数理解析研究所講究録
第 1837 巻 2013 年 119-131

119

A stability estimate and numerical curiosities
related to the flow of a curve by its binormal
curvature
Didier Smets
Universite Pierre & Marie Curie
These notes are companion to the talk given at the RIMS workshop “Progress in Variational Problems” on June 13th 2012 and entirely based on joint
work [9, 10] with R.L. Jerrard (Univ. of Toronto). The author expresses his
warmest gratitude to Professor M. Ishiwata (Fukushima Univ.) and Professor
F. Takahashi (Osaka Univ.) for their invitation to participate to this meetingand to Professor Y. Tsutsumi (Kyoto Univ.) for his invitation to Japan in thesecond week of June 2012.

1

Classical formulation of the binormal curvature flow

The equations for the evolution of a familyof smooth curves inaccording to their binormal curvature are written in terms of an arc-lengthparametrization:by$\mathbb{R}^{3}$$(\gamma_{t})_{t\in I}$$I\cross \mathbb{R}arrow \mathbb{R}^{3}$$\gamma$(1)$\partial_{t}\gamma=\partial_{s}\gamma\cross\partial_{ss}\gamma$

where $t\in I$ is the time variable,
is the arc-length parameter, anddenotes the vector product in. In geometric terms, equation (1) takes its
name from its equivalent form
$s\in \mathbb{R}$$\cross$$\mathbb{R}^{3}$$\partial_{t}\gamma=\kappa b$

where and are the curvature function and the binormal vector field along
respectively. Note that the arc-length parametrization conditionis always compatible with equation (1) since
whenever (1) is satisfied. In particular, closed curves
evolved by the binormal curvature flow equation (1) all have the same length.
$\kappa$$b$$\gamma_{t}$$|\partial_{s}\gamma(t, s)|^{2}=$$1$$2\partial_{s}\gamma\cdot(\partial_{s}\gamma\cross\partial_{sss}\gamma)=0$$\partial_{t}(|\partial_{S}\gamma|^{2})=2\partial_{s}\gamma$$\partial_{st}\gamma=$

120
Equation (1) first appeared in the 1906 Ph. . thesis of L.S. Da Rios [5],whose work was promoted in a series of lectures in 1931 in Paris by its advisor T. Levi-Civita [17]. The problem considered by Da Rios and Levi-Civita
goes back to the work of H. Helmholtz [7] in 1858 on the motion of a three
dimensional incompressible fluid in rotation. In part of [7], Helmholtz considered configurations called “vortex-filaments of indefinitely small cross-section”(at least in its translation [8] by P.G. Tait) : in that case, the vorticity fieldcurl(v) associated to the velocity field of the fluid at a given time
is concentrated along a closed oriented curve , parallel to it and vanishing rapidly away from it. Helmholtz could notl rigorously answer the question of the persistence in time of such vortex-filaments under the Euler flow
. Nevertheless, he obtained important contributionsin that direction.
few years later in 1867, Lord Kelvin announced in [12]and published in [13] (thirteen years later!) the first result on linear stabilityof circular vortex-filaments. The latter, also called vortex rings, correspondin the asymptotic of infinitely small cross-section to the travelling wave solu, whereis an arc-length
tions of equation (1) given byparametrization of a circle of radius in a plane perpendicular to the unitary. It is only in 1906, with the help of progress made in potential
vectortheory, that Da Rios formally obtained the speculated general motion law (1).
One hardly finds any reference to equation (1) in the literature in the intervalof time between Da Rios thesis in 1906 and the two papers [1] by R.J. Arms
and F.R. Hama and [3] by R. Betchov both in 1965. Actually, in alarge part of
the fluid dynamics community, equation (1) was long (mistakenly) attributed
to Betchov and $/or$ Arms and Hama. It is the author’s pleasure to mention
that not only Da Rios contribution was forgotten on those occasions, since it
appears that a formal derivation of (1) was also conducted in 1937 (We weretold it was presented as (3 pages) homework for students!) in the paper[19] by the Japanese mathematicians Y. Murakami, H. Takahasi, Y. Ukita
and S. Fujihara2 As will becomes clear in the remaining, Japan as played aprominent role in every perspective in the study of (1) so far.
$D$$\omega;=$

