On maximally mixed equilibria of two-dimensional perfect fluids

Abstract

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution ${\omega }_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of ${\omega }_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy ${\mathsf{E}}_0$ corresponding to ${\omega }_0$. The set of said vorticity fields, denoted by ${\overline{{\mathcal{O}}_{{\omega }_0}}}^{\ast }\cap \{\mathsf{E}={\mathsf{E}}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in ${\overline{{\mathcal{O}}_{{\omega }_0}}}^{\ast }\cap \{\mathsf{E}={\mathsf{E}}_0\}$ is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of ${\overline{{\mathcal{O}}_{{\omega }_0}}}^{\ast }\cap \{\mathsf{E}={\mathsf{E}}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit.

38 pages. Corrected the proof of Lemma 5.3 by introducing a refined result in Lemma 5.4. Updated the definition of minimal flows; minor typos fixed

References in corpus (4)

• Inviscid dynamical structures near Couette flow
• Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions
• Local structure of the set of steady-state solutions to the 2D incompressible Euler equations
• Late states of incompressible 2D decaying vorticity fields

Cited by in corpus (6)

• Geometric Hydrodynamics in Open Problems
• Two-dimensional fluids via matrix hydrodynamics
• A brief introduction to matrix hydrodynamics
• Entropy maximization in the two-dimensional Euler equations
• Decay of Time Correlations in Point Vortex Systems
• Statistical equilibrium principles in 2D fluid flow: from geophysical fluids to the solar tachocline