The question, whether finite time singularities in the incompressible
Navier-Stokes or Euler equations develop from smooth initial
conditions, is one of the outstanding open questions in classical
fluid dynamics. In the mathematics community, this question is one of
the Millenium prize problems [1] since solving
this problem mathematically will probably require completely new tools
in the theory of partial differential equations very different from
classical Sobolev type estimates. In physics, the question of
singularity formation in the Euler equation in particular is of great
relevance, since knowledge of the self-amplification process and the
resulting singular structures plays a fundamental role in
understanding small scale intermittency.

The question of finite time singularities has received a lot of
attraction in the last few years since the (ongoing) work in the group
of Thomas Hou at Caltech. In a series of papers [2], [3]
they try to relate
their mathematical work much closer to the results of numerical
simulations performed in this field. They found and claim that the
simulations of Richard Pelz [4], [5], [6]
and Robert Kerr [8], [9] do satisfy the assumptions
sufficient for smooth solutions without a finite time singularity.
These statements are heavily discussed in the community since R. Kerr
disagrees on the statements claimed by Thomas Hou. In a recent paper
by Cichowlas and Brachet [10] it is s argued
that resolutions in the range $16384^3 - 32768^3$ (which are in reach
with the Earth Simulator or with adaptive mesh codes) are needed to
really probe the Pelz singularity.

## Numerical simulations

The first relevant numerical simulation was done by Morf, Orszag, and
Frisch in 1980 [11]. They used analytical
continuation techniques on a temporal Taylor series expansion of the
solution beyond the convergence disk with the help of
Padé-approximations. Their conclusion at that time was that there is
a finite time singularity.

Brachet, Meiron, Nickel, Orszag, and Frisch [12]
repeated this numerical experiment in
1983 but with a higher order of the Padé-approximation. In addition
they performed direct numerical simulations using a pseudospectral
code. At that time they were very vague on the question of finite time
singularities.

In 1982 Chorin [13] applied a vortex-method for solving
the Euler equations. He claimed to observe a finite time singularity
and in addition he found that the Hausdorff-dimension of
$\epsilon$-support of vorticity is approximately $\approx$ 2.5.

Siggia (1985) [14] introduced a vortex-filament method
allowing the vortex to evolve with a variable core size. He observed
that the arclength of the vortex filament grows faster than
exponential. However, this method does not not allow vortex core
deformation.

In 1990, Pumir and Siggia [15] introduced
for the first time a simple adaptive grid to follow the evolution of
vorticity. They observed a tendency to form quasi-two-dimensional
structures. The maximum vorticity turned into only exponential growth.
However, their remeshing and smoothing operations produced a
substantial decrease in energy.

Bell and Marcus [16] studied in 1991 the evolution of
a perturbed vortex tube using a projection method. Using a resolution
of $128^3$ mesh points, they achieved an amplification of vorticity by
a factor 6. A complicated hairpin-like structure developed without the
effect of vortex core flattening. The evolution was fitted against a
growth of finite time singularity with the conclusion that a finite
time singularity is possible for this initial condition.

In 1992 Brachet, Meneguzzi, Vincent, Politano and Sulem
[17] performed a pseudospectral
simulation on the Taylor-Green vortex. Using the symmetries of the
vortex, a resolution of $864^3$ mesh points could be achieved. The
vorticity was amplified by a factor 5. The spontaneous emergence of
flat pancake like structures that shrink exponentially in time was
observed.

In 1993 Kerr [8], [9] studied the
interaction of perturbed antiparallel vortex tubes with smooth initial
profiles. Chebyshev polynomials are used in the direction
perpendicular to the symmetry plane. A resolution of $1024 \times 256
\times 128$ collocation points was reached. Vorticity was amplified by
a factor 18. Kerr's conclusions are in favor of a finite time
singularity.

Between 1994 and 2001, Boratav and Pelz [4], [5], [6]
studied a highly symmetric
configuration first introduced by Kida [7] of vortex
tubes which allowed high resolution simulations ($1024^3$ collocation
points) due to the symmetry. In addition, this configuration was the
first one, where the process of vorticity amplification was produced
by the strain of the neighboring vortices. Therefore, this was thought
to be a serious candidate for a finite time singularity.

