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On the Self-Similar Solutions of the 3D Euler and the Related Equations

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Abstract

We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases.

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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.

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Communicated by P. Constantin

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Chae, D. On the Self-Similar Solutions of the 3D Euler and the Related Equations. Commun. Math. Phys. 305, 333–349 (2011). https://doi.org/10.1007/s00220-011-1266-1

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  • DOI: https://doi.org/10.1007/s00220-011-1266-1

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