Stability of postulated, self-similar, hydrodynamic blowup solutions

Abstract

A solution with real time singularity is assumed to exist that is steady under a Leray-type normalization. This solution is further assumed to be reached asymptotically as t-->t(0) in the renormalized plane, and thus can be thought of as the leading behavior of an inner solution. Constraints due to conserved quantities like energy are shown to be weakened in this scenario. In the wake region that trails the collapsing structure, it is shown that eigenfunctions associated with initial conditions are stable and decay, allowing the attracting singular solution to be shielded from details of the initial conditions. The parameters of the normalization are t(0), r(0), v(0), lambda, and alpha, which are the critical time, the location of the singularity, the velocity of the singular point, a scaling factor, and the scaling exponent of the velocity (t(0)-t)(alpha). The stability of the eigenfunctions of this solution obtained from the perturbation of these parameters is also examined in this work. Perturbations in the critical time and location are shown to be unstable whereas perturbations in velocity and scaling are not. The condition that the amplitude of the unstable eigenfunctions vanishes determines the time and location of the singularity.

Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

“…The model is formulated in a form identical to the original Euler equations, but with the algebraic structure defined on the 3D logarithmic lattice. We show that the blowup in this model is associated with a chaotic attractor of a renormalized system, in accordance with some earlier theoretical conjectures [37][38][39][40]; one can also make an interesting connection with the chaotic Belinskii-Khalatnikov-Lifshitz singularity in general relativity [8,41]. A distinctive property of the attractor is its anomalous multiscale structure, which explains the diversity of the existing DNS results, discloses fundamental limitations of current strategies, and provides new guidelines for the original blowup problem.Model.…”

Chapter 1 Incompressible Euler Equations: The Blow-up Problem and Related Results

“…For the survey related the stability question please see for example [79] and references therein. For the results on the regularity of the Euler equations with uniformly rotating external force we refer [2], while for the numerical studies on the blow-up problem of the Euler equations there are many articles including [101,102,94,7,80,11,89,90,91,127]. For various mathematical and physical aspects of the Euler equations there are many excellent books, review articles including [1,8,45,47,49,79,86,115,116,118,29,152].…”

Development of high vorticity structures in incompressible 3D Euler equations

“…The blowup scenario based on the vortex lines breaking (or overturning) was analyzed in [26] in the framework of the integrable incompressible hydrodynamic model with the Hamiltonian |ω|dr and the same symplectic operator [27] as for the 3D Euler equations (such unusual Hamiltonian can be obtained from the 3D Euler equations in the so-called local induction approximation). Formation of a singularity in the framework of renormalization group formalism was discussed in [28][29][30][31].…”