In 1998, Grauer, Marliani and Germaschewski
[18] presented an adaptive mesh refinement
simulation of the Bell and Marcus [18] initial
condition. The achieved effective resolution was $2048^3$ mesh points
and an amplification of vorticity of a factor 21 was reached. The time
evolution of the maximum vorticity could be fitted to a finite time
singularity. Later, Germaschewski and Grauer (2001, not published)
redid the simulations of Boratav and Pelz but observed a strong vortex
flattening which stopped the initial dramatic growth of vorticity.
This seems to be in agreement with the analysis of Thomas Hou and
coworkers [2], [3].

In 2006, Hou and Li [19] presented a spectral very high
resolution simulation ($1536 \times 1024 \times 3072$ mesh points) of
Robert Kerr's initial conditions. They found vortex core flattening
and growth of vorticity not faster than double exponentially.

In 2007, Orlandi and Carnevale [20] started
with extended orthogonal colliding Lamb dipoles as initial conditions.
These are exact solutions of the two-dimensional Euler equations.
Their finding suggests a much stronger amplification of vorticity due
to the lack of vortex shedding over a long period of the simulation.

A slightly different approach is used by the French groups. Frisch,
Matsumoto and Bec (2003) [21], Cichowlas
and Brachet (2005) [10] and Pauls, Matsumoto,
Frisch and Bec (2006) [22] use the small
time asymptotics of the complex-space singularities to gain
information about possible real-space finite time singularities.

## Mathematical criteria for singularity formation

One of the most relevant criteria for singularity formation are the
conditions on the growth of maximum vorticity by Beale, Kato and Majda
(1984) [23] and Ponce (1985) [24].
The condition on the vorticity reads: Consider the incompressible Euler equations
\begin{eqnarray}
& \partial_t {\bf u} + {\bf u} \cdot \nabla {\bf u} + \nabla p = 0
\quad , & {\bf r} \in {\bf R}^d \quad , \quad d = 2,3 \\
& \nabla \cdot {\bf u} = 0
\end{eqnarray}
with initial conditions ${\bf u}^0({\bf r})$
\begin{eqnarray}
& {\bf u}(0,{\bf r}) = {\bf u}^0({\bf r}) \in H^s({\bf R}) \;\; .
\end{eqnarray}
Then there exists a global solution for $s \ge 3$
$$
{\bf u} = C([0,\infty];H^s) \cap C^1([0,\infty];H^{s-1})
$$
iff for the vorticity $\boldsymbol{\omega} = \nabla \times {\bf u}$ holds
$$
\int_0^T \| \boldsymbol{\omega}(t,\cdot) \|_{L^\infty} dt < \infty
$$
for every $T > 0$. This theorem states, that if one observes
numerically a finite time blow up of vorticity with a power law
behavior like $1/(T^* - t)^\alpha$, where $T^*$ is the singularity
time and $\alpha < 1$, then this singularity must be a numerical
artifact.

Ng and Bhattacharjee (1996) [25] formulated a
criterion for the Boratav-Pelz simulations. They showed that a
necessary condition for blowup is a positive value of the fourth-order
spatial derivative of the pressure ($p_{xxxx}$) at the origin.
However, due to the high order derivative it is very difficult to
evaluate this criterion. It is important to note that this criterion
changes in the Lagrangian framework to a criterion for the
second-order spatial derivative which must be positive.

Constantin, Fefferman, Majda (1996) [26]
were the first to look carefully
at geometric properties of the direction of the vorticity
$\boldsymbol{\xi}(\mathbf{x},t)$. They found that if
$$
\boldsymbol{\xi}(\mathbf{x},t) = \frac{\boldsymbol{\omega}(\mathbf{x},t)}{\| \boldsymbol{\omega}(\mathbf{x},t) \|}
$$
is smoothly directed in an $O(1)$ region, i.e. the maximum norm of
$\nabla \boldsymbol{\xi}$ is $L^2$ integrable in time from $0$ to $T$ in
that region, and the maximum norm of velocity in this region is
uniformly bounded, then no blow-up up exists up to time $T$.

Unfortunately, in all numerical simulations a smoothly directed $O(1)$
region was never observed and this theorem could not be applied to the
simulations.

