On the estimate of distance traveled by a particle in a disk-like vortex patch

TL;DR

This paper analyzes how the distance traveled by particles in a 2D vortex patch grows, showing linear growth for most particles when the patch is close to a disk in shape.

Contribution

It provides a new estimate on particle travel distance in vortex patches that are nearly circular, extending understanding of fluid particle trajectories in Euler flows.

Findings

Travel distance grows linearly for most particles in nearly circular vortex patches.

The result applies when the initial vortex patch is close to a disk in shape.

The analysis is based on the characteristic function of a bounded open set as initial vorticity.

Abstract

We consider the incompressible two-dimensional Euler equation in the plane in the case when its initial vorticity is the characteristic function of a bounded open set. We show that the travel distance grows linearly for most of fluid particles initially placed on the set when the area of the symmetric difference between the set and a disk is small enough.

Equations

$$\begin{array}{rl}{\partial }_t\theta +u\cdot \nabla \theta & =0\text{mbox}forx\in {\mathbb{R}}^2\text{mbox}andfort>0,\\ \theta {|}_{t=0}& ={\theta }_0\text{mbox}forx\in {\mathbb{R}}^2\end{array}$$

$$K(x):=\frac{1}{2\pi}\frac{x^{\perp}}{|x|^{2}}=\frac{1}{2\pi}\Big{(}-\frac{x_{2}}{|x|^{2}},\frac{x_{1}}{|x|^{2}}\Big{)}.$$

$$\frac{d}{dt}{\varphi }_x(t)=u(t,{\varphi }_x(t))\text{mbox}fort>0\text{mbox}and{\varphi }_x(0)=x.$$

$$d_x(t):={\int }_0^t|u(s,{\varphi }_x(s))|ds.$$

$$d_x(t)=\frac{|x|}{2}t\text{mbox}foranyx\in D\text{mbox}andforanyt>0.$$

$$d_x(t)\le Ct\text{mbox}foranyx\text{mbox}intheplaneandforanyt>0$$

$${\Omega }_0△B_r:=({\Omega }_0\backslash B_r)\cup (B_r\backslash {\Omega }_0)$$

$$D:=B_1$$

$$|D△{\Omega }_0|<{\delta }_0,$$

$$|H_{\delta }|\ge |{\Omega }_0|-C{\delta }^{1/4}$$

$$\underset{t\to \infty }{\mathrm{limsup}}\frac{d_x(t)}{t}\ge c{\delta }^{1/2}$$

$$|H_{\delta ,T}|\ge |{\Omega }_0|-C{\delta }^{1/4}$$

$$\frac{d_x(T)}{T}\ge c{\delta }^{1/4}$$

$$u(x)=\{\begin{array}{ll}& \frac{x^{\perp }}{2}\text{mbox}ifx\in D,\\ & \frac{x^{\perp }}{2|x{|}^2}\text{mbox}otherwise\end{array}$$

$$u_{tan}(x)=\{\begin{array}{ll}& \frac{|x|}{2}\text{mbox}ifx\in D,\\ & \frac{1}{2|x|}\text{mbox}otherwise.\end{array}$$

$$\frac{d_x(t)}{t}=\frac{|x|}{2}$$

$$\Vert {\text{mathbbm}1}_{{\Omega }_t}-{\text{mathbbm}1}_{B_r}{\Vert }_{L^1}^2\le 4\pi \cdot \underset{x\in {\Omega }_0△B_r}{\sup }||x{|}^2-r^2|\cdot \Vert {\text{mathbbm}1}_{{\Omega }_0}-{\text{mathbbm}1}_{B_r}{\Vert }_{L^1}\text{mbox}foranyt>0.$$

$$|{\Omega }_t△D{|}^2\le 4\pi \cdot \underset{x\in {\Omega }_0△D}{\sup }||x{|}^2-1|\cdot |{\Omega }_0△D|\text{mbox}foranyt>0$$

$$|{\Omega }_t△D|\le 2\sqrt{\pi (R^2+1){\alpha }_0}.$$

$$\Vert K\ast f{\Vert }_{L^{\infty }}\le C_0\Vert f{\Vert }_{L^1}^{1/2}\Vert f{\Vert }_{L^{\infty }}^{1/2}$$

