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Instability of unidirectional flows for the 2D α-Euler equations

  • *Corresponding author

    *Corresponding author 

Dedicated to Prof. Tomás Caraballo on the occasion of his 60-th birthday

Partially supported by the USA NSF grant DMS-171098

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  • We study stability of unidirectional flows for the linearized 2D $ \alpha $-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $ \mathbf p \in \mathbb Z^{2} $. We linearize the $ \alpha $-Euler equation and write the linearized operator $ L_{B} $ in $ \ell^{2}(\mathbb Z^{2}) $ as a direct sum of one-dimensional difference operators $ L_{B,\mathbf q} $ in $ \ell^{2}(\mathbb Z) $ parametrized by some vectors $ \mathbf q\in\mathbb Z^2 $ such that the set $ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $ covers the entire grid $ \mathbb Z^{2} $. The set $ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $ can have zero, one, or two points inside the disk of radius $ \|\mathbf p\| $. We consider the case where the set $ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $ has exactly one point in the open disc of radius $ \mathbf p $. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator $ L_{B, {\mathbf q}} $ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.

    Mathematics Subject Classification: Primary: 35Q31, 76E05, 47A10, 40A15; Secondary: 35Q35, 35B35, 35P99.


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  • Figure 1.  $ {\bf p} = (3,1) $; point $ {\bf q}_1 = (-1,2) $ is a point of type $ I_0 $ (green $ \Sigma_{{\bf q}_1} $), point $ {\bf q}_2 = (-1,1) $ is a point of type $ II $ (blue $ \Sigma_{{\bf q}_2} $), point $ {\bf q}_3 = (0,-2) $ is a point of type $ I_+ $ (red $ \Sigma_{{\bf q}_3} $), and point $ {\bf q}_4 = (2,-2) $ is a point of type $ I_- $ (brown $ \Sigma_{{\bf q}_4} $)

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