In the recent preprint with Jen Beichman we considered the 2D Euler evolution on the tube S=\mathbb{R}\times \mathbb{T}. Each rectangle \Omega_L=[-L,L]\times\mathbb{T} is a steady state. We proved that if L is sufficiently large, then these steady states are stable for all time. For example, if one takes a patch \Omega such that |\Omega\Delta\Omega_L| is small, then the Euler evolution \Omega(t) of this patch will  have |\Omega(t)\Delta \Omega_L| small for arbitrary t. 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

I_0=|\Omega(t)|,\,\, I_1=\int_{\Omega(t)} xdxdy,\,\, I_3=\int_{\Omega(t)}\psi dxdy

where \psi is a stream function given by \psi=\Delta^{-1}\chi_{\Omega(t)}. Then, we set up a variational problem with constraint

I_3\to\min,\,  I_2=0,\, I_1=4\pi L

We proved that the global minimizer is \Omega_L and if I_3(E) is close to the minimum value for some patch E, then E is close to \Omega_L in a weak topology. This essentially gives the required stability.

 

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