数理解析研究所講究録
1076 巻 1999 年 93-106

93

ON THE EVOLUTION OF A HIGHENERGY VORTICITY IN AN IDEAL FLUID

東京工業大学理学部

大塚浩史 (HIROSHI OHTSUKA)

l.Introduction.
The motion of a two dimensional incompressible homogeneous ideal

fluid is governed by the system of the partial differential equations called
the Euler equations. On the other hand, the vortex model is also usedby many researchers to study the motion of the fluid. In the vortex
model, the (scalar) vorticity is assumed to be concentrated in some points

evolving according to the system of the ordinary differential equations

called the Kirchhoff-Routh equations. (3.2) in section 3 is an example ofa system of the Kirchhoff-Routh equations.
However, if some parts of the vorticity concentrate in some points,

the fluid never has finite kinetic energy. Moreover, the solution of theKirchhoff-Routh equations does not constitute a solution of the Euler
equations even in such a weak sense as Definition 2.1 in this note.
Therefore, we want to understand the solutions of the Kirchhoff-

Routh equations in terms of the solutions of the Euler equations withhigh but finite kinetic energies.
To this purpose, we define the high energy vorticities and establish
the structure theorems of them (Theorem A and Theorem B). Then, we
Typeset by$A_{\mathcal{M}}S-\mathfrak{G}^{\mathfrak{c}}$94
consider the limit of the energy diverging sequence of the weak solutionsof the two-dimensional incompressible Euler equation (Theorem C).

Remark. The details of this note are in [O].
2.

$\mathrm{O}\mathrm{n}$two demensional incompressible ideal fluids.

Let

be a simply connected bounded domain with smooth$\Omega\subset R^{2}$

boundary$\partial\Omega$. We consider the motion of the incompressible homoge-

neous ideal fluid with unit density in$\Omega$.

The Euler equations are as follows:$\frac{\partial u}{\partial t}+(u\cdot\nabla)u=-\nabla p$in$\mathrm{d}\mathrm{i}\mathrm{v}u=\frac{\partial}{\partial x^{1}}u1+\frac{\partial}{\partial x^{2}}u^{2}=0$$u\cdot n=0$

on$\Omega\cross(0, T)$in,(2.1)$\Omega\cross(0, T)$$\partial\Omega\cross(0, T)$,(2.2)(2.3),

where $u(x, t)=(u^{1}(X, t),$ $u^{2}(x, t))$ is the velocity field and $p(x, t)$ is the(scalar) pressure.
Applying the curl operator to (2.1), we obtain the evolution equation:of the vorticity$\omega(x, t)=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}u(x, t)=\frac{\partial}{\partial x^{1}}u^{2}(x, t)-\frac{\partial}{\partial x^{2}}u^{1}(X, t)$.

(2.4)$\frac{\partial\omega}{\partial t}+(u\cdot\nabla)\omega=0’$

Because$\Omega$

is simply connected, it is well known that$u$satisfying (2.2)

and (2.3) admits the representation$u= \nabla^{\perp}c_{\omega=}(\frac{\partial}{\partial x^{2}}G\omega, -\frac{\partial}{\partial x^{1}}G\omega)$,(2.5)

where $G \omega(x, t)=\int_{\Omega}g(x, y)\omega(y, t)dy$ and $g(x, y)$ is the Green function of$-\Delta$

with -Dirichlet boundary condition.
$0$

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Using (2.5), we can eliminate$u$from (2.4) as follows:$\frac{\partial\omega}{\partial t}+(\nabla^{\perp}G\omega\cdot\nabla)\omega=0$

.

(2.6)

Therefore, the vorticity evolves by itself according to (2.6), which we callthe Euler equation for the vorticity, or simply, the Euler equation in thisnote. We consider solutions of (2.6).
It is well-known that a sufficiently smooth solution of the vorticity
equation conserves several quantities during the evolution. Especially,

the kinetic energy $(1/2) \int_{\Omega}|u(x, t)|^{2}dx$ , which is equal to $E(\omega(\cdot, t))=$$\frac{1}{2}\int_{\Omega}g(x, y)\omega(X, t)\omega(y, t)dXdy$tion$f(\cdot)$$f’(\omega)$, and$\int_{\Omega}f(\omega(x, t))dx$

are conserved. Indeed, multiply each side of (2.6) byrespectively and integrate them. Moreover,$q\leq\infty$

for any smooth func-

is conserved. Furthermore, if$\omega$$||\omega(\cdot, t)||_{L^{q}(\Omega)}$is positive initially, then$G\omega$

and

for

$1\leq$$\omega$is alsopositive at every following time.

