[Paper Review] Singularity formation in the incompressible Euler equation in finite and infinite time

Theodore D. Drivas, Tarek M. Elgindi|arXiv (Cornell University)|Mar 31, 2022

Navier-Stokes equation solutions4 citations

TL;DR

This paper investigates finite- and infinite-time singularity formation in the incompressible Euler equations using infinite-dimensional dynamical systems theory. It establishes a finite-time blowup example in infinite spatial dimensions via constant-pressure solutions and reviews mechanisms for singularity in 3D, including self-similar profiles and axi-symmetric no-swirl solutions, while posing key open problems on stability, blowup profiles, and critical regularity thresholds.

ABSTRACT

Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the lens of infinite-dimensional dynamical systems, and various open problems are listed and discussed.

Motivation & Objective

• To analyze singularity formation in the incompressible Euler equations through the lens of infinite-dimensional dynamical systems.
• To establish a finite-time blowup example from smooth initial data in infinite spatial dimensions using constant-pressure solutions.
• To review and unify recent advances on 3D finite-time blowup mechanisms, particularly self-similar solutions and axi-symmetric no-swirl configurations.
• To identify and formulate open problems on the existence, stability, and propagation of singularities in 3D and higher-dimensional settings.
• To explore the role of symmetry, angular weights, and linearized operators in constructing and analyzing blowup profiles.

Proposed method

• Uses the geometric formulation of fluid flow as geodesic motion on the group of volume-preserving diffeomorphisms, derived from the principle of least action.
• Applies the Euler-Poincaré equation on the configuration space of volume-preserving diffeomorphisms to derive the incompressible Euler equations.
• Analyzes the linearized operator $\mathcal{L}$ and designs angular weights to control nonlocal effects in self-similar blowup constructions.
• Constructs a class of global solutions in infinite dimensions with constant pressure to demonstrate finite-time blowup from smooth data.
• Employs self-similar ansatzes and asymptotic analysis to study blowup profiles in axi-symmetric no-swirl and 3D Euler systems.
• Uses Fourier integral operators and $L^p$-based regularity criteria to probe critical thresholds for singularity formation in 1D and 3D models.

Experimental results

Research questions

• RQ1 Can smooth initial data in infinite-dimensional spatial domains lead to finite-time singularity in the incompressible Euler equations?
• RQ2 Do self-similar solutions with $\nabla u \in \ring{C}^\infty$ exist that blow up in finite time under admissible symmetry classes?
• RQ3 Is there a universal critical $L^{p_{*}}$ norm threshold above which finite-time blowup is guaranteed in 3D Euler flows?
• RQ4 Can blowup occur in axisymmetric no-swirl Euler solutions on $\mathbb{R}^d$ for $d \geq 4$?
• RQ5 Does the singular set of self-similar solutions propagate as an expanding ring in the axisymmetric setting?

Key findings

• A finite-time blowup is demonstrated for the incompressible Euler equations in infinite spatial dimensions using smooth, compactly supported initial data with constant pressure.
• The construction relies on a class of global solutions in infinite dimensions where vorticity grows unboundedly in finite time despite smooth initial data.
• Self-similar blowup solutions are shown to exist for axi-symmetric no-swirl Euler equations under specific symmetry and weight conditions.
• The linearized operator $\mathcal{L}$ is analyzed to design angular weights that control nonlocal terms, enabling the construction of self-similar profiles.
• Conjectures are formulated suggesting that $L^{p_{*}}$ norms with $p_{*} < \infty$ must blow up if a solution becomes singular in finite time.
• It is conjectured that for $\alpha < \frac{1}{3}$, there exist $C^{\alpha}_{c}$ axi-symmetric no-swirl solutions that blow up in finite time, even with $C^{\infty}$ initial data in radial variables.