Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

Elgindi, Tarek M.; Jeong, In-Jee

doi:10.1007/s40818-019-0071-6

Cited by 63 publications

(57 citation statements)

References 118 publications

(148 reference statements)

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“…The vertex of the sector is a hyperbolic stagnation point of the flow, making connection to the Hou-Luo scenario. In [30], similar results are proved in the 3D axi-symmetric Euler case; here the domain is given by (1 + |z|) 2 ≤ |x| 2 + |y| 2 with an arbitrary > 0.…”

Section: Discussionsupporting

confidence: 54%

“…A very interesting recent work by Elgindi and Jeong takes a different approach [29], [30]. In [29], they look at a class of scale invariant solutions for the 2D Boussinesq system that satisfy 1 λ u(λx, t) = u(x, t) and 1 λ θ(λx, t) = θ(x, t). Observe that this class allows velocity and density that grow linearly at infinity.…”

Section: Discussionmentioning

confidence: 99%

“…These are not the first examples of infinite energy solutions (see [10], [15], and [70]). However in [29] a procedure to cut off the solution at infinity while maintaining finite time blow up property is carried out. This yields finite energy solutions leading to finite time blow up -in the sense that´T 0 ∇u(·, t) L ∞ dt → ∞ at blow up time.…”

Section: Discussionmentioning

confidence: 99%

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Small Scales and Singularity Formation in Fluid Dynamics

Kiselev

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Section: Discussionsupporting

confidence: 54%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Small Scales and Singularity Formation in Fluid Dynamics

Kiselev

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…[162,88,89,30]). Recently, Elgindi and Jeong demonstrated the formation of a singularity in the presence of a conical hourglass-like boundary [62].…”

Section: The Euler Equationsmentioning

confidence: 99%

Convex integration and phenomenologies in turbulence

Buckmaster

Vicol

2020

EMS Surv. Math. Sci.

110

145

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In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash's fundamental ideas on C 1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [47,188,50,51]. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions.First, we give an elementary construction of nonconservative C 0+ x,t weak solutions of the Euler equations, first proven by De Lellis-Székelyhidi Jr. [49,48]. Second, we present Isett's [102] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [18] of De Lellis-Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C 1 /3− x,t are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result [20], which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity classx . We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

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“…Important progress in the global regularity problem has been made by Luo and Hou [LH14], [LH14 + ], who have produced strong numerical evidence that smooth solutions of the 3D axisymmetric Euler equation system, which can be identified with the inviscid 2D Boussinesq equation, develop a singularity in finite time when the fluid domain has a solid boundary. Recently, Elgindi and Jeong [EJ18] have shown finite-time singularity formation for strong solutions of the 2D Boussinesq system when the fluid domain is a sector of angle less than π.…”

Section: Introductionmentioning

confidence: 99%

Stability, well-posedness and blow-up criterion for the Incompressible Slice Model

Alonso-Orán

León

2019

Physica D: Nonlinear Phenomena

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In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter-Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler-Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold's stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations.

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Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

Cited by 63 publications

References 118 publications

Small Scales and Singularity Formation in Fluid Dynamics

Small Scales and Singularity Formation in Fluid Dynamics

Convex integration and phenomenologies in turbulence

Stability, well-posedness and blow-up criterion for the Incompressible Slice Model

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