In our analysis, a vorticity should be a weak solution in the senseof Definition 2.1 below staying in the high energy class $E(q, s, K)$ defined in section 3. In particular, we consider the weak solutions in$L^{\infty}(\mathrm{O}, T;L^{q}(\Omega))$

Using

for $q>1$ .

$(\nabla^{\perp}G\omega\cdot\nabla)\omega=\mathrm{d}\mathrm{i}\mathrm{v}(\omega\nabla\perp G\omega)$, we have the following weak

expression of (2.6):$\int_{0}^{T}\int_{\Omega}[\omega\frac{\partial\varphi}{\partial t}+\omega(\nabla^{\perp}G\omega\cdot\nabla)\varphi]dXdt=0$

for every scalar function $\varphi\in D(\Omega\cross(0, T))$ .

.

(2.7)

From the elliptic reg-ularity theory and the Sobolev embedding, (2.7) has the meaning if$\omega\in L^{\infty}(\mathrm{o}, \tau;L^{4}/3(\Omega))$. However, we can extend (2.7) as follows:96
Definition 2.1. $\omega(x, t)\in L^{\infty}((0, T);L1(\Omega))$ is a weak solution of theEuler equation if$\omega(x, t)$satisfies$\int^{T}0[\int\Omega d\omega\frac{\partial\varphi}{\partial t}X+\int_{\Omega}\int_{\Omega}\hat{H}_{\varphi}(x, y, t)\omega(X, t)\omega(y, t)dxdy]dt=0$

for every scalar function$\varphi\in D(\Omega\cross(\mathrm{O}, T))$

(2.8)

, where$\hat{H}_{\varphi}(x, y, t)=H_{\varphi}(x, y, t)+\nabla_{x}^{\perp}h(x, y)\cdot\nabla\varphi(x, t)$,$H_{\varphi}(x, y, t)=- \frac{1}{4\pi}\frac{(x-y)^{\perp}\cdot[\nabla\varphi(x,t)-\nabla\varphi(y,t)]}{|x-y|^{2}}$,$h(x, y)=g(x, y)- \frac{1}{2\pi}\log|X-y|-1$(2.8) is equivalent to (2.7) if$\omega\in L^{\infty}(\mathrm{O}, T;L^{4}/3(\Omega))$. For the precise

derivation of (2.8) from (2.7), see [S] (or $[0]$ ).

We know the following facts on the existence of a weak solution ofthe Euler equation:

Facts 2.2. ($Lions[L]$ and YudovichlYl) For every$q\leq\infty$, there exists a weak solution$\omega\in C([0, \infty);L^{p}(\Omega))$

this$\omega$conserves$\omega_{0}\in L^{q}(\Omega)$$\omega\in L^{\infty}(\mathrm{O}, \infty;L^{q}(\Omega))$for $1<$such that. Moreover,
for all $1\leq p<q$ andfor every $1\leq r\leq q$ , and the positivity$E(\omega),$$\omega(\cdot, 0)=\omega_{0}(\cdot)$$||\omega||L^{r}(\Omega)$of .
$\omega$

Remark. For $q=\infty$ , Yudovich proves the above facts [Y]. For $1<q<\infty$ ,we obtain Facts 2.2 from the results by Lions [L], though his notion of
the weak solution of the Euler equations is different from us.

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3.

results.

$\mathrm{T}\mathrm{h}\mathrm{e}$We consider the following classes of the vorticities:$P(\Omega)=${$\omega\in L^{1}(\Omega)$:$\omega\geq 0$Let$I_{\epsilon}$$s$0\leq||\omega||_{Lq(}\Omega)\leq s\}$be the characteristic function of$x_{0}\in\Omega$$||\omega||L^{1}(\Omega)=1$

is a typical element of}.

for some $1<q\leq\infty$ .

$P_{q}(\Omega)=P(\Omega)\cap L^{q}(\Omega)$$P_{q}(\Omega, s)=\{\omega\in P_{q}(\Omega) :and$B_{\epsilon}(\mathrm{O})$for some $s>0$ .

