Monday, 17 February, 2014
SPEAKER: Prof. Juraj Foldes, IMA and UMN
TITLE: Ergodic and mixing properties of randomly forced equations
ABSTRACT: Due to sensitivity with respect to initial data and parameters, individual solutions of the basic equations of fluid mechanics are unpredictable and seemingly chaotic. However, some of their statistical properties of solutions are robust. As early as the 19th century it was conjectured that turbulent flow cannot be solely described by deterministic methods, and indicated that a stochastic framework should be used. In this framework, invariant measures of the stochastic equations of fluid dynamics presumably contain the statistics posited by the basic theories of turbulence.
We discuss the existence, uniqueness and attraction properties of an ergodic invariant measure for the Boussinesq Equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales. The central challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. Towards this aim we encounter a Lie bracket structure in the associated vector fields with a complicated dependence on solutions. This leads us to develop a novel Hormander-type condition for infinite-dimensional systems.