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A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

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Abstract

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter \(\alpha >0\). Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

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Acknowledgements

The work of E.S.T. was supported in part by ONR Grant Number N00014-15-1-2333, and by the NSF Grants Number DMS-1109640 and DMS-1109645.

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Authors and Affiliations

  1. University of Nebraska-Lincoln, 203 Avery Hall, PO Box 880130, Lincoln, NE, 68588-0130, USA

    Adam Larios

  2. CCS-2, MS B296, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Mark R. Petersen

  3. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX, 77843-3368, USA

    Edriss S. Titi

  4. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Edriss S. Titi

  5. University of Exeter, College of Engineering, Mathematics and Physical Sciences Harrison Building, University of Exeter, Streatham Campus, North Park Road, Exeter, EX4 4QF, UK

    Beth Wingate

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  1. Adam Larios
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Correspondence to Adam Larios.

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Communicated by Oleg V. Vasilyev.

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Larios, A., Petersen, M.R., Titi, E.S. et al. A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization. Theor. Comput. Fluid Dyn. 32, 23–34 (2018). https://doi.org/10.1007/s00162-017-0434-0

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