$v$

$t$

$\gamma_{t}$$\partial_{t}\omega+v$$\nabla\omega=\omega$$\nabla\omega$$A$$\gamma(t, s)=\gamma_{r,\vec{e}}(s)+\frac{t}{r}\vec{e}$$\gamma_{r,\vec{e}}$$r$$e^{arrow}\in \mathbb{R}^{3}$

$a$

2

$A$

new form of (in)stability estimate

In [10], one of our goals with R.L. Jerrard was to extend the classical formulation (1) to a larger class the regular parametrized curves. The motivation
to do so was two-fold: first, since it deals with parametrized curves it is necessarily insensitive to self-intersections, by which we mean lack of injectivityof the map. This is surely unsatisfactory if one believes that such flows$\gamma(t, \cdot)$lAs anybody so far!
2The author would actually be grateful to receive some copy of it.

121arise as limits from three dimensional fluid dynamics. Our formulation in [10]in principle would be able to detect such self-intersections, as well as possible
collisions between elements of disconnected vortex filaments and changes of
topology. Second, there are presumably important configurations of curves
which are too singular to be considered under formulation (1). Invoking distributional derivatives on can give a meaning to equation (1) in a variety ofspaces, but those spaces just fail to include the case of curves which are at most
Lipschitz. On the other hand, in numerical simulations of the Euler equationor the Gross-Pitaevskii equation for quantum fluids, it is observed (see e.g.
S. Kida and M. Takaoka [15] (still Japan!) or J. Koplik and H. Levine [16]that vortex-filaments often tend to recombine by exchanging strands in casesof collisions or self-intersections. Those recombinations, when the intersections
are transverse, inevitably create curves which are at most Lipschitz.
$)$

It is not our purpose here to present the details of the new formulation
proposed in [10], these involve a number constructions based on notions related
to Geometric Measure Theory which are not appropriate for a short survey.
Instead, we will focus on the main ingredient in the corresponding (weakstrong) uniqueness theory which happens to have a counterpart with its own
interest also in the framework of parametrized solutions of (1). As far as weknow, it is the first rigorous stability (actually: control of instabilities) estimate
for filament flows.
In order to measure distances between curves, without attaching too much
importance in their parametrizations but still, we define the following notionof distance.
Definition 1.
curve in:such that$A$$\mathbb{R}^{3}$parametrized by arc-length is a Lipschitz map$\mathbb{R}arrow \mathbb{R}^{3}$$\gamma$for a.e.

$|\gamma’(s)|=1$

A closed curve inand such that$\mathbb{R}^{3}$of period $l>0$ is a curve in$\gamma(s+\ell)=\gamma(s)$Given two closed curves$\gamma$and$\Gamma$in$\mathbb{R}^{3}$,$s\in \mathbb{R}.$$\mathbb{R}^{3}$parametrized by arc-length$\forall s\in \mathbb{R}.$, we define the parametric distance$d_{\mathcal{P}}( \Gamma, \gamma):=\inf_{p\in p(L,\ell)}\sup_{s\in \mathbb{R}}|\Gamma(s)-\gamma(p(s))|_{p\in \mathcal{P}}=int_{L)}^{\sup_{s\in \mathbb{R}}}|\gamma(s)-\Gamma(p(s))|,$

where$\ell$

and$L$

are the periods of$\gamma$and$\Gamma$respectively, and for $0<s_{0},$$\mathcal{P}(s_{0}, s_{1})=\{p\in \mathcal{C}(\mathbb{R}, \mathbb{R}) s.t.

