Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations

Details Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations

2011-08-17

arxiv.org/abs/1108.3416v1

Mathematics

Analysis of PDEs

21 pages, 2 figures

Scientific paper

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. In this paper 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 as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. Using the so-called hypocoercive properties of the linearized operator, it is shown that there is an invariant subspace on which this operator produces decay at a rate much faster than the viscous convergence rate. Thus, we provide rigorous justification in this context for the existence of multiple time-scales in the evolution of the two-dimensional incompressible Navier-Stokes equation on the torus and for the role that stationary solutions of the Euler equations play in serving as metastable states.

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