Tobias Grafke — Existence of Finite Time Singularities in the Euler equations

Summary: There has been a history of claims for and against possible numerical evidence for a finite-time singularity of the incompressible three-dimensional Euler equations. Addressing the shortcomings of simply monitoring peak vorticty to discern a singularity, this project aims to instead numerically test the assumptions posed by analytic blowup criteria connecting geometric properties of Lagrangian vortex line segments, like curvature and spreading, to possible finite-time singular behavior.

The Euler equations for incompressible fluids,

$$\begin{cases} \partial_t u + u\cdot\nabla u + \nabla p = 0\\ \nabla\cdot u=0 \end{cases}$$

are known for more than 250 years. Nevertheless, the question of the global in time existence of smooth solutions for smooth initial conditions is not answered. What is known is local in time existence,

Local Existence:

For initial conditions $u_0 \in H^m$, $\nabla\cdot u_0=0$, $m\ge 7/2$, there exists a time $T \le 1/c\|u_0\|_{H^m}$, such that there exists a unique solution of the Euler equations with
$$ u \in C([0,T];C^2(\mathbb{R}^3)) \cap C^1([0,T]; C(\mathbb{R}^3)). $$

With the advent of scientific computing, analytical results are accompanied by numerical simulations, which are used to gain insight into the evolution of flow structures and the interplay between physical quantities such as strain and vorticity, and may serve as clue, pointing in the general direction to be taken for analytical work. Most important in this direction is the classical result by Beale-Kato-Majda [BKM],

Beale-Kato-Majda:

If $T$ is the maximal time of existence of a unique, smooth solution of the Euler equations, then the vorticity $\omega=\nabla\times u$ fulfills
$$ \int_0^T \|\omega(\cdot,t)\|_{L^\infty}\,dt = \infty $$

Therefore, if the maximum of the vorticity, $\Omega(t)=\|\omega(\cdot,t)\|_{L^\infty}$ scales in time like $\Omega(t)\sim 1/(T-t)^\gamma$, then there is a finite time singularity if $\gamma \ge 1$. Historically, numerical simulations have tried to prove or disprove finite time singularities by measuring the exponent $\gamma$. Physically, it is worthwhile asking if such a growth is reasonable. Generally, a singularity in the Euler equations is believed to be point-wise and supposedly locally self-similar. Analytical results investigating the bounds of such scenarios are quite scarce for the Euler equations, yet recently several cases relevant to numerical simulations have been ruled out. By far the most important process in the formation of singularities in finite time is the coupling of vorticity and strain. Starting from the vorticity formulation of the Euler equations,

$$\partial_t \omega + u \cdot \nabla \omega = \omega \cdot \nabla u,$$

the vortex stretching term $\omega \cdot \nabla u$ (which is notably absent in 2D) might provide a mechanism for critical growth: If the strain grows alongside the vorticity, $\nabla u \sim \omega$, then along an advected fluid volume one would have $D/Dt\, \omega \sim \omega^2$, resulting in the critical scaling $\omega(t) \sim 1/(T-t)$. It is not known to date, whether internal mechanisms render such amplification impossible. Yet, it is known from turbulence research that the mentioned alignment is at least unlikely in a natural context. The open question remains whether it is possible to design initial conditions which exhibit and maintain, despite its inherent instability, a period of vorticity-strain coupling long enough to cause the blowup.

Several such candidates are proposed in the literature. A perturbed vortex tube as well as anti-parallel vortex tubes are afflicted with the inconvenience to require the curvature in the plane of symmetry to blow up alongside the vorticity. This is due to the fact that an amplification of the strain with the same growth rate as the vorticity can be connected via the Biot-Savart law to the requirement to kink infinitely at the location of maximum vorticity. On the other hand, due to axial stretching of the vortex tube, a growing strain reduces the curvature in the plane of symmetry. These counteracting processes may limit the ability of the aforementioned initial conditions to maintain vorticity-strain coupling over a period of time long enough for the formation of a singularity in finite time. The class of high-symmetry flows, and most notably among them the vortex dodecapole configuration pictured below, does not suffer from the above mentioned disadvantage: Here, the strain imposed on the vortex tubes in the plane of symmetry is induced by the rotational images. Axial strain is not dictated by the curvature and the above canceling does not take place. This renders the vortex dodecapole initial condition to be one of the most promising in terms of singularity formation known today.

Kida-Pelz vortex dodecapole

Next to classical results such as BKM, recently the geometric analysis of the flow has played a role in distinguishing finite-time singularities from flows that exhibit merely fast accumulation of vorticity. This approach, applied to numerical simulations, may provide clearer insight into the possible formation of the singularity. The following theorem is a simple example of such techniques:

Deng-Hou-Yu (2006):

Let $x(t)$ be the position of maximum vorticity and $y(t)$ a point on the same vortex line $c(s)$ as $x(t)$. If
$$ \left| \int_{x(t)}^{y(t)} (\nabla \cdot \xi) (c(s),t)\,ds \right| \le C$$
for $\xi=\omega/|\omega|$ being the direction of the vorticity, and $\int_0^T |\omega(y(t),t)|\,dt<\infty$, then there is no blowup up to time $T$

Most notably, theorems like this imply numerical techniques to distinguish between a point-wise blowup and the blowup of a whole vortex line segment: Monitoring a vortex line segment which maintains a fixed convergence of neighboring lines, $\int \nabla \cdot \xi\,ds=C$, the absence of the segment's collapse must coincide with a blowup of vorticity along the whole segment, assuming the formation of a finite-time singularity. Furthermore, using additional theorems, if a blowup of curvature and $\int \nabla \cdot \xi \,ds$ is not observed, then components of the velocity have to scale like $1/(T-t)$. Since in numerical simulations, velocity growth is usually far from that, this argument can be used against critical accumulation of vorticity much more clearly than the usual approach via BKM .

Relevant publications

T. Grafke and R. Grauer, "Finite-Time Euler singularities: A Lagrangian perspective", Appl. Math. Letters 26 (2013), 500
T. Grafke and R. Grauer, "Lagrangian and geometric analysis of finite-time Euler singularities", Procedia IUTAM 9 (2013), 32
T. Grafke, H. Homann, J. Dreher and R. Grauer, "Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods", Physica D 237 (2008), 1932
T. Grafke, "Finite-time Euler Singularities: A Lagrangian perspective", PhD thesis (Jun 2012)