$s_{1},$

p(s+s_{0})=p(s)+s_{1}, \forall s\in \mathbb{R}\}.$$\square$122
Clearly,$d_{\mathcal{H}}(\Gamma, \gamma)\leq d_{\mathcal{P}}(\Gamma, \gamma)$,

is the Hausdorff distance between sets.
is a periodic curve of period $\ell>0$ andIn the sequel,the minimal radius ofis a periodic curve of period $L>0$ . We denote bycurvature of :where

$d_{\mathcal{H}}$$\Gamma$$\gamma\in C^{2}(\mathbb{R}, \mathbb{R}^{3})$$r_{\gamma}$$\gamma$$r_{\gamma}:=( \max_{s\in[0,\ell]}|\gamma"(s)|)^{-1}\in(0, +\infty],$and by$C_{\gamma}$

the tubular neighborhood :$C_{\gamma}:=\{x\in \mathbb{R}^{3}, s.t.

Forandthere exists a uniquesuch that$x_{0}\in \mathbb{R}^{3}$$s_{0}\in \mathbb{R}$d(x, \gamma)<r_{\gamma}/8\}.$such that $|x_{0}-\gamma(s_{0})|<r_{\gamma}/8$ , it can be proved that(the “best” orthogonal projection)$\xi_{0}=:\xi(\mathcal{S}_{0}, X_{0})\in \mathbb{R}$

1. $\xi_{0}\in(s_{0}-r_{\gamma}/4, s+r_{\gamma}/4)$ ,2.

$(x_{0}-\gamma(\xi_{0}))$.

$\gamma’(\xi_{0})=0.$

Moreover, the function

:

$\xi$

$\Xi_{\gamma}arrow \mathbb{R},$$\Xi_{\gamma}:=\{(s, x)\in \mathbb{R}^{4}s.t.

is of class$C^{1}$

on$\Xi_{\gamma}$

and for every

defined on the open set$(s, x)\mapsto\xi(s, x)$|x-\gamma(s)|<r_{\gamma}/8\}$$(s, x)\in\Xi_{\gamma}$

we have

3.

$(s+\ell, x)\in\Xi_{\gamma}$

4.

$\partial_{S}\xi(s, x)=0,$5.

$D_{x} \xi(s, x)=(1+\frac{(x-\gamma(\xi(s,x)))\cdot\gamma"(\xi(s,x))}{1-(x-\gamma(\xi(s,x)))\cdot\gamma’(\xi(s,x))})\gamma’(\xi(s, x))$and $\xi(s+\ell, x)=\xi(s, x)+\ell,$.

We have
Corollary 2. Assume that$d_{\mathcal{P}}(\Gamma, \gamma)<r_{\gamma}/8$

and let$p\in \mathcal{P}(L, \ell)$be such that$\sup_{s\in \mathbb{R}}|\Gamma(s)-\gamma(p(s))|<r_{\gamma}/8.$

The function :function which satisfies, for any$\sigma$$\mathbb{R}arrow \mathbb{R},$$s\mapsto\sigma(s)$1. $\sigma(s+L)=\sigma(s)+\ell,$

2. $|\Gamma(s)-\gamma(\sigma(s))|<r_{\gamma}/8,$

$s$

in

$:=\xi(p(s), \Gamma(s))$$\mathbb{R},$is the unique continuous

123
3. $(\Gamma(s)-\gamma(\sigma(s)))\cdot\gamma’(\sigma(s))=0,$
4. $|\sigma(s)-p(s)|<r_{\gamma}/4.$aA continuous function that satisfies 1, 2 and 3 in Corollary 2 is called
has aa reparametrization of for . Notice that when each point inunique orthogonal projection $P(x)$ on (this can happen only if has no selfintersection), then there exist a unique reparametrization of for (modulo
a constant multiple of ), and it is determined by $\gamma(\sigma(s))=P(\Gamma(s))$ .
The following discrepancy measure between and is better suited to ourneeds than$\sigma$$x$$\Gamma$$\gamma$$C_{\gamma}$$\gamma$$\gamma$$\Gamma$$\sigma$$\gamma$$P$$\Gamma$$\gamma$$d_{\mathcal{P}}.$

