Collaborative Research: Deterministic and Statistical Relations Between the Navier-Stokes Equations and Its Determining Forms

University Of Maryland Baltimore County, Baltimore MD

Investigators

Abstract

In this project, the principal investigators study key open problems concerning the fundamental equations governing the atmosphere and ocean with a view towards applications in meteorology and weather forecasting. These equations describe fluid motions for gases and liquids under quite general conditions, from laminar to turbulent flows, on scales ranging from below a millimeter to astronomical lengths. Consequently, their study has wide-ranging applications in aeronautical sciences, in meteorology, in the petroleum industry, in plasma physics, and more recently, in biophysical fluid dynamics. The inherent nonlinear and multi-scale nature of these equations make the problem of weather forecasting and understanding climate evolution challenging. The investigators develop analytical and computational tools to study the long-term behavior of the fundamental governing equations of the atmosphere and climate. A main focus is to connect recently developed techniques of data assimilation with the theory of statistical solutions of these equations. This in turn facilitates the incorporation of vast amounts of weather/climate data, collected over a long period of time, into mathematical models of weather and climate, potentially leading to improved statistical prediction. Graduate students are engaged in the work of the project. The investigators study several aspects of turbulent flows, including the time of existence of smooth solutions, statistical properties, and the long-term dynamics on the strong or weak global attractor, in a new class of functional spaces that is contained in all Sobolev classes. A key feature of this class is that the nonlinear evolution equation satisfies a differential inequality that is almost linear. This functional class has the property that it can absorb derivatives, which has the effect of making the nonlinear term milder, with potential applications to a wide class of nonlinear evolution equations. For certain types of initial data, this yields a better existence time for the Navier-Stokes equations than some well-known Sobolev functional classes. This new class of functional spaces is employed to study the finite-dimensional properties of statistical solutions of the Navier-Stokes equations, via the determining modes, nodes, and determining forms. Furthermore, the investigators develop data assimilation techniques for statistical solutions of hydrodynamic equations. Statistical solutions are directly related to the average physical quantities with which engineers are concerned. They provide a bridge from the time average of physical observables such as energy, enstrophy, and heat transfer to ensemble measures on the phase space. The techniques developed in this project should have immediate engineering and scientific applications.