In 2001, Cordoba and Fefferman [27]
presented an improvement of the above theorem which reads (in a rather
simplified way): Vortex tubes with $O(1)$ length that don't twist or
bend enough are ruled out if the infinity norm of velocity in a
neighborhood of that region is integrable in time. The criterion on
the velocity field is a strong improvement, but still, vortex tubes
with $O(1)$ length are not observed in numerical simulations.

Also in 2001, Cordoba and Fefferman [28]
presented a theorem for incompressible ideal magnetohydrodynamic
flows, where for the first time the assumptions are in agreement with
the observations from numerical simulations. Their theorem states:

*
If a current sheet has a potato chip like structure and the infinity
norm of velocity is integrable in time near this structure, then no
finite time blow-up of vorticity or current density can occur. The
adaptive mesh MHD simulations of Grauer and Marliani (2000)
[29] show exactly this behavior: a potato chip
like current sheet with an exponential growth of current density and
vorticity.
*
In the last two years (starting 2005)
[2], [3], the group of Thomas Hou from
Caltech concentrated even more on geometric properties of
incompressible Euler flows. They formulated a few theorems on the
non blowup where the assumptions of those theorems are close to the
situation in the numerical experiments. The first theorem
[2] states that if
$$
\mid\!\! \int_{s_1}^{s_2} (\nabla \cdot \boldsymbol{\xi}) \; \!\!\mid
\le \mbox{const}(T)
\;\; , \;\;\; 0 \le t \le T
$$
along a vortex line segment $[s_1,s_2]$ which contains the point of
maximum vorticity, then no point singularity is possible up to time
$T$. If the vorticity blows up at one point on this vortex line
segment then it must blow up simultaneous on the entire line segment.
The authors claim that this theorem could apply to the simulations of
R. Pelz with the conclusion that there is no finite time blow up in
his simulations. They argue that one observes partial regularity of
the vortex lines in the inner core. However, this question
seems to be more complicated since the length of this inner core is
shrinking in time and the important and strong divergence of the
direction of the vorticity happens at the end of the vortex tube.
This question has to be investigated by looking at the detailed motion
of vortex lines and their spreading at both ends of the vortex.

The second theorem [2] is an improvement of the
theorems of Constantin \textit{et al.} (1996)
[26] and Cordoba and Fefferman (2001)
[27]. To formulate this theorem we use
exactly the notation of [3]. Consider a family of
vortex line segments $L_t$, along which the vorticity is
comparable to the maximum vorticity $\| \boldsymbol{\omega} \|_\infty$. Let
$L(t)$ denote the arc length of $L_t$, $\boldsymbol{\xi} =
\boldsymbol{\omega}/|\boldsymbol{\omega}|$, $\mathbf{n}$ the unit normal vector and $\kappa$
the curvature of the vortex line. Define
$$
U_{\boldsymbol{\xi}}(t) = \max_{x,y \in L_t}
|(\mathbf{u}\cdot\boldsymbol{\xi}(\mathbf{x},t)
- \mathbf{u}\cdot\boldsymbol{\xi}(\mathbf{y},t)| \; ,
$$
$ U_{\mathbf{n}}(t) = \max_{L_t} |\mathbf{u}\cdot\mathbf{n}|$ and
$M(t) = \max
(\|\nabla\cdot\boldsymbol{\xi}\|_{L^\infty(L_t)},\|\kappa\|_{L^\infty(L_t)})$.
Let $A,B \in (0,1)$ with $B<1-A$ and $C_0$ be a positive constant. If

- $U_{\boldsymbol{\xi}}(t) + U_{\mathbf{n}}(t) M(t) L(t) \leq (T-t)^{-A}$,
- $M(t) L(t) \leq C_0$,
- $L(t) \geq (T-t)^B$,

then there will be no blowup up to time $T$. R. Kerr's and our
computations indicate that the maximum velocity blows up like
$(T-t)^{-1/2}$. In Kerr's simulations the length of the vortex tube
scales like $(T-t)^{1/2}$. This implies $B=A=1/2$ which is the
critical case not covered by the theorem above. The improved theorem in
[3] includes the critical case $B = 1-A$.

In 2002, J.D. Gibbon [30] gave a description of the
Euler equations using quaternions. For Lagrangian tracers, he and
coauthors [31] could formulate the dynamics of
an associated quaternionic frame and could relate it to the standard
frame involving curvature and tension of the vortex line. This
treatment aims to identify the local amplification or depletion
mechanisms.

## References