. Then$\lambda I_{\epsilon}(x-X_{0})$for sufficiently large$P_{q}(\Omega, s)$

if$s$

for some$\lambda,$

and$\epsilon$that satisfy$\pi\epsilon^{2}\lambda=1$

and$\pi\epsilon^{2}\lambda^{q}=s^{q}$,i.e.,$\epsilon=(\pi s^{q’})-1/2$

where $q’=q/(q-1)$ for $1<q<\infty$ and $q’=1$ for $q=\infty$ .

of this note,$\mathrm{i}.\mathrm{f}\epsilon_{i}$$\lambda,$$\epsilon$

and

$s$

$\ln$

(3.1)

the rest

always satisfy these relations (3.1), Moreover,is given for example,$\pi\epsilon_{i}^{2}\lambda_{i}q=S_{i}q$,$\lambda_{i}$and$s_{i}$are determined by$\pi\epsilon_{i}^{2}\lambda_{i}=1$

and.

The following facts are standard:
Fact 3.1.
$E(q, s):=$$\sup$$E(\omega)<\infty$and$E(q, s)arrow\infty$as$sarrow\infty$.

$P_{q}(\Omega, s)$as$\omega\in P_{q}(\Omega,s)$

Now, we define the class of the high energy vorticities of$E(q, s, K):=\{\omega\in P_{q}(\Omega, S) : E(q, s)-K\leq E(\omega)\leq E(q, s)\}$for some $K>0$ . Then, we obtain the following results:,

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Theorem A. There exists a constant$s\geq\tilde{s}_{0}$such that for every$\tilde{s}_{0}=\tilde{s}_{0}(\Omega)$, every $\omega\in E(q, s, K)$ , and every $\gamma>0$ ,,$r_{\omega}( \gamma):=\inf\{r>0:\mathrm{s}\mathrm{u}\mathrm{p}x\in\Omega\int_{\Omega\cap B()}x\omega(ry)dy\geq 1-\gamma\}\leq\epsilon\exp(c_{0}/\gamma)$where$C_{0}$

is a positive constant independent of$s,$$\omega_{f}$and .
$\gamma$

Theorem A implies that every $\omega\in E(q, s, K)$ concentrates near its

center$\overline{x}_{\omega}:=\int_{\Omega}x\omega(X)dX$, for example.

Theorem B. Fix 1 $<q\leq\infty$ and $K>0$ . There exists a constantsatisfying the following properties. If we choose any sequence$\tilde{K}\geq K$

as $narrow\infty$ and $\omega_{n}\in E(q, s_{n}, K)$ such that there exists a
as $narrow\infty$ , thenlimiting center$s_{n}arrow\infty$$\overline{x}_{n}(:=\overline{x}_{\omega_{n}})arrow\overline{x}_{\infty}\in\overline{\Omega}$$\overline{x}_{\infty}\in\Omega_{\tilde{K}}:=\{x\in\Omega :,\max_{x\in\Omega}H(X)-\tilde{K}\leq H(x)\leq\max_{x\in\Omega}H(x)\}$where $H(x)=(1/2)h(x, X)$ and $h(x, y)=g(x, y)-(1/2\pi)\log|x-y|^{-1}$ .
Theorem C. If we choose any sequence $s_{n}arrow\infty$ as $narrow\infty$ and$\omega_{n}(x, t)\in L^{\infty}(0, T;L^{1}(\Omega))$that is a weak solution the Euler equationsuch that $\omega_{n}(\cdot, t)\in E(q, s_{n}, K)$ for a. . $t\in(0, T)$ . Furthermore, if therein
is a limiting path of center$e$$\overline{x}_{n}(t)(:=\overline{x}_{\omega n(\cdot,t)})arrow\overline{x}_{\infty}(t)$$weakly*asnarrow\infty$ , then,for almost every , this$t$$\overline{x}_{\infty}(t)$$L^{\infty}(0, T)^{2}$is equal to asolution of$\frac{dz}{dt}=\nabla^{\perp}H(z)$staying in$\Omega_{\tilde{K}}$

(3.2)

.

