Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

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

Journal Article · · Theoretical and Computational Fluid Dynamics
 [1];  [2];  [3];  [4]
  1. Univ. of Nebraska, Lincoln, NE (United States)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. Texas A & M Univ., College Station, TX (United States); Weizmann Inst. of Science, Rehovot (Israel)
  4. Univ. of Exeter (United Kingdom)
+ Show Author Affiliations
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.
Research Organization:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR) (SC-21)
Grant/Contract Number:
AC52-06NA25396
OSTI ID:
1357141
Report Number(s):
LA--UR-17-23192
Journal Information:
Theoretical and Computational Fluid Dynamics, Journal Name: Theoretical and Computational Fluid Dynamics Journal Issue: 1 Vol. 32; ISSN 0935-4964
Publisher:
SpringerCopyright Statement
Country of Publication:
United States
Language:
English

References (50)

An inviscid regularization for the surface quasi‐geostrophic equation journal October 2008
On NAVIER-STOKES and KELVIN-VOIGT Equations in Three Dimensions in Interpolation Spaces journal January 1992
Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) journal January 1969
The Taylor-Green vortex and fully developed turbulence journal March 1984
Remarks on a paper by J. T. Beale, T. Kato, and A. Majda journal September 1985
Remarks on the breakdown of smooth solutions for the 3-D Euler equations journal March 1984
On the blow-up of solutions of the 3-D Euler equations in a bounded domain journal July 1993
Onsager's conjecture on the energy conservation for solutions of Euler's equation journal October 1994
A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range journal February 2014
On Admissibility Criteria for Weak Solutions of the Euler Equations journal December 2008
On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations journal January 2014
The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations journal March 2002
Onsager’s Conjecture Almost Everywhere in Time journal January 2015
Dissipative continuous Euler flows journal October 2012
Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations journal August 2006
Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations journal November 2008
The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up? journal June 2013
Global attractors and determining modes for the 3D Navier-Stokes-Voight equations journal September 2009
Global existence for a regularized magnetohydrodynamic-α model journal June 2009
Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer journal November 1994
Finite-time Euler singularities: A Lagrangian perspective journal April 2013
Numerical approximation of the Voigt regularization for incompressible Navier–Stokes and magnetohydrodynamic flows journal October 2012
Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows journal April 2005
Viscosity versus vorticity stretching: Global well-posedness for a family of Navier–Stokes-alpha-like models journal June 2007
The three-dimensional Euler equations: Where do we stand? journal August 2008
Numerical simulations of possible finite time singularities in the incompressible Euler equations: Comparison of numerical methods journal August 2008
Blowup or no blowup? The interplay between theory and numerics journal August 2008
Calculation of complex singular solutions to the 3D incompressible Euler equations journal December 2009
3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis journal June 2011
Small-scale structure of the Taylor–Green vortex journal May 1983
Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations journal May 2009
Evidence for a singularity of the three‐dimensional, incompressible Euler equations journal July 1993
The dynamics of unforced turbulence at high Reynolds number for Taylor–Green vortices generalized to MHD journal April 2010
On relaxation times in the Navier-Stokes-Voigt model journal March 2013
Geometric Properties and Nonblowup of 3D Incompressible Euler Flow journal April 2005
Energy conservation and Onsager's conjecture for the Euler equations journal April 2008
Numerical solution of the Navier-Stokes equations journal January 1968
The Navier-Stokes-Voight model for image inpainting journal March 2012
Model Equations for Long Waves in Nonlinear Dispersive Systems
  • Benjamin, T. B.; Bona, J. L.; Mahony, J. J.
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 272, Issue 1220 https://doi.org/10.1098/rsta.1972.0032
journal March 1972
Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem journal December 2012
Self-truncation and scaling in Euler-Voigt- α and related fluid models journal July 2015
Onsager and the theory of hydrodynamic turbulence journal January 2006
Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation journal January 2014
journal January 1993
Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit journal April 2010
Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations journal April 2010
Anomalous dissipation for 1/5-Hölder Euler flows journal July 2015
Dissipative Euler flows and Onsager's conjecture journal January 2014
Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models journal January 2006
On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model journal January 2010

Similar Records

Numerical methods for the Euler equations of fluid dynamics
Book · Mon Dec 31 23:00:00 EST 1984 · OSTI ID:6960769
Geometric constraints on potentially singular solutions for the 3-D Euler equations
Journal Article · Mon Dec 30 23:00:00 EST 1996 · Communications in Partial Differential Equations · OSTI ID:441146
An adaptive projection method for the incompressible Euler equations
Conference · Wed Jun 09 00:00:00 EDT 1993 · OSTI ID:10166132

Related Subjects

97 MATHEMATICS AND COMPUTING
Euler-Voigt
Inviscid Regularization
Inviscid Regularization
Turbulence Models
Mathematics
Navier-Stokes-Voigt
Turbulence Models