Definition 3. For $r>0$ , we set$F_{\gamma,r}( \Gamma) :=\inf_{\sigma}F_{\gamma,\sigma,r}(\Gamma)\equiv\inf_{\sigma}\int_{0}^{L}\mathcal{F}(\Gamma, \gamma, \sigma, r, s)ds$

(2)

,

where
,

$\mathcal{F}(\Gamma, \gamma, \sigma, r, s):=1-f(|\Gamma(s)-\gamma(\sigma(s))|^{2})\gamma’(\sigma(s))\cdot\Gamma’(s)$the function$f\equiv f_{r}$: $[0, +\infty)arrow[0, +\infty)$ is given by$f(d^{2}):=\{\begin{array}{ll}1 -- (\frac{d}{r})^{2}, for 0\leq d^{2}\leq r^{2},0, for d^{2}\geq r^{2},\end{array}$and the infimum in (2) is taken over all possible reparametrizations of$\gamma$for$\Gamma.$, which plays a central role in the following (in)staThe functional
bility estimate, can be related to more standard discrepancy measures, likedistance between the
between the curves, or theor
the distancesparametrizations of their tangents (see [9] Section 3 for details).
We come back now to equation (1) and assume that
$F_{\gamma,r}$$d_{\mathcal{H}}$$L^{2}$$d_{\mathcal{P}}$$\gamma\in L^{\infty}(I, H_{1oc}^{4}(\mathbb{R}, \mathbb{R}^{3}))$be two solutions3 of the binormal curvature flowand, where $I=(-T, T)$ for some $T>0$ . Assume moreover
equation (1) onthat and are both periodic with respective periods $\ell>0$ and $L>0$ . For$t\in I$ , we set$\Gamma\in L^{\infty}(I, H_{1oc}^{\frac{3}{2}}(\mathbb{R}, \mathbb{R}^{3}))$$I\cross \mathbb{R}$$\Gamma$$\gamma$$\gamma_{t}:=\gamma(t, \cdot) , \Gamma_{t}:=\Gamma(t, \cdot)$,

and define4$r_{\gamma} := \max_{t\in\overline{I}}(\max_{s\in \mathbb{R}}|\partial_{\mathcal{S}S}\gamma(t, s)|)^{-1}\in(0, +\infty].$and
3Then necessarily
4We have defined above a quantity called$\gamma\in C(\overline{I}, C^{3}(\mathbb{R}, \mathbb{R}^{3}))$$\Gamma\in \mathcal{C}(I, C(\mathbb{R}, \mathbb{R}^{3}))$.

for a curve with no time dependence. One
is the natural extension of the other and there shouldn’t be any possible confusion.
$r_{\gamma}$

124
Assume that at time zero we have$d_{\mathcal{P}}(\Gamma_{0}, \gamma_{0})<r_{\gamma}/8,$

be a reparametrization ofand let therefore
being ensured by Corollary 2).
$\sigma_{0}$for$\gamma_{0}$$\Gamma_{0}$(the existence of which

Theorem 4. Fix $0<r\leq r_{\gamma}/8$ and assume that(3)

$F(0):=F_{\gamma 0,\sigma 0^{\gamma}},( \Gamma_{0})<F_{r}:=r(\sqrt{2}+\frac{r}{L})^{-1}$

