Corpus ID: 237513385

On the role of continuous symmetries in the solution of the 3D Euler fluid equations and related models

@inproceedings{Bustamante2021OnTR,
  title={On the role of continuous symmetries in the solution of the 3D Euler fluid equations and related models},
  author={M. Bustamante},
  year={2021}
}
We review the continuous symmetry approach and apply it to find the solution, via the construction of constants of motion and infinitesimal symmetries, of the 3D Euler fluid equations in several instances of interest, without recourse to Noether’s theorem. We show that the vorticity field is a symmetry of the flow and therefore one can construct a Lie algebra of symmetries if the flow admits another symmetry. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows… Expand

References

SHOWING 1-10 OF 41 REFERENCES
Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions
Motivated by the work on stagnation-point-type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (Physica D, vol. 132 (4), 1999, pp. 497–510) and the subsequentExpand
Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp.Expand
Lagrangian structures, integrability and chaos for 3D dynamical equations
In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrableExpand
Further Properties of a Continuum of Model Equations with Globally Defined Flux
To develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier toExpand
Mixed Lagrangian–Eulerian description of vortical flows for ideal and viscous fluids
It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid is equivalent to the equations of motion of a charged compressible fluid moving due to a self-consistent electromagneticExpand
3D Euler in a 2D Symmetry Plane
Initial results from new calculations of interacting anti-parallel Euler vortices are presented. The objective is to understand the origins of singular scaling presented by Kerr (1993) with differentExpand
Dynamically stretched vortices as solutions of the 3D Navier—Stokes equations
Abstract A well known limitation with stretched vortex solutions of the 3D Navier–Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which isExpand
Stretching and folding diagnostics in solutions of the three-dimensional Euler and Navier-Stokes equations
Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on the dynamics of the gradient of potential vorticity ($q = \bom\cdot\nabla\theta$) associated withExpand
Numerical study of singularity formation in a class of Euler and Navier–Stokes flows
We study numerically a class of stretched solutions of the three-dimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a EulerExpand
Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem.
  • M. Bustamante, M. Brachet
  • Mathematics, Physics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
TLDR
The main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time interval on which a well-resolved power-law behavior of δ(t) takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay. Expand
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