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Osgood’s Lemma and Some Results for the Slightly Supercritical 2D Euler Equations for Incompressible Flow

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Abstract

We investigate the (slightly) super-critical two-dimensional Euler equations. The paper consists of two parts. In the first part we prove well-posedness in C s spaces for all s > 0. We also give growth estimates for the C s norms of the vorticity for \({0 < s \leqq 1}\). In the second part we prove global regularity for the vortex patch problem in the super-critical regime. This paper extends the results of Chae et al. where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi–Constantin on the vortex patch problem to the slightly supercritical case. Both problems we study in the setting of the whole space.

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Authors and Affiliations

  1. The Courant Institute, New-York University, 251 Mercer Street, New York, USA

    Tarek Mohamed Elgindi

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  1. Tarek Mohamed Elgindi
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Correspondence to Tarek Mohamed Elgindi.

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Communicated by V. Šverák

T.M. Elgindi is partially supported by NSF Grant DMS-0807347.

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Elgindi, T.M. Osgood’s Lemma and Some Results for the Slightly Supercritical 2D Euler Equations for Incompressible Flow. Arch Rational Mech Anal 211, 965–990 (2014). https://doi.org/10.1007/s00205-013-0691-z

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  • DOI: https://doi.org/10.1007/s00205-013-0691-z

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