Define then$T_{r}:= \frac{1}{K}\log(F)$

,

where
$K:= \frac{8}{r^{2}}+\frac{32}{r_{\gamma}^{2}}+(2+32\frac{r}{r_{\gamma}})\Vert\partial_{sss}\gamma\Vert_{L^{\infty}(I\cross \mathbb{R})},$and set $J_{r}=[-T_{r}, T_{r}]\cap(-T, T)$ . There exists a uniquesuch thatis a reparametrization of for , for
and
every
by $F(t)$. The function defined on
is Lipschitz
the inequality
continuous on
and satisfies for almost every$\sigma\in C(J_{r}\cross \mathbb{R}, \mathbb{R})$$\sigma(0, \cdot)=\sigma_{0}$$\sigma_{t}$$:=\sigma(t, \cdot)$$F$$t\in J_{r}$$\Gamma_{t}$$\gamma_{t}$$J_{r}$$:=F_{\gamma_{t},\sigma_{t},r}(\Gamma_{t})$$t\in J_{r}$$J_{r}$$|F’(t)|\leq KF(t)$ .

Therefore by Gronwall Lemma$F(t)\leq\exp(K|t|)F(O) , \forall t\in J_{r},$and in particular$d_{p}(\Gamma_{t}, \gamma_{t})<r, \forall t\in J_{r}.$Since, as mentionned above, $F$ can be related to more usual forms of discrepancy measures (like Haussdorff disntance e.g.), the final estimate in theprevious theorem allows to control the “distance” between a reference smooth
solution and a rough solution by constants which only depend on the smooth
solution.

3

short excursion into two related worlds(after H. Hasimoto)$A$

If
is a solution to the binormal curvature flow equation(1), then the map, parametrizing the evolution
in time of the unit tangent vectors to the curves, satisfies$\gamma\in L^{\infty}(I, H_{loc}^{3/2}(\mathbb{R}, \mathbb{R}^{3}))$$u$$:=\partial_{s}\gamma\in L^{\infty}(I, H_{loc}^{1/2}(\mathbb{R}, S^{2}))$$\gamma(t, \cdot)$$\partial_{t}u=\partial_{t}\partial_{s}\gamma=\partial_{s}\partial_{t}\gamma=\partial_{s}(u\cross\partial_{s}u)$

(4)

125
in the sense of distributions on. In other words, is a solution of the
Schr\"odinger map equation (4) for maps from to , and the binormal curvature fl $ow$ equation is therefore a primitive equation of the Schr\"odinger map$I\cross \mathbb{R}$$u$$S^{2}$$\mathbb{R}$equation.

Conversely, let
map equation (4) and define the function$u\in L^{\infty}(I, H_{loc}^{1/2}(\mathbb{R}, S^{2}))$be a solution to the Schr\"odinger
by$\Gamma_{u}\in L^{\infty}(I, H_{loc}^{3/2}(\mathbb{R}, \mathbb{R}^{3}))$$\Gamma_{u}(\cdot t, s):=\int_{0}^{s}u(t, z)dz$

In the sense of distributions on$I\cross \mathbb{R}$(5).

, we have$\partial_{s}(\partial_{t}\Gamma_{u}-\partial_{s}\Gamma_{u}\cross\partial_{ss}\Gamma_{u})=0$(6).

all have their base point
By construction, the primitive curves
fixed at the origin. For smooth solutions, equation (6) would directly imply
depending only time only and such thatthe existence of a function$\Gamma_{u}(t, 0)$$\Gamma_{u}(t, \cdot)$$c_{u}$$\gamma_{u}(t, s):=\Gamma_{u}(t, s)+c(t)$is a solution to the binormal curvature flow equation (1). The function
indeed represents the evolution in time of the actual base point of the curves.
For solutions barely inthe same conclusion holds with slightly more
involved argument.
Besides the above sort of equivalence between the binormal curvature flow
equation and the Schr\"odinger map equation presented just above, H. Hasimoto [6] (Japan once more!) exhibited in 1972 an intimate relation between
the binormal curvature flow equation (1) and the cubic focusing nonlinear
Schr\"odinger equation. Let :be a smooth and biregular solutionof the binormal curvature flow equation (1), and denote by and respectively the curvature and torsion functions of . Then, the function definedby the Hasimoto transformon$c_{u}$$H^{3/2}$$I\cross \mathbb{R}arrow \mathbb{R}^{3}$$\gamma$$\kappa$$T$$\Psi$$\gamma$$I\cross \mathbb{R}$$\psi(t, s) :=\kappa(t, s)\exp(i\int_{0}^{s}T(t, z)dz)$

