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Bifurcation of Critical Points for Solutions of the 2D Euler and 2D Quasi-geostrophic Equations

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Abstract

We consider the 2D Euler and 2D quasi-geostrophic equations with periodic boundary conditions. For both systems we will use the stream-function formulation and study the bifurcation problem for the critical points of the stream function. In a small neighborhood of the origin, we construct a set of initial data such that initial critical points of the stream function bifurcate from 1 to 2 and then to 3 critical points in finite time. For the quasi-geostrophic equation the whole bifurcation process takes place strictly within the lifespan of the constructed local solution.

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Notes

  1. Note that the minus sign in front of ψ corresponds to reversing the direction of the velocity field u.

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Acknowledgements

The financial support from NSF, grant DMS 0908032 is highly appreciated.

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  1. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada, V6T1Z2

    Dong Li

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  1. Dong Li
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Correspondence to Dong Li.

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Li, D. Bifurcation of Critical Points for Solutions of the 2D Euler and 2D Quasi-geostrophic Equations. J Stat Phys 149, 92–107 (2012). https://doi.org/10.1007/s10955-012-0583-x

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