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Construction of unstable concentrated solutions of the Euler and gSQG equations

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... This logarithmic bound is optimal in general, see [6]. However, in the same article, Buttà and Marchioro observed that for some special configurations of point-vortices, this bound can be improved. ...
... As stated in Remark 2.7, our methods provide a unique γ N such that the long-time confinement result holds, but we do not claim that the confinement result is false for other values of γ N within the range given by Theorem 2.5. However, outside of that range, if the configuration is unstable then the instability prevents the long time confinement from holding, see [6]. ...
... The main difference in the proof of Theorem 3.4 compared to [3] and [7] is that the control of the center of vorticity is much more delicate. Indeed, should for instance the point-vortex configuration be unstable, then the long-time confinement result would necessarily fail as the center of vorticity of the blobs would suffer from the same instability (see [6] for an example of this situation). This is why we have to assume Hypothesis 3.3 which is the stability of the configuration. ...
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Interacting motion of rectilinear geostrophic vortices
  • G K Morikawa
  • E V Swenson
G. K. Morikawa and E. V. Swenson, Interacting motion of rectilinear geostrophic vortices, Physics of Fluids, 14 (1971), 1058-1073.