$$\Vert K\ast ({\text{mathbbm}1}_{{\Omega }_t}-{\text{mathbbm}1}_D){\Vert }_{L^{\infty }}\le C_0\Vert {\text{mathbbm}1}_{{\Omega }_t}-{\text{mathbbm}1}_D{\Vert }_{L^1}^{1/2}\Vert {\text{mathbbm}1}_{{\Omega }_t}-{\text{mathbbm}1}_D{\Vert }_{L^{\infty }}^{1/2}\le C_1{\alpha }_0^{1/4}\text{mbox}where$$

$$C_1=C_1(R):=C_0\sqrt{2\sqrt{\pi (R^2+1)}}>0.$$

$$|u_{rad}(t,x)|\le C_1{\alpha }_0^{1/4}$$

$$|u_{tan}(t,x)|\ge \frac{|x|}{2}-C_1{\alpha }_0^{1/4}.$$

$$\int_{\Omega_{0}}\Big{(}\frac{1}{T}\int_{0}^{T}{\mathbbm{1}}_{B_{\epsilon}\cup B_{1}^{C}}(\phi_{x}(t))dt\Big{)}dx\leq 2\epsilon^{2}\pi.$$

$$F(t,x)={\text{mathbbm}1}_{B_{ϵ}\cup B_1^C}({\varphi }_x(t))=\{\begin{array}{l}1\text{mbox}ifeither|{\varphi }_x(t)|<ϵ\text{mbox}or|{\varphi }_x(t)|\ge 1,\\ 0\text{mbox}otherwise.\end{array}$$

$$\begin{array}{rl}{\int }_{{\Omega }_0}F(t,x)dx& ={\int }_{{\Omega }_0}{\text{mathbbm}1}_{B_{ϵ}\cup B_1^C}({\varphi }_x(t))dx={\int }_{{\Omega }_t\cap B_{ϵ}}1dx+{\int }_{{\Omega }_t\cap B_1^C}1dx\\ & \le {\int }_{B_{ϵ}}1dx+{\int }_{{\Omega }_t△D}1dx\le |B_{ϵ}|+|{\Omega }_t△D|\\ & \le {ϵ}^2\pi +2\sqrt{\pi (R^2+1){\alpha }_0}\le 2{ϵ}^2\pi \end{array}$$

$$C_1{\alpha }_0^{1/4}\le C_1{\delta }^{1/4}\le \frac{ϵ}{4}$$

$$2\sqrt{\pi (R^2+1){\alpha }_0}\le 2\sqrt{\pi (R^2+1)\delta }\le {ϵ}^2\pi .$$

$$\int_{\Omega_{0}}\psi_{n}dx=\int_{\Omega_{0}}\Big{(}\frac{1}{n}\int_{0}^{n}{\mathbbm{1}}_{D\backslash{B_{\epsilon}}}(\phi_{x}(t))dt\Big{)}dx=\int_{\Omega_{0}}\frac{1}{n}\int_{0}^{n}\Big{(}1-{\mathbbm{1}}_{{B_{\epsilon}}\cup B_{1}^{C}}(\phi_{x}(t))\Big{)}dtdx\geq|\Omega_{0}|-2\epsilon^{2}\pi.$$

Full text

1. Introduction

We consider the incompressible 2D Euler equation in vorticity form in the whole plane:

[TABLE]

where the Biot-Savart law is given by $u=(K*\theta)$ with

[TABLE]

When $\theta_{0}$ lies on $L^{1}\cap L^{\infty}$, the existence and uniqueness of a global-in-time weak solution is due to Yudovich [21].

In this paper, we are interested in estimating the distance traveled by a fluid particle. More precisely, we consider the case when the initial data $\theta_{0}$ is the characteristic function ${\mathbbm{1}}_{\Omega_{0}}$ of a bounded open set $\Omega_{0}$ in $\mathbb{R}^{2}$. Then the corresponding solution $\theta$ is given by Yudovich theory, and it has the form of $\theta(t)={\mathbbm{1}}_{\Omega_{t}}$ where $\Omega_{t}$ is defined by $\Omega_{t}=\{\phi_{x}(t)\in\mathbb{R}^{2}\,|\,x\in\Omega_{0}\}$ and $\phi_{x}(\cdot)$ is the particle trajectory of the particle whose initial position is at $x\in\mathbb{R}^{2}$, which can be obtained by solving the following system of the ordinary differential equations:

[TABLE]

This is well-defined since the velocity field $u$ allows a log-Lipschitz estimate (e.g. see the modern texts [15], [17]). For $t>0$ and $x\in\mathbb{R}^{2}$, we say that $d_{x}(t)$ is the distance traveled by the particle, whose initial position is at $x$, up to time $t$. More precisely, we define $d_{x}(t)$ by

[TABLE]

For instance, when $\Omega_{0}$ is the unit disk $D$ centered at the origin, we easily compute (see Subsection 1.1)

[TABLE]

For a general bounded open set $\Omega_{0}$, we have, at least, a trivial linear upper bound

[TABLE]

where the above constant $C$ depends only on the Lebesgue measure $|\Omega_{0}|$ of the set $\Omega_{0}$. Indeed, the conservation of the total mass $|{\Omega_{t}}|$ in time due to the incompressibility of the fluid gives a uniform bound for $u$ (e.g. see estimate (9) with the substitution $f={\mathbbm{1}}_{\Omega_{0}}$).

The main result of this paper says that, for most of particles placed on the initial vortex patch in the beginning, the travel distance actually grows linearly in time when the initial patch is disk-like in the sense that the measure of the initial symmetric difference

[TABLE]

is small enough. Here we denote $B_{r}:=\{x\in\mathbb{R}^{2}\,|\,|x|<r\}$ for $r>0$. Without loss of generality, we will compare the initial patch $\Omega_{0}$ only with the unit disk

[TABLE]

thanks to the scaling of the Euler equation.

Theorem 1.1**.**

For any $R>0$, there exist constants $\delta_{0}>0$, $C>0$ and $c>0$ such that if $\Omega_{0}\subset B_{R}$ and if

[TABLE]

*then the solution of (1) for the initial data $\theta_{0}={\mathbbm{1}}_{\Omega_{0}}$ satisfies the following two properties:

$(I)$ For any $\delta>0$ satisfying $|D\bigtriangleup\Omega_{0}|\leq\delta\leq\delta_{0}$, there exists a set $H_{\delta}\subset\Omega_{0}$ such that*

[TABLE]

and

[TABLE]

*for every $x\in H_{\delta}$.

$(II)$ For any $\delta>0$ satisfying $|D\bigtriangleup\Omega_{0}|\leq\delta\leq\delta_{0}$ and for each $T>0$, there exists a set $H_{\delta,T}\subset\Omega_{0}$ such that*

[TABLE]

and

[TABLE]

for every $x\in H_{\delta,T}$.

In the proof, we use the $L^{1}$-stability result in [19]. Indeed, when the initial patch is disk-like, we can show that for each fixed time moment, most of particles, which are initially placed on the initial set $\Omega_{0}$, should be detected in the annulus region $\{\epsilon\leq|x|<1\}$ thanks to the incompressibility of the fluid. Here $\epsilon>0$ will be taken small depending on $\delta>0$. This gives Lemma 2.2. Then we prove that most of particles spend most of their life time in the annulus (e.g. see (15)). Since the speed induced by the exact disk patch is non-trivial within the region, we get the conclusion by using the stability result again.

For the vortex patch problem, there are many other interesting results including persistence of boundary regularity [4], [2] (or see the textbook [5] and references therein, also see [14] for a blow-up result in a modified SQG patch equation), existence of rotating patches [7], [8], [3], [11], which are so-called “V-states”, and stability of circular patches [20], [9], [19] (also see [1] for rectangular patches in a 2D infinite cylinder). Confinement of patch evolution, or support of positive vorticity evolution, is also interesting (*e.g. *see [16], [13], [18], [6], [12]).

Before proving our theorem, we present the simplest example for travel distance.

1.1. When $\Omega_{0}=D$

If we consider the initial data $\theta_{0}={\mathbbm{1}}_{D}$, then the radial symmetry of the data implies that corresponding solution is stationary. The velocity induced by the stationary vorticity $\theta_{t}={\mathbbm{1}}_{D}$ is

[TABLE]

(e.g. see [15]). If we decompose into its radial part and tangential part: $u=u_{rad}\frac{x}{|x|}+u_{tan}\frac{x^{\perp}}{|x|},$ then we get $u_{rad}=0$ and

[TABLE]

It says that each particle in the disk just rotates in a constant angular velocity. Thus we simply have, for any $x\in D$ and for any $t>0$,

[TABLE]

and $d_{x}(\infty)=\infty$ unless $x=0$.

