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

## 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 xed values of the regularization parameter α > 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 suficient 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.

- Authors:

- Univ. of Nebraska, Lincoln, NE (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Texas A & M Univ., College Station, TX (United States); Weizmann Inst. of Science, Rehovot (Israel)
- Univ. of Exeter (United Kingdom)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1357141

- Report Number(s):
- LA-UR-17-23192

Journal ID: ISSN 0935-4964

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Theoretical and Computational Fluid Dynamics

- Additional Journal Information:
- Journal Volume: 32; Journal Issue: 1; Journal ID: ISSN 0935-4964

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Mathematics; Euler-Voigt; Navier-Stokes-Voigt; Inviscid Regularization, Turbulence Models; Inviscid Regularization; Turbulence Models

### Citation Formats

```
Larios, Adam, Petersen, Mark R., Titi, Edriss S., and Wingate, Beth.
```*A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization*. United States: N. p., 2017.
Web. doi:10.1007/s00162-017-0434-0.

```
Larios, Adam, Petersen, Mark R., Titi, Edriss S., & Wingate, Beth.
```*A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization*. United States. doi:10.1007/s00162-017-0434-0.

```
Larios, Adam, Petersen, Mark R., Titi, Edriss S., and Wingate, Beth. Sat .
"A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization". United States.
doi:10.1007/s00162-017-0434-0. https://www.osti.gov/servlets/purl/1357141.
```

```
@article{osti_1357141,
```

title = {A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization},

author = {Larios, Adam and Petersen, Mark R. and Titi, Edriss S. and Wingate, Beth},

abstractNote = {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 xed values of the regularization parameter α > 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 suficient 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.},

doi = {10.1007/s00162-017-0434-0},

journal = {Theoretical and Computational Fluid Dynamics},

number = 1,

volume = 32,

place = {United States},

year = {Sat Apr 29 00:00:00 EDT 2017},

month = {Sat Apr 29 00:00:00 EDT 2017}

}

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