New Well-Posedness Results for the Stochastic Equations of Geophysical Fluid Dynamics
Department of Mathematics
The Primitive Equations are widely regarded as a fundamental description of geophysical scale ﬂuid ﬂow. They provide the analytical core of large General Circulation Models (GCMs) that are at the forefront of numerical simulations of the earths ocean and atmosphere. In view of the wide progress made in com- putation the need has appeared to better understand and model some of the uncertainties which are contained in these GCMs. This is the so called problem of parametrization. Besides all of the physical forms of parametrization, stochastic modeling has appeared as one of the major modes in the contemporary evolution of the ﬁeld. In this context there is a clear need to better understand the numerical and analytical underpinnings of stochastic partial differential equations.
Although the study of well posedness for nonlinear stochastic evolution systems such as the Navier-Stokes equations goes back to the 1970s, many basic questions are still open. In particular the case of nonlinear multiplicative noise presents challenging problems since the equations are not easily transformed into more classical PDE. In this talk we introduce some recently developed techniques which may be used to circumvent the diﬃculty of compactness. In particular our techniques have led to novel local and global existence results concerning pathwise solutions for both the Navier-Stokes and Primitive Equations.
This is joint work with M. Ziane and R. Temam.