数理解析研究所講究録
第 1905 巻 2014 年 53-60

53

On the Motion of a Vortex Filament
in an External Flow
Masashi Aiki
Department of Mathematics, Tokyo Institute of Technology
Tatsuo Iguchi
Department of Mathematics, Keio University
Abstract

We consider a nonlinear model equation describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. In the present problem
setting, we also take into account the eﬀect of external flow. We prove the uniquesolvability, locally in time, of an initial value problem posed on the one dimensional
torus. The problem describes the motion of a closed vortex filament.
. 1

Introduction

A vortex filament is a space curve on which the vorticity of the fluid is concentrated.
Vortex filaments are used to model very thin vortex structures such as vortices that trailoﬀ airplane wings or propellers. In this paper, we prove the solvability of the following
initial value problem which describes the motion of a closed vortex filament.
$x_{t}= \frac{x_{\xi}\cross x_{\xi\xi}}{|x_{\xi}|^{3}}+F(x, t) , \xi\in T, t>0,$(1.1)$x(\xi, 0)=x_{0}(\xi) , \xi\in T,$

where $x(\xi, t)=(x_{1}(\xi, t), x_{2}(\xi, t), x_{3}(\xi, t))$ is the position vector of the vortex filamentparametrized by at time , the symbol
the exterior product in the three dimensional$F$
Euclidean space,t) is a given external flow field, is the one dimensional torus $R/Z,$
and subscripts are diﬀerentiations with the respective variables. Problem (1.1) describes
the motion of a closed vortex filament under the influence of external flow. Such a setting
can be seen as an idealization of the motion of a bubblering in water, where the thicknessof the ring is taken to be zero and some environmental flow is also present. Many otherphenomena can be modeled by a vortex ring or a closed vortex filament and are importantin both application and theory. Here, we make the distinction between a vortex ring anda closed vortex filament. A vortex ring is a closed vortex tube, in the shape of a torus,
which has a finite core thickness. A closed vortex filament is a closed curve, which can be
regarded as a vortex ring with zero core thickness.
$\xi$

$t$

$\cross is$

$T$

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The equation in problem (1.1) is a generalization of a equation called the Localized
Induction Equation (LIE) given by$x_{t}=x_{s}\cross x_{ss},$

which is derived by applying the so-called localized induction approximation to the BiotSavart integral. Here, is the arc length parameter of the filament. The LIE was first
derived by Da Rios in 1906 and was re-derived twice independently by Murakami etal. in 1937 and by Arms and Hama in 1965. Many research has been done on the
LIE and many results have been obtained. Nishiyama and Tani [11, 12] proved theunique solvability of the initial value problem in Sobolev spaces. Koiso considered ageometrically generalized setting in which he rigorously proved the equivalence of the LIE
and a nonlinear Schr\"odinger equation. This equivalence was first shown by Hasimoto[6] in which he studied the formation of solitons on a vortex filament. He defined atransformation of variable known as the Hasimoto transformation to transform the LIE
into a nonlinear Schr\"odinger equation. The Hasimoto transformation was proposed by
Hasimoto [6] and is a change of variable given by$s$$\psi=\kappa\exp(i\int_{0}^{s}\tau ds)$,

where is the curvature and is the torsion of the filament. Defined as such, it is wellknown that satisfies the nonlinear Schr\"odinger equation given by$\tau$$\kappa$$\psi$(1.2)$i\frac{\partial\psi}{\partial t}=\frac{\partial^{2}\psi}{\partial s^{2}}+\frac{1}{2}|\psi|^{2}\psi.$

The original transformation proposed by Hasimoto uses the torsion of the filament in its
definition, which means that the transformation is undefined at points where the curvatureof the filament is zero. Koiso [9] constructed a transformation, sometimes referred to as
the generalized Hasimoto transformation, and gave a mathematically rigorous proof of theequivalence of the LIE and (1.2). More recently, Banica and Vega [2, 3] and Guti\’errez,
Rivas, and Vega [4] constructed and analyzed a family of self-similar solutions of the LIE
which forms a corner in finite time. The authors [1] proved the unique solvability of an
initial-boundary value problem for the LIE in which the filament moved in the threedimensional half space. Nishiyama and Tani [11] also considered initial-boundary value
problems with diﬀerent boundary conditions. These results fully utilize the property thata vortex filament moving according to the LIE doesn’t stretch and preserves its arc lengthparameter. This is not the case when we consider external flow.
The LIE can be naturally generalized to take into account the eﬀect of external flow.
The model equation is given by(1.3)$x_{t}= \frac{x_{\xi}\cross x_{\xi\xi}}{|x_{\xi}|^{3}}+F(x, t)$.

