Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice

Abstract

The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem.

arXiv comment7 pages, 5 figures, 1 supplemental video

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Cited by in corpus (19)

• Suppressing thermalization and constructing weak solutions in truncated inviscid equations of hydrodynamics: Lessons from the Burgers equation
• Fluid dynamics on logarithmic lattices
• Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows
• On the thermalization of the three-dimensional, incompressible, Galerkin-truncated Euler equation
• Development of high vorticity structures and geometrical properties of the vortex line representation
• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Statistical determinism in non-Lipschitz dynamical systems
• Lagrangian dynamics and regularity of the spin Euler equation
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• Inverse cascade anomalies in fourth-order Leith models
• Logarithmic lattice models for flows with boundaries
• Shell Models on Recurrent Sequences: Fibonacci, Padovan and Other Series
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• Behind the mirror: the hidden dissipative singular solutions of ideal reversible fluids on log-lattices
• Spontaneous stochasticity in the fluctuating Navier-Stokes equations on a logarithmic lattice
• "Life after death" in ordinary differential equations with a non-Lipschitz singularity