The three-dimensional Euler equations: singular or non-singular?

Gibbon, John; Bustamante, Miguel D.; Kerr, Robert M.

doi:10.1088/0951-7715/21/8/t02

Cited by 29 publications

(37 citation statements)

References 57 publications

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“…Initial conditions of that kind are relevant for the problem of developed turbulence, i.e., the inviscid limit of the incompressible Navier-Stokes equations. Finite-time blowup in the 3D incompressible Euler equations remains an open problem [21,22,24]: numerical simulations suggest the blowup at a physical boundary [30], while nearly exponential vorticity growth is typical for generic initial conditions with periodic boundary conditions [7,1]. The latter means that the viscous range gets excited within logarithmic times ∝ log Re with respect to the Reynolds number, which makes the inviscid formulation with rough (weak) velocity fields physically relevant [16].…”

Section: Discussionmentioning

confidence: 99%

Spontaneously stochastic solutions in one-dimensional inviscid systems

Mailybaev

2016

Nonlinearity

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In this paper, we study the inviscid limit of the Sabra shell model of turbulence, which is considered as a particular case of a viscous conservation law in one space dimension with a nonlocal quadratic flux function. We present a theoretical argument (with a detailed numerical confirmation) showing that a classical deterministic solution before a finitetime blowup, t < t b , must be continued as a stochastic process after the blowup, t > t b , representing a unique physically relevant description in the inviscid limit. This theory is based on the dynamical system formulation written for the logarithmic time τ = log(t−t b ), which features a stable traveling wave solution for the inviscid Burgers equation, but a stochastic traveling wave for the Sabra model. The latter describes a universal onset of stochasticity immediately after the blowup.

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Section: Discussionmentioning

confidence: 99%

Spontaneously stochastic solutions in one-dimensional inviscid systems

Mailybaev

2016

Nonlinearity

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“…Indeed, there exists numerical evidence to suggest that the 3D Euler equations may develop a singularity [137,136] (cf. [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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“…This so-called "blow-up problem" is one of the key open questions in mathematical fluid mechanics and, in fact, its importance for mathematics in general has been recognized by the Clay Mathematics Institute as one of its "millennium problems" (Fefferman, 2000). Questions concerning global-intime existence of smooth solutions remain open also for a number of other flow models including the 3D Euler equations (Gibbon et al, 2008) and some of the "active scalar" equations (Kiselev, 2010).…”

Section: Introductionmentioning

confidence: 99%

Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows

Ayala

Protas

2017

J. Fluid Mech.

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In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy E 0 which maximize the instantaneous rate of growth of enstrophy dE/dt. We provide an analytic characterization of these extreme vortex states in the limit of vanishing enstrophy E 0 and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of dE/dt in this limit. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu & Doering (2008) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound dE/dt < C E 3 , for some constant C > 0. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy E 0 . However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.

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The three-dimensional Euler equations: singular or non-singular?

Cited by 29 publications

References 57 publications

Spontaneously stochastic solutions in one-dimensional inviscid systems

Spontaneously stochastic solutions in one-dimensional inviscid systems

Convex integration and phenomenologies in turbulence

Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows

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