Remark 3.1. The equations (3.2) are called the Kirchhoff-Routh equations of one vortex with unit intensity (i.e., a vorticity consists of a Diracmeasure). See,$\mathrm{e}.\mathrm{g}.,[\mathrm{T}2]$.

99
Remark 3.2. It is easy to see that there exists a sequence that satisfies
the hypothesis of Theorem$\mathrm{B}$and Theorem$\mathrm{C}$, because$\Omega$

is bounded andwe know Facts 2.2.
Remark 3.3. Our result. relate to Turkington’s results on$\mathrm{s}$$E(\infty, S, \mathrm{o})$in[T1], closely. See also [T2].

4.Sketch of the proofs.

4.1. Theorem A.
Instead of $E(q, s, K)$ , we consider$F(q, s, K)=\{\omega\in P_{q}(\Omega, s) :F(q, s)-K\leq F(\omega)\leq F(q, s)\}$,where$F(q, S)=$$\sup$$F(\omega)(<\infty)$,$\omega\in P_{q}(\Omega,S)$$F( \omega)=\frac{1}{2}\int_{\Omega}N\omega(x)\omega(X)d_{X}$,$N \omega(x)=\frac{1}{2\pi}\int_{\Omega}\log\frac{1}{|x-y|}\omega(y)dy$.

Then, we prove the following theorem:

Theorem 4.1. There exists $s_{1}>0$ depending only on

every$s\geq s_{1}$$C_{1}$

such that for, every $K>0$ , every $\omega\in F(q, s, K)$ , and every $\gamma>0$ ,$r_{\omega}(\gamma)\leq\epsilon\exp(C_{1}/\gamma)$

where$\Omega$,

is a constant depending on $K$ but independent of$\omega_{f}s$, and .
$\gamma$Now, Theorem A follows the fact that $E(q, s, K)\subset F(q, s, K’)$ forsome $K’>0$ , which we can see from the following energy estimate:

100
Facts 4.2.

(1) $E(\omega)\leq F(\omega)+C_{2}$ ,$E(q, s)\leq F(q, s)+C_{2}$ ,especiallyfor sufficiently large(2) $F(q, s)+x \in\Omega\sup_{1}H(x)-$$C|3\leq E(q, s)$$s,$$(4.1)$

.

where

$C_{2}$

and$C_{3}$are constants independent of .
$s$

The estimate (1) is easily obtained because $h(x, y)$ is bounded fromthe above. On the other hand, the estimate of (2) is obtained by calculating the energy of the typical element$x_{0}\in\Omega$$\lambda I_{\epsilon}(x-x\mathrm{o})\in P_{q}(\Omega, s)$, wheresatisfies $H(x \mathrm{o})=\sup_{x\in\Omega}H(x)$ .

The following estimate of the Newton potential $N\omega(x)$ ofis the$\omega(x)$key in the proof of Theorem 4.1:

Lemma 4.3. For every$1<q\leq\infty$$\epsilon>0$, every $R\geq 1$ and every$\omega\in P_{q}(\Omega)$,$N \omega(x)\leq\frac{1}{2\pi}\log\frac{1}{\epsilon}+\frac{C_{4}}{2\pi}\epsilon|2/q|\omega||_{Lq(\Omega)}-\frac{1}{2\pi}\mathrm{l}\mathrm{o}\mathrm{g}\prime R\int_{\Omega\backslash B_{R}(x}8)\omega(y)dy$where$C_{4}$

foris a constant depending only on .
$q$

Proof. We have the following decomposition of$N\omega(x)$:$N \omega(_{X)=}\frac{1}{2\pi}[\log\frac{1}{\epsilon}+\int_{\Omega\cap B_{\Xi}(x})\log\frac{\epsilon}{|x-y|}\omega(y)dy$$+ \int_{\Omega\backslash B_{\epsilon}(}x)\log\frac{\epsilon}{|x-y|}\omega(y)dy]$

lt is easy to see that$\int_{\Omega\cap B_{\epsilon}(}x)\log\frac{\epsilon}{|x-y|}\omega(y)dy\leq\in^{2}/q’c4||\omega||_{L^{q}}(\Omega)$

,.