2. Proof

2.1. $L^{1}$-stability of a disk patch

The paper [19] showed the following $L^{1}$-stability of a circular vortex patch:

Lemma 2.1** (Theorem 3 in [19]).**

For any bounded open set $\Omega_{0}\subset\mathbb{R}^{2}$ and for any $r>0$, we have

[TABLE]

In [19], the authors used conservation of mass, momentum and moment of inertia to prove the above result.

We note $\|{\mathbbm{1}}_{A}-{\mathbbm{1}}_{B}\|_{L^{1}}=\int_{A\bigtriangleup B}1dx=|A\bigtriangleup B|$ for any bounded open sets $A$ and $B$. In this paper, we will use the result (6) of the above lemma with the choice $r=1$:

[TABLE]

in the following way:

Let $R>0$. Assume $\Omega_{0}\subset B_{R}$ with $\alpha_{0}:=|\Omega_{0}\bigtriangleup D|>0.$ Consider the initial data $\theta_{0}={\mathbbm{1}}_{\Omega_{0}}$ with its solution $\theta(t)={\mathbbm{1}}_{\Omega_{t}}$. Since $\sup_{\Omega_{0}\bigtriangleup D}||x|^{2}-1|\leq(R^{2}+1)$, the stability result (7) implies

[TABLE]

On the other hand, we observe that there exists a constant $C_{0}>0$ such that

[TABLE]

for any $f\in(L^{1}\cap L^{\infty})(\mathbb{R}^{2}).$ This can be proved by using the obvious estimate $|K(x)|\leq\frac{C}{|x|}$ of the Biot-Savart kernel (2) (or see Lemma 2.1. in [13]). In our setting, we have

[TABLE]

[TABLE]

Since the vector field $(K*{\mathbbm{1}}_{D})$ has no radial component by (5) and the velocity $u(t)$ from $\theta(t)={\mathbbm{1}}_{\Omega_{t}}$ can be decomposed: $u(t)=K*[\theta(t)]=K*{\mathbbm{1}}_{\Omega_{t}}=K*{\mathbbm{1}}_{D}+K*({\mathbbm{1}}_{\Omega_{t}}-{\mathbbm{1}}_{D}),$ we obtain

[TABLE]

for any $x\in\mathbb{R}^{2}$ and for any $t>0$. For the tangential component, we have, for $x\in D$ and for $t>0$,

[TABLE]

In this setting, we can prove the following lemma:

Lemma 2.2**.**

Let $0<\epsilon<1$. Suppose $\Omega_{0}\subset B_{R}$ and $2\sqrt{\pi(R^{2}+1)\cdot|\Omega_{0}\bigtriangleup D|}\leq\epsilon^{2}\pi.$ Then for each finite $T>0$, we have

[TABLE]

Proof.

We define

[TABLE]

For each $t>0$, since the velocity $u$ is divergence-free, the flow map $X(t,x):=\phi_{x}(t)$ is area preserving so that we have

[TABLE]

by (8) and by the assumption in this lemma. Let $T>0$. Then, by Fubini, we get $\int_{\Omega_{0}}{\int_{0}^{T}F(t,x)dt}dx=\int_{0}^{T}\int_{\Omega_{0}}F(t,x)dxdt\leq 2\epsilon^{2}\pi T.$

∎

2.2. Proof of $(I)$ in Theorem 1.1: Travel distance in infinite time

Proof of $(I)$ in Theorem 1.1.