Here, the parametrization of the filament has been changed to because, unlike the LIE,
a vortex filament moving according to (1.3) stretches in general and the arc length is nolonger preserved. It is worth mentioning that if the Jacobi matrix of $F$ is skew-symmetric,
$\xi$

55
which amounts to assuming that the eﬀect of external flow consists only of translation
and rigid body rotation, then the solvability for (1.3) can be considered in the same way
as for the LIE. This is because if the Jacobi matrix is skew-symmetric, then the filament
no longer can stretch, and the techniques used in the analysis of the LIE can be utilizedfor (1.3). Thus, in what follows, we don’t assume any structural conditions on $F.$
Regarding the solvability of (1.3), Nishiyama [10] proved the existence of weak solutions to initial and initial-boundary value problems in Sobolev spaces. The solutions
obtained by Nishiyama are weak in the sense that the uniqueness of the solution is notknown, but the equation is satisfied in the point wise sense almost everywhere. The result
presented in this paper is an extension of Nishiyama’s result for the initial value problem,
and we proved the unique solvability in higher order Sobolev spaces.
The contents of the rest of the paper are as follows. In Section 2, we define notations
used in this paper and state our main theorem. In Section 3, we give a brief description
for the construction of the solution, and in Section 4, we give the main part of the proofof the theorem, which is to obtain energy estimates of the solution in $C([0, T];H^{m}(T))$ ,in more detail.

2

Function Spaces, Notations, and Main Theorem

We define some function spaces that will be used throughout this paper, and notations
associated with the spaces. For a non-negative integer $m$ , and $1\leq p\leq\infty,$ $W^{m,p}(T)$ is
the Sobolev space containing all real-valued functions that have derivatives in the senseof distribution up to order $m$ belonging to $L^{p}(T)$ . We set $H^{m}(T)$ $:=W^{m,2}(T)$ as the
Sobolev space equipped with the usual inner product. The norm in $H^{m}(T)$ is denoted by
and we simply writefor. Otherwise, for a Banach space $X$ , the norm in$X$ is written as. The inner product in $L^{2}(T)$ is denoted by
For $0<T<\infty$ and a Banach space $X,$ $C^{m}([0, T];X)$ denotes the space of functions
that are $m$ times continuously diﬀerentiable in with respect to the norm of $X$ , and$L^{2}(0, T;X)$ is the space of functions with the norm
being finite.
For any function space described above, we say that a vector valued function belongs
to the function space if each of its components does.
$\Vert\cdot\Vert_{m}$$\Vert\cdot\Vert$$\Vert\cdot\Vert_{0}$$\Vert\cdot\Vert_{X}$$t$$\int_{0}^{T}\Vert u(t)||_{X}^{2}dt$

Now we state our main theorem regarding the solvability of (1.1).

Theorem 2.1 For $T>0$ and natural number $m\geq 4$ , if the initial filamentsatisfies, and the external flow $F$ satisfies $F\in C([O, T];W^{m,\infty}(R^{3}))$ ,andthen there exists $T_{0}\in(0, T] such that a$ unique solution $x(\xi, t)$ of (1.1) exists and satisfies$x_{0}$$x_{0}\in H^{m}(T)$$|x_{0\xi}|\equiv 1$$x\in C([0, T_{0}];H^{m}(T))\cap C^{1}([0, T_{0}];H^{m-2}(T))$

The above theorem gives the time-local unique solvability of (1.1). We note that Nishiyama
proved the existence of the solution in $C([O, T];H^{2}(T))$ for any $T>0$ , and comparing
with our result, we notice that the case $m=3$ is missing. So far, we don’t know whether
solvability can be shown in this case.

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3Construction of Solution

In this section, we give a brief explanation regarding the construction of the solution.
The method shown in this section is due to Nishiyama [10]. We construct the solution toin the following regularized problem.
problem (1.1) by passing to the limit$\epsilonarrow+0$(3.1)$\{\begin{array}{ll}x_{t}=-\epsilon x_{\xi\xi\xi\xi}+\frac{x_{\xi}\cross x_{\xi\xi}}{|x_{\xi}|^{3}+\epsilon^{\alpha}}+F(x, t) , \xi\in T, t>0,x(\xi, 0)=x_{0}(\xi) , \xi\in T,\end{array}$and with $0<\alpha<8/3$ are real parameters. The solution of problem (3.1)
wherecan be constructed by an iteration scheme based on the solvability of the following linear
problem.
$\epsilon>0$(3.2)$\alpha$$\{\begin{array}{ll}x_{t}=-\epsilon x_{\xi\xi\xi\xi}+G, \xi\in T, t>0,x(\xi, 0)=x_{0}(\xi) , \xi\in T.\end{array}$