,

101
where $C_{4}=||\log|x|||L^{Q}’(B_{1})<\infty$ . On the other hand, as $R\geq 1$ and$\omega\geq 0$, we obtain$\int_{\Omega\backslash B_{\epsilon}()}x\log\frac{\epsilon}{|x-y|}\omega(y)dy\leq\int_{\Omega\backslash B_{R\epsilon}()}x\log\frac{\epsilon}{|x-y|}\omega(y)dy$$\leq-\log R\int_{\Omega\backslash B_{R\epsilon}(x)}\omega(y)dy$

Corollary 4.4. For every sufficiently large$s$and every.

$\omega\in P_{q}(\Omega, s)$$F( \omega)\leq\frac{1}{4\pi}\log\frac{1}{\epsilon}+\frac{C_{5}}{4\pi}-\frac{1}{4\pi}\log R\inf_{x\in\Omega}\int_{\Omega\backslash B_{R\epsilon}(x})y\omega(y)d$Proof.

$F( \omega)\leq\sup_{x\in\Omega}N\omega(X)$Proof of Theorem 4.1. Fix

provided$x_{0}\in\Omega$$\omega\in P_{q}(\Omega)$. Then.

$\square$.

,(4.2)$\square$$\lambda I_{6}(x-x_{0})\in P_{q}(\Omega, s)$forsufficiently large . Therefore, we have$s$$F(q, s) \geq F(\lambda I_{\epsilon}(x-x_{0}))=\frac{1}{4\pi}\log\frac{1}{\epsilon}+\frac{C_{6}}{4\pi}$.

(4.3)

Using (4.2) and (4.3), for every $R\geq 1$ and every $\omega\in F(q, s, K)$ , we have$\log R\inf_{x\in\Omega}\int_{\Omega\backslash B_{R}(x})\omega 6(y)dy\leq 4\pi[\frac{1}{4\pi}\log\frac{1}{\epsilon}-F(\omega)]+c_{5}$$\leq 4\pi[F(q, s)-F(\omega)]-C6+c_{5}$$\leq 4\pi K+c_{5}-\mathit{0}_{6}$

.

(4.4)

Now, we take $C_{1}> \max\{4\pi K+C_{5}-C_{6},0\}$ . Then, if $R>1$ , we canrewrite (4.4) as$\sup_{x\in\Omega}\int_{\Omega\cap}B_{R\epsilon}(x)\omega(y)dy\geq 1-C_{1}/\log$

R.

(4.5)

For every $\gamma>0$ , let $R$ satisfy $\gamma=C_{1}/\log R$ , that is, $R=\exp(C_{1}/\gamma)$ .
Then, $R>1$ , since $C_{1}>0$ . Therefore, using (4.5), we obtain$\inf\{r>0:\sup_{x\in\Omega}\int_{\Omega\cap B_{r}(x}))\omega(X)dx\geq 1-\gamma\}\leq R\epsilon=\epsilon\exp(C_{1}/\gamma$.

$\square$102
4.2. Theorem B.
Let $\gamma=\gamma(s)=-C_{0}’(\log_{\mathcal{E}})-1$ for some fixed $C_{0}’>C_{0}$ . Then$\gamma(s)arrow \mathrm{O}$

Therefore,and$r(s):=\epsilon\exp(C_{0}/\gamma(s))arrow 0$$\omega_{n}arrow\delta(x-\overline{x}_{\infty})$as$sarrow\infty$.

weakly in the sense of the measures.

On the other hand, using the energy estimate (4.1), we have, forsufficiently large ,$n$$F(q, s_{n})+ \sup_{\Omega x\in}H(_{X)}-C3-K$$\leq E(q, s_{n})-K$$\leq E(\omega_{n})=F(\omega_{n})+\frac{1}{2}\int_{\Omega}\int_{\Omega}h(x, y)\omega n(x)\omega_{n}(y)dxdy$$\leq F(q, s_{n})+\frac{1}{2}\int_{\Omega}\int_{\Omega}h(x, y)\omega_{n}(x)\omega n(y)dXdy$,

that is,$\sup_{x\in\Omega}H(X)-K-c_{3}\leq\frac{1}{2}\int_{\Omega}\int_{\Omega}h(x, y)\omega n(x)\omega_{n}(y)dxdy$.

, we can see that
Since $H(x)=(1/2)h(x, X)arrow-\infty$ as
becauseThen, the righthand side of (4.6) converges to$xarrow\partial\Omega$$H(\overline{x}_{\infty})$$\delta(x-\overline{x}_{\infty})$. Therefore, we obtain Theorem$\mathrm{B}$(4.6)$\overline{x}_{\infty}\in\Omega$.