is a solution to$\partial_{t}\psi+\partial_{ss}\psi+\frac{1}{2}(|\psi|^{2}-A(t))\psi=0$

where$A(t):=(2 \frac{\partial_{ss}\kappa-\kappa T^{2}}{\kappa}+\kappa^{2})(t, 0)$

If isunless$\gamma$$2\pi$.

is only quasiperiodic, then-periodic in , that is if :. Nevertheless, it is possible to recover a -periodic$s$$\int_{0}^{2\pi}T(t, z)dz\in 2\pi \mathbb{Z}$$I\cross T^{1}arrow \mathbb{R}^{3}$$\Psi$$\gamma$$2\pi$

126
function by means of a Galilean transform. One can also get rid of the $A(t)$factor by means of a phase shift. More precisely, the function$\Psi$$\Psi(t, s) :=\psi(t, s-\frac{b}{2}t)\exp(i(bs-b^{2}t-\int_{0}^{t}\frac{1}{2}A(z)dz))$

,

where
$b:=1- \frac{1}{2\pi}\int_{0}^{2\pi}T(O, z)dz=1-\frac{1}{2\pi}\int_{0}^{2\pi}T(t, z)dz$

is well-defined on
Schr\"odinger equation$I\cross T^{1}$and is a solution to the cubic focusing nonlinear(7)$\partial_{t}\Psi+\partial_{ss}\Psi+\frac{1}{2}|\Psi|^{2}\Psi=0$

on$I\cross T^{1}$

Equation (7) is integrable and known to be solvable by the inverse scattering
method since the works of Zakharov and Shabat [20] in 1971 for the vanishingcase and Ma and Ablowitz [18] in 1981 for the periodic case. Therefore thebinormal curvature flow equation (1) and the Schr\"odinger map equation (4) arealso integrable, in the weak sense that they can be mapped to an integrable
equation. In particular, solitons of NLS give rise to solitons on a filament(Hasimoto’s original title). Notice however that the inverse of the Hasimototransform, whenever it is well defined5, involves a big deal of nonlinearity.

4

beautiful class of special solutions(after S. Kida)$A$

Besides the soliton solutions of Hasimoto, another special class of solutions of(1) was singled out in 1981 by S. Kida [14] (guess what?!). He studied the setof initial curves infor which the solution map of the binormal curvature
flow equation reduces to a family of rigid motions. Using the symmetries of (1)and its conservations laws, such motions are necessarily the superposition of aconstant speed rotation around a fixed axis (which we may always assume tobe the -axis after a fixed rotation) and a constant speed translation parallelto that same axis. Following [14], we denote by and the speeds of rotation
aroundand of translation along respectively, and by $C$ the speed of theslipping motion of the curve along itself6 , so that$\mathbb{R}^{3}$$x_{3}$$\Omega$$e_{3}$$V$$e_{3}$$\gamma(t, s)=\gamma(0, s-Ct)\cdot(-\sin(\Omega t)\cos(\Omega t)0\cos(\Omega t)sin(\Omega t)0001)+V(0,0, t)$$\overline{5Vanishing}$of yields underdetermination.
6Even though the speed given by the binormal curvature flow equation is perpendicular$\Psi$

to the tangent vector, in a non-orthogonal frame likethe component $C$ ofalong
may not be zero. For the rigid motions we consider, $C$ is a constant functionof space and time.
$(\partial_{s}\gamma, \gamma\cross e_{3}, e_{3})$$\partial_{t}\gamma$$\partial_{s}\gamma$

127
and therefore$\gamma$satisfies the additional equation(8).