Finally, the unique existence of the solution to (3.2) in $C([0, T];.H^{m}(T))\cap C^{1}([0, T];H^{m-2}(T))$
for any $T>0$ and $m\geq 2$ is known from the standard theory of parabolic equations.
Hence, by iteration, we can prove the solvability of problem (3.1) in the same function
space. It is shown in [10] that a solution of (3.1) belonging to $C([0, T];H^{2}(T))$ satisfies$|x_{\xi}(\xi, t)|\geq c_{0}>0$ for some positive constantfor all $\xi\in T$ and $t\in[0, T]$ . We also makeuse of this property in the next section.
$c_{0}$

4

Energy Estimates of the Solution

Our next and final step is to derive energy estimates for the solution to (3.1) which are
and finish the. This will allow us to pass to the limit
uniform with respect toproof of Theorem 2.1. We do this by deriving suitable energies that allow us to estimate
the solution in the appropriate function space. The derivation of such energy is the mostimportant part of the proof and thus, we go into more detail. For simplicity, we deriveenergy estimates for the solution to our original problem (1.1) because the arguments forthe uniform estimates of the solution to (3.1) are the same.
Our objective is to derive energy estimates for the solution of$\epsilonarrow+0$$\epsilon>0$(4.1)$\{\begin{array}{ll}x_{t}=\frac{x_{\xi}\cross x_{\xi\xi}}{|x_{\xi}|^{3}}+F(x, t) , \xi\in T, t>0,x(\xi, 0)=x_{0}(\xi) , \xi\in T,\end{array}$withbelonging to $C([O, T];H^{m}(T))\cap C^{1}([0, T];H^{m-2}(T))$ on some time interval$the$$T_{0}\in(0, T]. The$ diﬃculty arises fromfact that solution $of (4.1)$ stretches, i.e.
,many useful properties of the solution can. Wheneven if
be utilized to obtain energy estimates, but these properties are not at our disposal in thepresent problem setting.
$[0, T_{0}]$$a$$|x_{\xi}|\not\equiv 1$$|x_{0\xi}|\equiv 1$$|x_{\xi}|\equiv 1$

57
To overcome this, we modify the Sobolev norm to obtain a suitable form of energy
which allow us to derive the necessary estimates. First, we set
and take thederivative of (4.1) to rewrite the equation in terms of
$v$

$\xi$

$:=x_{\xi}$

$v.$

(4.2)

$\{\begin{array}{ll}v_{t}=fv\cross v_{\xi\xi}+f_{\xi}v\cross v_{\xi}+(DF)v, \xi\in T, t>0,v(\xi, 0)=v_{0}(\xi) , \xi\in T,\end{array}$where we have set
Since the$:=x_{0\xi},$ $f=1/|v|^{3}$ , and omitted the arguments of $F$ .
energy estimate for the solution in $C([0, T];H^{2}(T))$ is already obtained in Nishiyama [10],we only show the higher order estimates. Following standard procedures, we diﬀerentiate
the equation in (4.2) with respect totimes for a fixed satisfying $3\leq k\leq m$ and set
to obtain$v_{0}$$\xi,$$k$$k$$v^{k}:=\partial_{\xi}^{k}v$(4.3)$v_{t}^{k}=fv\cross v_{\xi\xi}^{k}+kfv_{\xi}\cross v_{\xi}^{k}+(k+1)f_{\xi}v\cross v_{\xi}^{k}+G^{k},$whereare terms that contain derivatives of up to order . From here on, we
regard terms with derivatives of up to order as lower order and disregard the precise
expression of the terms. We can do this because these terms are harmless in terms ofregularity when estimating the solution, although the nonlinearity of these terms are highin general and cause the estimates to become time-local. In what follows, we will use thesymboldenote that two sides are equal modulo lower order terms such as. For
example, (4.2) can be expressed as$G^{k}$

$k$

$v$

$k$

$v$

$G^{k}$$\sim to$$v_{t}^{k}\sim fv\cross v_{\xi\xi}^{k}+kfv_{\xi}\cross v_{\xi}^{k}+(k+1)f_{\xi}v\cross v_{\xi}^{k}.$Now that we have derived (4.3), the standard method would be to take the innerproduct ofand (4.3) and integrate overwith respect to to estimate the time
evolution of. This is not possible for our equation because the terms with derivativesof cause a loss of regularity. To avoid such loss, we employ a series of change of variables
to derive a modified energy from which we can derive the necessary estimates. The key
idea is to decompose
into two parts. More precisely, we decompose
as$v^{k}$

$T$

$\xi$

$\Vert v^{k}\Vert$$v^{k}$$v^{k}$(4.4)$v^{k}$$v^{k}= \frac{(v\cdot v^{k})}{|v|^{2}}v-\frac{1}{|v|^{2}}v\cross(v\cross v^{k})$.

