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.
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)