$\partial_{t}\gamma=-C\partial_{s}\gamma+\Omega e_{3}\cross\gamma+Ve_{3}$Combining (1) and (8) we write$\partial_{s}\gamma\cross\partial_{ss}\gamma=-C\partial_{S}\gamma+\Omega e_{3}\cross\gamma+Ve_{3}$Taking the scalar product of (9) with$\partial_{s}\gamma$(9).

yields(10)$-C+\Omega r^{2}\theta’+Vz’=0$ ,where we wrote. Taking the vector product of (9) with$(\gamma^{1}(0, \cdot), \gamma^{2}(0, \cdot))=:(r(\cdot)\cos(\theta(\cdot)), r(\cdot)\sin(\theta(\cdot)))$$z(\cdot)$$\partial_{s}\gamma$

and$\gamma^{3}(0, \cdot)=$:

instead yields$z”=- \Omega rr’=-\frac{\Omega}{2}R’$

(11)

,

where $R=r^{2}$ After integration, (11) leads to(12)$z’= \frac{\Omega}{2}[A-R]$

for some integration constant(12) we obtain$A\in \mathbb{R}$. Assume that$\Omega\neq 0$$\theta’=\frac{1}{2}V+(C-\frac{1}{2}AV\Omega)/(\Omega R)$. Combining (10) and

,

(13)

and then combining (1) with (12) and (13) we finally obtain$(R’)^{2}+f(R)=0$ ,

(14)

where
$f(R)= \Omega^{2}R^{3}+(V^{2}-2A\Omega^{2})R^{2}+(4V(C-\frac{1}{2}AV\Omega)/\Omega+\Omega^{2}A^{2}-4)R$(15)$+4(C- \frac{1}{2}AV\Omega)^{2}/\Omega^{2}$

Playing on the many parameters $A,$ $C,$
, it is then possible to identify asubclass leading to smooth closed curves. Nice examples and pictures can befound in the original paper [14]. In [9], we have used this family in order
to construct counter-examples to some form of continuity properties for the
Schr\"odinger map equation (4) mentionned in the previous section.
$V,$$\Omega$128$5A$ puzzling class of special solutions
Since open problems are particularly welcomed in RIMS proceedings, we shallend this short survey by numerical curiosities which presumably have interesting theoretical roots (and may be related to integrability).
As mentionned in the second section, our existence theory in [10] allows to
consider initial curves for (1) (or rather for its extended formulation) that arebarely Lipschitz, and in particular polygonal lines. In order to get some insighton the corresponding solutions, we performed numerical simulations according
to an algorithm due to Buttke [4] (after all even smooth curves when discretized
become polygonal lines ). If is a solution to (1), the corresponding tangent:
satisfies the Schr\"odinger map equation
vector(4). Buttke’s algorithm simulates the binormal curvature flow equation (1) by
the Crank-Nicolson type discretization$\ldots$$u:=\partial_{s}\gamma$$\gamma$$I\cross(\mathbb{R}/I\mathbb{Z})arrow S^{1}$$\frac{u_{n}^{j+1}-u_{n}^{j}}{\triangle t}=(\frac{u_{n}^{j}+u_{n}^{j+1}}{2})\cross(\frac{u_{n-1}^{j}+u_{n+1}^{j}}{2(\triangle x)^{2}}+\frac{u_{n-1}^{j+1}+u_{n+1}^{j+1}}{2(\triangle x)^{2}})$of (4), and numerical integration to recover from . The implicit scheme forcan be resolved by a fixed point method if
for some explicit$\sigma>0$ ;it has the advantage that the constraint, the mean
, andthe discrete squared
quantities
norm
are conservedof thescheme.
Doing so, we have observed some phenomena which we did not expect, andfor which we have no clear explanation7.
In the following pictures, we present the shape of the simulated solution at
different (well chosen) times for a 5000 points discretization of a unit squareparallel to the $xy$-plane as initial datum.
$\gamma$$u$$\Delta t<\sigma(\triangle x)^{2}$$u$$|u_{n}^{j}|=1$$\dot{H}^{1}$$r$

