The three-dimensional Euler equations: Where do we stand?

Gibbon, John

doi:10.1016/j.physd.2007.10.014

Cited by 166 publications

(197 citation statements)

References 104 publications

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“…These results were based on earlier work by McIntyre and Roulstone, and were reviewed by them in McIntyre & Roulstone (2002). It has also been shown that the three-dimensional Euler equations has a quaternionic structure in the dependent variables (Gibbon 2002) and that this idea can be used to discuss the evolution of orthonormal frames on particle trajectories: see Gibbon et al (2006), Gibbon (2007Gibbon ( , 2008a and Gibbon & Holm (2007). The use of different sets of dependent and independent variables in geophysical models of cyclones and fronts has facilitated some remarkable simplifications of otherwise hopelessly difficult nonlinear problems: see Hoskins & Bretherton (1972).…”

Section: Equations For An Incompressible Fluidmentioning

confidence: 99%

A geometric interpretation of coherent structures in Navier–Stokes flows

et al. 2009

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The pressure in the incompressible three-dimensional Navier-Stokes and Euler equations is governed by Poisson's equation: this equation is studied using the geometry of threeforms in six dimensions. By studying the linear algebra of the vector space of three-forms L 3 W Ã where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of L 3 W Ã to the sign of the Laplacian of the pressure-and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Dp, satisfies DpO0, the three-form associated with Poisson's equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Dp!0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.

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Section: Equations For An Incompressible Fluidmentioning

confidence: 99%

A geometric interpretation of coherent structures in Navier–Stokes flows

et al. 2009

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“…The 3D Euler equations have a rich mathematical theory, for which the interested readers may consult the excellent surveys [2,18,24] and the references therein. This paper primarily concerns the existence or nonexistence of globally regular solutions to the 3D Euler equations, which is regarded as one of the most fundamental yet most challenging problems in mathematical fluid dynamics.…”

Section: Introduction the Celebrated Three-dimensional (3d) Incomprementioning

confidence: 99%

“…Other interesting pieces of work are [10,47], which studied axisymmetric Euler flows with complex initial data and reported singularities in the complex plane. A more comprehensive list of interesting numerical results can be found in the review article [24].…”

Section: Introduction the Celebrated Three-dimensional (3d) Incomprementioning

confidence: 99%

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

Luo¹,

Hou²

2014

Multiscale Model. Simul.

106

150

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Abstract. Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 × 10 12 ) 2 near the point of the singularity, we are able to advance the solution up to τ 2 = 0.003505 and predict a singularity time of ts ≈ 0.0035056, while achieving a pointwise relative error of O(10 −4 ) in the vorticity vector ω and observing a (3 × 10 8 )-fold increase in the maximum vorticity ω ∞ . The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

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“…Standing open for more than 250 y and closely related to the Clay Millennium Prize Problem on the Navier-Stokes equations, the problem has received great attention from not only the mathematics but also the physics and engineering communities, where the formation of singularities in inviscid (Euler) flows is believed to be relevant to the creation of small scales in viscous turbulent flows (1)(2)(3). The finite-time blowup problem has been studied extensively from both mathematical and numerical points of view.…”

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Potentially singular solutions of the 3D axisymmetric Euler equations

Luo

Hou

2014

Proc. Natl. Acad. Sci. U.S.A.

144

166

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The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(t s − t) −2.46 , where t s ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ 2 = 0.003505 at which time the vorticity amplifies by more than (3 × 10 8 )-fold and the maximum mesh resolution exceeds (3 × 10 12 ) 2 . The vorticity vector is observed to maintain four significant digits throughout the computations.W hether initially smooth solutions to the 3D incompressible Euler equationscan develop a singularity in finite time is one of the most fundamental problems in mathematical fluid dynamics. Standing open for more than 250 y and closely related to the Clay Millennium Prize Problem on the Navier-Stokes equations, the problem has received great attention from not only the mathematics but also the physics and engineering communities, where the formation of singularities in inviscid (Euler) flows is believed to be relevant to the creation of small scales in viscous turbulent flows (1-3). The finite-time blowup problem has been studied extensively from both mathematical and numerical points of view. On the mathematical side, a number of useful blowup/nonblowup criteria have been obtained over the years, which have greatly facilitated the numerical search of a finite-time singularity (4-10). On the numerical side, interesting numerical simulations suggesting the existence of a finite-time singularity have been reported from time to time (see, for example, refs. 11-17), but in essentially every case, evidence against a singularity was found in subsequent studies (see, for example, refs. 18-21). This casts doubt on the validity of the claimed singularity and leaves the question of finite-time blowup an unresolved puzzle. By focusing on flows with rotational symmetry and other special properties, we have identified, through careful numerical studies, a class of potentially singular solutions to the 3D axisymmetric Euler equations in a radially bounded, axially periodic cylinder (see Eqs. 2 and 3 below). The simulations take advantage of the reduced computational complexity in cylindrical geometries, and use a specially designed adaptive mesh to achieve a maximum mesh resolution of over (3 × 10 12 ) 2 near the point of the singularity. This results in a computed vorticity vector with four digits of accuracy (up to the stopping time) and with a (3 × 10 8 )-fold increase in magnitude. The numerical data are checked against all major blowup/nonblowup criteria to confirm the validity of the singularity. A careful local analysis also suggests the existence of a self-similar blowup in the meridian plane.We...

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The three-dimensional Euler equations: Where do we stand?

Cited by 166 publications

References 104 publications

A geometric interpretation of coherent structures in Navier–Stokes flows

A geometric interpretation of coherent structures in Navier–Stokes flows

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

Potentially singular solutions of the 3D axisymmetric Euler equations

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