Abstract

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows where they often emerge on time-scales much shorter than the viscous time scale, and then dominate the dynamics for very long time intervals. We give a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed in a variety of settings. Using the so-called hypocoercive properties of the linearized operator, we show that there is an invariant subspace in which there is fast decay. Thus, we provide rigorous justification for the existence of multiple time-scales and for the role that stationary solutions of the Euler equations play in serving as metastable states. This is joint work with C. Eugene Wayne (Boston University).