$\omega_{n}arrow$with $\tilde{K}\geq K+C_{3}$ .

4.3. Theorem C.
The following proof is essentially equal to that of Theorem 3.2 in [T2].
lnstead of considering the motion of the center of the vorticity

we consider the motion of more regular function$\tilde{x}_{n}(t)=\int_{\Omega}x\xi(x)\omega n(x, \mathrm{t})dX$,$\overline{x}_{n}(t)$,103
whereHere$\xi(x)\in C_{0}^{\infty}(\Omega)$$L_{1}>\tilde{K}$is a fixed function satisfyingis a fixed constant.

lt is easy to see thatas$||\overline{x}n-\tilde{X}n||L^{\infty}(0,T)^{2}=o(1)$$narrow\infty$

.

(4.7)

Moreover, we have the following fact:

Lemma 4.5. Let
equation. Then$\omega\in L^{\infty}(\mathrm{o}, \tau_{;}L1(\Omega))$

be a weak solution of the Euler$\tilde{x}(t)=\int_{\Omega}x\xi(X)\omega(X, t)dt\in W^{1,\infty}(\mathrm{o}, \tau)^{2}$

in$\frac{d}{dt}\tilde{x}^{i}=\int_{\Omega}\int_{\Omega}\hat{H}_{x^{i}\xi}(x, y)\omega(_{X}, t)\omega(y, t)dxdy$

Proof. Insert a test function$\eta(t)x^{i}\xi(x)$for. Especially,$D’(0, T)$$\eta(t)\in D(\mathrm{O}, T)$.

into (2.8).

The following theorem is the main part of the proof of Theorem$\square$$\mathrm{C}$:

Theorem 4.6. For every $0<T<\infty$ and every $\sigma>0$ , there exists aconstant$s_{2}$

depending on$T$and$\sigma$that satisfies following properties. Leta weak solution of the vorticity equation $\omega(x, t)\in L^{\infty}(0, T;L1(\Omega))$ satisfy. Then there exists
for a. . $t\in(0, T)$ for some$z(t)$ that is a solution
such thatof (3.2) staying in$\omega(\cdot, t)\in E(q, s, K)$$s\geq s_{2}$$e$$\Omega_{L_{1}}$$||\tilde{x}(t)-z(t)||W^{1,\infty}(0,T)^{2}\leq\sigma$.

Sketch of the proof. At every $t\in(0, T)$ such that $\omega(\cdot, t)\in E(q, s, K)$ , wemay assume that

$B_{r(s)}(\tilde{X}(t))\subset\Omega_{L_{1}}$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ if is sufficiently
$s$

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large. Now, we take $T_{1}\in(0, T)$ such that $\omega(x, T_{1})\in E(q, s, K)$ , and $z(t)$that is a solution of (3.2) satisfying $z(T_{1})=\tilde{x}(T_{1})$ . Then, $z(t)$ stays in$\Omega_{L_{1}}$because $z(t)$ conserves $H(z(t))$ . Furthermore, for a.
$| \frac{d}{dt}z^{i}(t)-\frac{d}{dt}\tilde{X}^{i}(t)|\leq J_{1}+J_{2}+J_{3}$$\mathrm{e}.\mathrm{t}$, we have,where$J_{1}=|(\nabla^{\perp}H)i(z(t))-(\nabla^{\perp}H)i(\tilde{x}(t))|$,$J_{2}=|(\nabla^{\perp}H)^{i}(\tilde{x}(t))$$t)x^{i}\xi\omega(y, t)dxdy|$ ,$- \int_{B_{r(s)}}(\tilde{x}(t))\int_{B_{r(s)(\tilde{x}(}}t))(_{X}\hat{H}(x, y)\omega,$$t)x(\omega y, t)dxdy$$J_{3}=| \int_{B_{r(s)(\tilde{x}(t}}))\int_{B_{r(s)}}(\tilde{x}(t))(_{X}\hat{H}i\xi(x, y)\omega,$$y\omega(_{X}, t)\omega(y, t)dxdy|$$- \int_{\Omega}\int_{\Omega}\hat{H}_{x^{i}\xi()}x,$.

