the numerical and analytical underpinnings of stochastic partial differential equations (SPDEs) in general and in relation to the basic equations of fluid dynamics. This project seeks to further the development of numerical and analytical tools for SPDEs with a view towards novel applications in climate and weather modeling and for the study of turbulence.
The work will focus in particular on three emerging areas of nonlinear SPDEs. Firstly we will address the theory of invariant measures and statistically steady states. In particular we will develop tools to study inviscid limits in the class of invariant measures for the Navier-Stokes equations and for related models arising in geophysical and turbulence applications. We will also identify and characterize noise regimes leading to ergodic and mixing properties for some previously unaddressed classes of nonlinear SPDEs. A second portion of the project will develop parameter estimation and inverse modeling techniques for noisy nonlinear PDEs. The third portion of the project is devoted to the numerical analysis of nonlinear SPDEs and to developing related analytical tools needed for this purpose.
