DFG Project: GR 967/3-1

Finite time Euler singularities: a Lagrangian perspective
Navigator: DFG GR 967/3-1:: Motivation
Saturday, 24. March 2018


Finite time Euler singularities

A short history of the Euler singularity problem

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


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