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Title:
On maximally mixed equilibria of two-dimensional perfect fluids
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Abstract:
The motion of a two-dimensional incompressible and inviscid fluid can be described as an area-preserving rearrangement of the initial vorticity that preserves the kinetic energy. In the infinite time limit, some irreversible mixing can occur and predicting what structures can persist is an issue of fundamental importance. 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 a set containing Euler's end states, is maximally mixed. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, in the straight channel, we give examples of open sets of initial data which can be arbitrarily close to any shear flow in L1 of vorticity but do not weakly converge to them in the long time limit.
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