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Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

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Abstract

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in \(\mathbb{R}^3\). By a similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in \(\mathbb{R}^n\). This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.

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Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Korea

    Dongho Chae

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  1. Dongho Chae
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Correspondence to Dongho Chae.

Additional information

Communicated by P. Constantin

The work was supported partially by the KOSEF Grant no. R01-2005-000-10077-0, and KRF Grant (MOEHRD, Basic Research Promotion Fund).

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Chae, D. Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations. Commun. Math. Phys. 273, 203–215 (2007). https://doi.org/10.1007/s00220-007-0249-8

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