Ftow Te$\sum_{n}u_{n}^{j}$$\sum_{n}|u_{n}^{j}-u_{n+1}^{j}|^{2}$$m\infty$ Tr$\triangleright\alpha$

7L. Vega (Univ. of Bilbao) recently told us that he believes to have some kind of explanation and should publish it shortly. See also [2] for related questions.

129$b\mathfrak{a}om\cdot 1R,\approx 0079\Phi$$e_{\{\mathfrak{d}\circ m}\cdot|FMT\cdot 0(oe\theta 1$$a\cap\pi m\cdot I$

Flow$T\cdot 0$

15878

As it may suggests, at some times close to 0.05296, 0.07948, 0.10591 and
0.15878, the (or ”) solution seems to turn again polygonal. Notice thatthe symmetries of the square are preserved (intermediate shapes have 8 or 12$a$sides), and that the square in the last picture is rotated by $\pi/4$ with respect
to the initial one. At times intermediate between those special moments thesimulated solution looks quite irregular and has not been represented. Also,
running the simulation further in time suggests that this sequence is reproduced in (quasi)periodic manner. It is of course tempting to believe thatsolitons could play a role here (notice in particular the ratios of those special
times); on the other hand polygons are the worst possible examples for the
Hasimoto transform (the solution is not smooth and the curvature vanishes
almost everywhere!).
$a$This kind of phenomena seems rather robust to some changes in the initial
polygon. This history of (1) so far should convince japanese mathematicians
to spend some time on that problem!

References[1] R.J. Arms and F.R. Hama, Localized-induction concept on a curved vortex
and motion of an elliptic vortex ring, Phys. Fluids 8 (1965), 553-559.
[2] V. Banica and L. Vega, On the stability of a singular vortex dynamics,
Comm. Math. Phys. 286 (2009), no. 2, 593-627.

130[3] R. Betchov, On the curvature and torsion of an isolated vortex filament,
J. Fluid Mech. 22 (1965), 471-479.

[4] T. F. Buttke, A numerical study of superfluid turbulence in the selfinduction approximation. Thesis, University of California, Berkeley, 1986.
[5] L. S. Da Ri os, Sul moto di un filetto vorticoso di forma qualunque, Rend.
del Circolo Mat. di Palermo 22 (1906), 117-135 and 29 (1910), 354-368.

[6] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972),

477-485.
[7] H. Helmholtz, Uber Integmle der hydrodynamischen Gleichungen welche
den Wirbelbewegungen entsprechen, J. Reine Angew. Math. 55 (1858),25-55.
[8] H. Helmholtz (translated by P.G. Tait), On the integrals of the hydrodynamical equations which express vortex-motion, Phil. Mag. Series 4, 33(1867), 485-512.

[9] R.L. Jerrard and D. Smets, On Schr\"odinger maps fromscientifiques de I’ENS 45, fascicule 4 (2012), 637-680.

$T^{1}$

to$S^{2}$, Annales[10] R.L. Jerrard and D. Smets, The motion of a curwe by its binormal curvature, submitted.

[11] T. Kambe and T. Takao, Motion of Distorted Vortex Rings, Jour. Phys.
Soc. Japan 31 (1971), no. 2, 591-599.
[12] Lord Kelvin, On Vortex Atoms, Proceedings of the Royal Society of Edinburgh 6 (1867), 94-105.

[13] Lord Kelvin, Vibrations of a columnar vortex, Phil. Mag. 10 (1880), 155-

168.

[14] S. Kida, A vortex filament moving without change of form, J. Fluid Mech.
112 (1981), 397-409.

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