Let $R>0$. First, we take \delta_{0}:=\Big{(}\min\{\frac{1}{8C_{1}},\frac{1}{2}\sqrt{\frac{\pi}{2\sqrt{\pi(R^{2}+1)}}}\}\Big{)}^{4}>0 where $C_{1}=C_{1}(R)>0$ is defined in (10). Assume $\Omega_{0}\subset B_{R}$ and $\alpha_{0}:=|\Omega_{0}\bigtriangleup D|<\delta_{0}$ and take any $\delta>0$ satisfying $\alpha_{0}\leq\delta\leq\delta_{0}$. Define $\epsilon=\epsilon(\delta):=C_{2}\cdot\delta^{1/4}$ where $C_{2}:=\max\{4C_{1},\sqrt{\frac{2\sqrt{\pi(R^{2}+1)}}{\pi}}\}>0$. Clearly, we have $0<\epsilon\leq 1/2$,

[TABLE]

and

[TABLE]

For each $n\in\mathbb{N}$ and for $x\in\Omega_{0}$, we define $\psi_{n}(x):=\frac{1}{n}\int_{0}^{n}{\mathbbm{1}}_{D\backslash{B_{\epsilon}}}(\phi_{x}(t))dt.$ We observe $0\leq\psi_{n}\leq 1$. By Lemma 2.2 due to (13), we have

[TABLE]

We put $f_{n}(x)=\sup_{m\geq n}\psi_{m}(x)$ for each $n$. Then we notice that $0\leq f_{n}\leq 1$ and $\{f_{n}\}_{n=1}^{\infty}$ decays pointwise to some function $f\geq 0$. By Dominated Convergence Theorem (e.g. see [10]), we have $\int_{\Omega_{0}}fdx=\lim_{n\to\infty}\int_{\Omega_{0}}f_{n}dx$. From $\int_{\Omega_{0}}f_{n}dx\geqslant\int_{\Omega_{0}}\psi_{n}dx\geq|\Omega_{0}|-2\epsilon^{2}\pi$, we get

[TABLE]

By putting $H:=\{x\in\Omega_{0}\,|\,f(x)>\epsilon\}$, we have

[TABLE]

Thus we get

[TABLE]

for some $C>0$ by $\epsilon\sim\delta^{1/4}$. We observe

[TABLE]

Thus for each $x\in H$, there is an increasing sequence $\{n_{k}(x)\}_{k=1}^{\infty}$ in $\mathbb{N}$ such that $\lim_{k\to\infty}n_{k}(x)=\infty$ and

[TABLE]

On the other hand, whenever $(t,x)$ satisfies $\epsilon\leq|\phi_{x}(t)|<1$, we have

[TABLE]

by (11) and (12). Thus, for any $x\in H$ and for any $k\in\mathbb{N}$, we have

[TABLE]

which implies $\limsup_{t\to\infty}\frac{d_{x}(t)}{t}\geq\frac{\epsilon^{2}}{4}$ for each $x\in H$. Since $\epsilon\sim\delta^{1/4}$, we get (3).

∎

2.3. Proof of $(II)$ in Theorem 1.1: Travel distance in finite time

Proof of $(II)$ in Theorem 1.1.

We take the same $\delta_{0},\alpha_{0},\delta,\epsilon$ as in the proof of Theorem 1.1 so that we have $0<\epsilon\leq 1/2$, (12) and (13).

Let $T>0$. By Lemma 2.2 due to (13), we have

[TABLE]

By Chebyshev (e.g. see [10]), we get $|\{x\in\Omega_{0}\,|\,G_{T}(x)\geq\epsilon\}|\leq{2\epsilon\pi}.$

We put $H_{T}:=\{x\in\Omega_{0}\,|\,G_{T}(x)<\epsilon\}$. Then we observe $|H_{T}|\geq|\Omega_{0}|-{2\epsilon\pi}\geq|\Omega_{0}|-C\delta^{1/4}$ for some $C>0$ by $\epsilon\sim\delta^{1/4}$. We also observe that for any $x\in H_{T}$,

[TABLE]

Thus for each $x\in H_{T}$, we get

[TABLE]

By (11) and (12), if $\epsilon\leq|\phi_{x}(t)|<1$, then we get $|u(t,\phi_{x}(t))|\geq\frac{\epsilon}{4}$ as in (14). Thus, for any $x\in H_{T}$, we get $d_{x}(T)\geq(1-\epsilon)T\cdot\frac{\epsilon}{4}\geq T\cdot\frac{\epsilon}{8}$ because of $\epsilon\leq 1/2$. Since $\epsilon\sim\delta^{1/4}$, we obtain (4).

∎

acknowledgement

KC was supported by the National Research Foundation of Korea (NRF-2018R1D1A1B07043065) and by the POSCO Science Fellowship of POSCO TJ Park Foundation. We thank Prof. S. Denisov for many helpful discussions.