A continuation principle for the $3$-D Euler equations for incompressible fluids in a bounded domain

Shirota, Taira; Yanagisawa, Taku

doi:10.3792/pjaa.69.77

Cited by 24 publications

(21 citation statements)

References 6 publications

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“…Note that this error was derived from the pointwise error of the velocity field and hence is likely an overestimate of the true error. The existence of a finite-time singularity is confirmed using the well-known Beale-Kato-Majda (BKM) criterion (4,6,7). It asserts that a smooth solution of the 3D Euler equations blows up at time t s if and only if the maximum vorticity kωk ∞ accumulates so fast in time that…”

Section: Numerical Resultsmentioning

confidence: 94%

“…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)(5)(6)(7)(8)(9)(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.…”

mentioning

confidence: 99%

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Potentially singular solutions of the 3D axisymmetric Euler equations

Luo

Hou

2014

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

139

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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Section: Numerical Resultsmentioning

confidence: 94%

mentioning

confidence: 99%

Potentially singular solutions of the 3D axisymmetric Euler equations

Luo

Hou

2014

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

139

166

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“…The main difficulty in the analysis lies in the presence of the nonlinear vortex stretching term and the lack of a regularization mechanism, which implies that even the local well-posedness of the equations can only be established for sufficiently smooth initial data (see, for example, [37]). Despite these difficulties, a few important partial results [3,44,22,46,19,20,25] have been obtained over the years which have led to improved understanding of the regularity properties of the 3D Euler. More specifically, the celebrated theorem of Beale, Kato, and Majda [3] and its variants [22,46] characterize the regularity of the 3D Euler equations in terms of the maximum vorticity, asserting that a smooth solution u of (1.1) blows up at t = T if and only if…”

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.

104

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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“…However, the methods in [1][2][3][4][5][6] could not be used here directly. We will use a well-known logarithmic Sobolev inequality in [8,9] to complete our proof. We will prove:…”

Section: Introductionmentioning

confidence: 99%

“…From now on we will drop the subscript e and throughout this section C will be a constant independent of > 0. First, we recall the following two lemmas in [8][9][10].…”

Section: Introductionmentioning

confidence: 99%