Special Issue Editorial: Small Scales and Singularity Formation in Fluid Dynamics

Kiselev, Alexander

doi:10.1007/s00332-018-9452-3

Cited by 4 publications

(3 citation statements)

References 41 publications

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“…Despite their wide range of applications, the question regarding the global regularity of the 3D Euler equations has remained open. The interested readers may consult the excellent surveys [1,9,15,18,25,31] and the references therein. The main difficulty associated with the regularity properties of the 3D Euler equations is due to the presence of vortex stretching, which is absent in the 2D Euler equations.…”

Section: Introductionmentioning

confidence: 99%

“…The singularity scenario reported in [29,30] has generated great interests and has inspired a number of subsequent developments, see e.g. [7,23,24] and the excellent survey article [25]. Despite all the previous efforts, there is still lack of theoretical justification of the finite time singularity for the 3D axisymmetric Euler equations reported in [29,30].…”

Section: Introductionmentioning

confidence: 99%

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Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Chen

Hou

2021

Commun. Math. Phys.

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Inspired by the numerical evidence of a potential 3D Euler singularity by 30] and the recent breakthrough by Elgindi [10] on the singularity formation of the 3D Euler equation without swirl with C 1,α initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with C 1,α initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [29,30] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use C 1,α initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [10]. We also use some strategy proposed in our recent joint work with Huang in [6] and adopt several methods of analysis in [10] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with C 1,α initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Chen

Hou

2021

Commun. Math. Phys.

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“…Roughly speaking, this demonstrates that although the symmetry group Õ provides a strong stability on the solution (analogously to the 2D case), it can still yield finite time blowup. This is a clear manifestation of the following general principle, seemingly counter-intuitive: the more drastic growth one wants to prove, the more stability is required on the solution (see [38,36,37]).…”

Section: Octahedral Symmetrymentioning

confidence: 99%

The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain

Elgindi¹,

Jeong²

2020

Preprint

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We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: {(x 1 , x 2 , x 3 ) : 0 < x 3 < x 2 < x 1 }. In this domain, we prove local well-posedness for C α vorticities not necessarily vanishing on the boundary with any 0 < α < 1, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R 3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R 3 with bounded and piecewise smooth vorticities. Contents

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The incompressible Euler equations under octahedral symmetry: Singularity formation in a fundamental domain

Elgindi

Jeong

2021

Advances in Mathematics

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Special Issue Editorial: Small Scales and Singularity Formation in Fluid Dynamics

Cited by 4 publications

References 41 publications

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain

The incompressible Euler equations under octahedral symmetry: Singularity formation in a fundamental domain

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