Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

Finite number of determining parameters for the Navier-Stokes equations with applications into feedback control and data assimilation

Edriss Titi

The Weizmann Institute and Texas A&M

In this talk we will implement the notion of finite number of determining parameters for the long-time dynamics of the Navier-Stokes equations (NSE), such as determining modes, nodes, volume elements, and other determining interpolants, to design finite-dimensional feedback control for stabilizing their solutions. The same approach is found to be applicable for data assimilation of weather prediction. In addition, we will show that the long-time dynamics of the NSE can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named {it determining form}, which is governed by a globally Lipschitz vector field. The NSE are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs.

This is a joint work with A. Azouani, H. Bessaih, C. Foias, M. Jolly, R. Kravchenko and E. Olson.