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"Finite-dimensionality in Fluids and Applications"

Fluids can be described by partial differential equations (PDEs). As such, their solutions are typically given by functions of several variables. In the modern theory of PDEs, solutions are viewed as belonging to vector spaces whose elements are functions; these spaces are typically infinite-dimensional. For instance, if one insists on solutions having “finite energy,” then it is often natural to assume that solutions should be elements of the vector space of square-integrable functions. This talk will present the fundamental equations of incompressible fluids in a simplified setting that allows one to see its infinite-dimensional nature, but nevertheless demonstrate that their long-time behavior is, in some sense, finite-dimensional. We will then discuss the importance of this property in the context of turbulence and weather prediction.