The above decomposes
into the sum of its component and the component orthogonal
to . The decomposition is well-defined since we know that $|v|\geq c_{0}>0$ . The principle
part of the components arerespectively, and we define two new variables
and$v^{k}$$v$$v$$v\cdot v^{k}$$v\cross v^{k}$$h^{k}:=v\cdot v^{k},$$Z^{k}:=V\cross v^{k},$and estimate them separately.

58
4.1

Estimate of$h^{k}$

We first derive an equation for$h^{k}$. Taking the inner product of and equation (4.2) yields$v$$v\cdot v_{t}=v\cdot((DF)v)$

Diﬀerentiating$k$times with respect to

$\xi$

.

further yields$v\cdot v_{t}^{k}+kv_{\xi}\cdot v_{t}^{k-1}\sim 0.$with $m\geq 4$ , we can regard
Since we are estimating the solution inlower order, and thus, we can further calculate$H^{m}$$\Vert v\Vert_{W^{3,\infty}(T)}$as$0\sim v\cdot v_{t}^{k}+kv_{\xi}\cdot v_{t}^{k-1}$$\sim[v\cdot v^{k}+kv_{\xi}\cdot v^{k-1}]_{t}$$=[h^{k}+kv_{\xi}\cdot v^{k-1}]_{t}.$Finally, since$kv_{\xi}\cdot v^{k-1}$

is lower order, we obtain$\frac{1}{2}\frac{d}{dt}\Vert h^{k}+kv_{\xi}\cdot v^{k-1}\Vert^{2}\leq C(1+\Vert v\Vert_{k})^{n(k)}\Vert v\Vert_{k}^{2},$where $n(k)$ is an integer depending on that is greater than in general. $Rom$ the above
estimate, we see that there is a $T_{1}\in(0, T$] such that for some constant $C_{*}>0$ dependingonandsatisfies$k$$\Vert v_{0}\Vert_{k}$$T_{1},$$0$$h^{k}$$\Vert h^{k}(t)\Vert^{2}\leq C_{*}$

for any $t\in(0, T_{1}$ ].

Estimate of
Next we consider . Directly calculating the derivative of
4.2$z^{k}$$z^{k}$

(4.5)

$t$

$z^{k}=v\cross v^{k}$yields$z_{t}^{k}\sim fv\cross z_{\xi\xi}^{k}+(k-2)fv\cross(v_{\xi}\cross v_{\xi}^{k})+(k+1)f_{\xi}v\cross z_{\xi}^{k}$First we notice that$v\cross(v_{\xi}\cross v_{\xi}^{k})=(v\cdot v_{\xi}^{k})v_{\xi}-(v\cdot v_{\xi})v_{\xi}^{k}$(4.6)$\sim h_{\xi}^{k}v_{\xi}-(v\cdot v_{\xi})v_{\xi}^{k}.$To proceed further, we must express
decomposition as in (4.4) and obtain$v_{\xi}^{k}$in terms of$h^{k}$

and$z^{k}$$v_{\xi}^{k}= \frac{(v\cdot v_{\xi}^{k})}{|v|^{2}}v-\frac{1}{|v|^{2}}v\cross(v\cross v_{\xi}^{k})$$\sim\frac{h_{\xi}^{k}}{|v|^{2}}v-\frac{1}{|v|^{2}}v\cross z_{\xi}^{k}.$. Specifically, we apply the

59
Substituting this into (4.6) yields$v\cross(v_{\xi}\cross v_{\xi}^{k})\sim h_{\xi}^{k}v_{\xi}-(v\cdot v_{\xi})v_{\xi}^{k}$$\sim h_{\xi}^{k}(v_{\xi}-\frac{(v\cdot v_{\xi})}{|v|^{2}}v)+\frac{(v\cdot v_{\xi})}{|v|^{2}}v\cross z_{\xi}^{k}$$=- \frac{h_{\xi}^{k}}{|v|^{2}}[v\cross(v\cross v_{\xi})]+\frac{(v\cdot v_{\xi})}{|v|^{2}}v\cross z_{\xi}^{k}$

Substituting this back into (4.5) yields$z_{t}^{k} \sim fv\cross\{z_{\xi\xi}^{k}-(k-2)\frac{h_{\xi}^{k}}{|v|^{2}}v\crossv_{\xi}\}+\{(k-2)f\frac{(v\cdot v_{\xi})}{|v|^{2}}+(k+1)f_{\xi}\}v\cross z_{\xi}^{k}.$Next we focus on first term on the right-hand side

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