It is easy to see that
$J_{1}\leq C_{7}|Z(t)-\tilde{X}(t)|$where$C_{7}$

is a constant depending only on

formly continuous over$\Omega_{L_{1}}$$L_{1}$, because$\nabla^{\perp}H(x)$is uni-. lt is also easy to see that,$J_{3}\leq C_{8}(||\omega||_{L}1(\Omega)+||\omega||_{L}1(Br(s)(\overline{x}(t))))||\omega||_{L}1(\Omega\backslash B)r(s(\tilde{x}(t)))\leq 2C_{8}\gamma=o(1)$where a constant$C_{8}$

and $o(1)$ is also independent of and$t$$\omega$.

By the way, we have$\int_{B_{r(s)(\tilde{x}(t}}))\int_{B_{r(s)(\overline{x}(t}}))\hat{H}_{x}i\xi(X, y)\omega(_{X}, t)\omega(y, t)dxdy$,$= \int_{B_{r(s)(\tilde{x}(}}t))\int_{B_{r(s)(\tilde{x}(}}t))((\nabla^{\perp_{h}})^{i}(_{X}, y)\omega(X, t)\omega yx’)tdxdy$

105

becauseif$H_{x^{i}\xi}(X, y)\equiv 0$$x,$ $y\in B_{r(s)}(\tilde{X}(t))\subset\Omega_{L_{1}}$. Therefore$J_{2}=|(\nabla^{\perp_{H}})^{i}(\tilde{x}(t))$$y\omega(X,$- \int_{B_{r(s}(})\tilde{x}(t))\int_{B_{r(s)(\tilde{x}(}}t)))(\nabla^{\perp}hx)i(_{X},$$=o(1)$

as$sarrow\infty$t)\omega(y, t)dxdy|$,

where $o(1)$ is also independent of and$t$$\omega$.

We can summarize the above calculations as follows:$| \frac{dz}{dt}(\mathrm{t})-\frac{d\tilde{x}}{dt}(t)|\leq C_{9}|z(t)-\tilde{x}(t)|+o(1)$for$\mathrm{a}.\mathrm{e}$.

$t\in(\mathrm{O}, T)$as$sarrow\infty$(4.8), where $o(1)$ is independent of and .
$t$$\omega$Then, using the Gronwall inequality, we obtain Theorem 4.6.

$\square$Now, we sketch the proof of Theorem C. We may assume $T<\infty$ .
Using (4.7) and Theorem 4.6, we can construct $\{z_{n}(t)\}$ , which are thesolutions of (3.2) and$z_{n}arrow\overline{x}_{\infty}$in$L^{\infty}(\mathrm{O}, T)^{2}$

tions of (3.2), it is easy to see that thefor a.

$\mathrm{e}.t\in(\mathrm{O}, T)$$\overline{x}_{\infty}$. As$\{z_{n}(t)\}$are the solu-is equal to a solution of (3.2). Therefore, we obtain Theorem C.
REFERENCES[L]

P. L. Lions, ”Mathematical topics in fluid mechanics, Volume 1. IncompressibleModels”, Oxford University Press, Oxford, 1996.

[O]H. Ohtsuka, On the evolution of a high energy vorticity in an ideal fluid, Kyushu

J. Math. (to appear).
[S]

S. Schochet, The weak vorticity formulation of the 2-D Euler equations andconcentration-cancellation, Comm. Partial Differential Equations 20 (1995),

1077-1104.

106[T1] B. Turkington, On steady vortex flow in two dimensions, I, Comm. Partial Dif-

ferential Equations 8 (1983), 999-1030.
[T2] –, On the evolution of a concentrated vortex in an ideal fluid, Arch. Ra-tional Mech. Anal. 97 (1987), 75-87.
[Y]

V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, $Zh$ . Vychisl
Mat.

$i$

Mat. $Fiz$ .

$3$(1963), 1032-1066 (Russian); Comput. Math. Math. Phys. 3(1963), 1407-1456. (English)
HIROSHI OHTSUKA
DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE,
TOKYO INSTITUTE OF TECHNOLOGY,
2-12-1 OH-OKAYAMA MEGURO-KU TOKYO, 152-8551, JAPAN$E$ -mail address: ohtsuka@math.titech.ac.jp