In the recent preprint with Jen Beichman we considered the 2D Euler evolution on the tube . Each rectangle is a steady state. We proved that if is sufficiently large, then these steady states are stable for all time. For example, if one takes a patch such that is small, then the Euler evolution of this patch will have small for arbitrary . This result generalizes analogous statement for the stability of the disc on the plane (proved by Sideris-Vega). As Sideris and Vega, we used the method of V. Arnold. The idea of this method is to study the variational problem associated to the conserved quantities. In our case, these are

where is a stream function given by Then, we set up a variational problem with constraint

We proved that the global minimizer is and if is close to the minimum value for some patch , then is close to in a weak topology. This essentially gives the required stability.

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