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A Lower Bound on Blowup Rates for the 3D Incompressible Euler Equation and a Single Exponential Beale-Kato-Majda Type Estimate

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Abstract

We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in \({H^{s}(\mathbb {R}^3)}\) , for \({s>\frac{5}{2}}\) . Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on \({\|u(t)\|_{H^s}}\) involving the length parameter introduced by Constantin in (SIAM Rev. 36(1):73–98, 1994). In particular, we derive lower bounds on the blowup rate of such solutions.

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  1. Department of Mathematics, University of Texas at Austin, University Station C1200, Austin, TX, 78712, USA

    Thomas Chen & Nataša Pavlović

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  1. Thomas Chen
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  2. Nataša Pavlović
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Correspondence to Thomas Chen.

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Communicated by P. Constantin

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Chen, T., Pavlović, N. A Lower Bound on Blowup Rates for the 3D Incompressible Euler Equation and a Single Exponential Beale-Kato-Majda Type Estimate. Commun. Math. Phys. 314, 265–280 (2012). https://doi.org/10.1007/s00220-012-1523-y

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  • DOI: https://doi.org/10.1007/s00220-012-1523-y

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