Geometric constraints on potentially singular solutions for the 3-D Euler equations

  title={Geometric constraints on potentially singular solutions for the 3-D Euler equations},
  author={Peter Constantin and Charles Fefferman and Andrew J. Majda},
  journal={Communications in Partial Differential Equations},
We discuss necessary and sufficient conditions for the formation of finite time singularities (blow up) in the incompressible three dimensional Euler equations. The well-known result of Beale, Kato and Majda states that these equations have smooth solutions on the time interval (0,t) if, and only if lim/t{r_arrow}T {integral}{sup t}{sub 0} {parallel}{Omega}({center_dot},s){parallel}{sub L}{sup {infinity}} (dx)dx < {infinity} where {Omega} = {triangledown} X u is the vorticity of the fluid and u… 

Vortex Stretching by a Simple Hyperbolic Saddle

We study solutions to the 3D Euler vorticity equation of the form \(\omega = \tilde \omega (x,t)\left( {\frac{{\partial t}}{{\partial {x_2}}},\frac{{\partial t}}{{\partial {x_1}}},0} \right) \) in a

The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?

It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B=∇q×∇θ, provided B has no null points initially.

An Example of Finite-time Singularities in the 3d Euler Equations

Abstract.Let $$\Omega = {\user2{\mathbb{R}}}^{3} \backslash \overline{B} _{1} (0)$$ be the exterior of the closed unit ball. Consider the self-similar Euler system $$\alpha u + \beta y \cdot

Some advances on the geometric non blow-up criteria of incompressible flows

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a

A new path to the non blow-up of incompressible flows

  • L. Agélas
  • Mathematics
    Annales de l'Institut Henri Poincaré C, Analyse non linéaire
  • 2019

A sufficient condition for a finite-time $L_2 $ singularity of the 3d Euler Equations

  • Xinyu He
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2005
A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this

Finite Time Blowup for Lagrangian Modifications of the Three-Dimensional Euler Equation

In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as $$\begin{aligned} \partial _t \omega + {\mathcal {L}}_u

A Lower Bound on Blowup Rates for the 3D Incompressible Euler Equation and a Single Exponential Beale-Kato-Majda Type Estimate

We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $${H^{s}(\mathbb {R}^3)}$$ , for $${s>\frac{5}{2}}$$ . Instead of double

Reconnection and the road to turbulence (Coherent Vortical Structures : Their Roles in Turbulence Dynamics)

It will be noted that several initial conditions besides anti-parallel vortices give varying degrees of evidence for a singularity of the incompressible Euler equations. The primary test is whether



Evidence for a Singularity of the Three Dimensional, Incompressible Euler Equations

Three‐dimensional, incompressible Euler calculations of the interaction of perturbed antiparallel vortex tubes using smooth initial profiles in a bounded domain with bounded initial vorticity are

Geometric Statistics in Turbulence

The author presents results concerning scaling exponents in turbulence and estimates the average dissipation rate, the average dimension of level sets, and a class of two-dimensional equations that are useful models of incompressible dynamics.

Geometric and analytic studies in turbulence

Theories of turbulence ([1, 2] as well as [3] and [4]) are statistical. There exists also a relevant mathematical framework [5-7]—that of statistical solutions of the Navier–CStokes equations.

Singular front formation in a model for quasigeostrophic flow

A two‐dimensional model for quasigeostrophic flow which exhibits an analogy with the three‐dimensional incompressible Euler equations is considered. Numerical experiments show that this model

Vorticity, Turbulence, and Acoustics in Fluid Flow

This paper presents recent and ongoing research in mathematical fluid dynamics and emphasizes the interdisciplinary interaction of ideas from large-scale computation, asymptotic methods, and mathem...

Remarks on the breakdown of smooth solutions for the 3-D Euler equations

The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially

Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical