Planar vortices in a bounded domain with a hole

1.   Introduction

In this paper, we consider a planar incompressible flow in a bounded smooth domain

where $O_0$, $\Omega_0$ are two bounded simply-connected open subsets of $\mathbb{R}^2$, such that $\bar{O}_0\subset \Omega_0$. A simple model describing this flow is

where $\psi$ and $\omega$ are the stream function and the vorticity of this flow, respectively, and $\nabla^\bot \psi: = (\partial_2\psi,-\partial_1\psi)$. For a detailed presentation of this model, we refer the readers to [9].

An existence result obtained by Smets and Schaftingen [15] via a variational method shows that (2) has a solution $(\psi_\lambda,\omega_\lambda)$, such that $\psi_\lambda = \rho_\lambda$ on $\partial O_0$,

and $\omega_\lambda = \lambda(\psi_\lambda-q_\lambda)_+^p,$ where $1<p<+\infty$, $q_\lambda = q+\frac{\kappa}{4\pi}\ln \lambda$ with $\kappa>0$ and $q$ is a harmonic function in $\Omega$. Moreover, as $\lambda\to+\infty$, the total vorticity

and the vorticity set $\{y\in\Omega|\,\omega_\lambda(y)>0\}$ shrinks to a point in $\Omega$, which is a critical point of the Kirchhoff-Routh function corresponding to $\kappa$ and $q$. For flows past obstacles, we refer the readers to [11,12,13,14] for other results.

Note that in [15], the value $\rho_\lambda$ of $\psi_\lambda$ on $\partial O_0$ is as a lagrangian multiplier, which is unknown. In this paper, we assume that $\rho_\lambda$ is a prescribed constant, and we remove the condition

The first equation in (2) suggests that $\psi$ and $\omega$ are functionally dependent. So, for simplicity, we consider the following elliptic equation

where $1<p<+\infty$, $\kappa>0$ is a constant, and $\rho_\lambda>0$ is a constant, depending on $\lambda$.

In this paper, we mainly focus on the solvability of (3), and the effect from the constant $\rho_\lambda$ on the location of the vorticity set $\Omega_\lambda: = \{y\in \Omega|\, \psi(y)>\frac{\kappa}{4\pi}\ln\lambda\}$. We expect that for large $\rho_\lambda>0$, the vorticity set $\Omega_\lambda$ concentrates near the boundary of $O_0$, as $\lambda\to+\infty$.

Let $\eta$ be the unique solution of the following problem

Making the change of $\psi = \frac{\ln\lambda}{4\pi}u+\rho_\lambda\eta$, $\varepsilon = \lambda^{-\frac{1}{2}}\left(\frac{\ln\lambda}{4\pi}\right)^{\frac{1-p}{2}}$ and $\lambda_\varepsilon = \frac{4\pi }{\ln\lambda}\rho_\lambda$, (3) can be changed into

Now we want to find a solution to (3) by constructing a solution for (5), whose vorticity set is close to the boundary of $O_0$. For this purpose, the following assumption can be imposed on $\rho_\lambda$:

$(H_\lambda)$ There is a small constant $\sigma>0$, such that

It is easy to see that if $\rho_\lambda$ satisfies the condition $(H_\lambda)$, then $\lambda_\varepsilon$ satisfies the condition:

$(H_\varepsilon)$ There are constants $\gamma_1$ and $\gamma_2$ with $0<\gamma_1<\gamma_2<1$, such that

Before we state the main result, we give the following definition.

Definition 1.1. Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a continuous function. We call $I = [a.b]$, $a\leq b$ a minimum interval of $f$, if $f(t_1) = f(t_2)$ for any $t_1,t_2\in I$, and there is $\sigma_0>0$ such that for $0<\sigma<\sigma_0$, $f(a-\sigma)> f(a)$ and $f(b+\sigma)>f(b)$.

Our main result of this paper can be stated as follows.

Theorem 1.2. Let $\kappa$ be a given positive number. Suppose that $\rho_\lambda$ satisfies $(H_\lambda)$. Then there is a constant $C_0>0$, such that for any $\lambda>C_0$, (3) has a solution $\psi_\lambda$, such that

where $x_\lambda\in \Omega$ and $L>0$ is a constant independent of $\lambda$.

Moreover, as $\lambda\to+\infty$,

In particular, if $I$ is a minimum interval of $\frac{\partial \eta(x(s))}{\partial\nu}$ defined for $x(s)\in \partial\Omega$, then as $\lambda\to+\infty$,

As a result of Theorem 1.2, we obtain a flow in $\Omega$ with single non-vanishing anti-clockwise vortex, which concentrates on the boundary of $O_0$. Let us point out that we can also construct a planar Euler flow with single clockwise vortex, which nears the boundary of $\Omega_0$. This is carried out by considering the problem (3) with nonlinearity $\lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p$ replaced by $-\lambda(-\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p$.

Theorem 1.2 is proved via the following theorem.

Theorem 1.3. Let $\kappa$ be a given positive number. Suppose that $\lambda_\varepsilon$ satisfies $(H_\varepsilon)$. Then there is a constant $\varepsilon_0>0$, such that for any $0<\varepsilon<\varepsilon_0$, (5) has a solution $u_\varepsilon$, such that

where $x_\varepsilon\in \Omega$ and $L>0$ is a constant independent of $\varepsilon$. Moreover, as $\varepsilon\to0$,

In particular, if there is a minimum interval $I$ of $\frac{\partial \eta(x(s))}{\partial\nu}$ defined for $x(x)\in \partial\Omega$, then as $\varepsilon\to0$,

To prove Theorem 1.3, we will use a finite reduction argument as in [6,7,8,9]. Since we consider the vortex nears the boundary, it turns out that more delicate estimates are needed in the proof of Theorem 1.3 than those estimates in [6,7,8,9]. In particular, we need the estimates of the functions $\eta$ and $G$ near the boundary, where $G$ is the Green's function for $-\Delta$ in $\Omega$ with zero boundary condition, written as

and $H(y,x)$ is the regular part of the Green's function. Recall that the Robin's function is defined by $\varphi(x) = H(x,x)$.

The stationary incompressible Euler equations have been studied by many authors, see for instance [1,3,4,5,6,7,8,9,10,15,16,17] and references therein. Roughly speaking, there are two commonly used methods to study the existence of the stationary incompressible Euler equations: the vorticity method and the stream-function method. The vorticity method was first established by Arnold [2] and further developed by Burton [4,5] and Tukington [16]. This argument roughly consists in maximizing the kinetic energy under a constrained sublevel set of $\omega$. In this paper, we will use the stream-function method.

This paper is organized as follows. In section 2, we construct approximate solutions for (5). We will carry out a reduction procedure in section 3 and the results of existence will be proved in section 4. In appendix A, we give the estimates for the radius of vortex core.

2.   Approximate solutions

In this section, following [10], we will construct an approximate solution for (5).

For $p>1$, there is a unique solution $\phi$ for the following problem:

Moreover, $\phi$ is a radial function and satisfies

Let $R>0$ be a large constant, such that for any $x\in \Omega$, $\Omega\subset\subset B_R(x)$. Now we consider

where $a>0$ is a constant. Then, (8) has a solution $U_{\varepsilon,a}$, which can be represented by

where and $\phi(y)$ is a radial solution of (7), and $s_\varepsilon$ is a constant, such that $U_{\varepsilon,a}\in C^1(B_R(0))$. So, $s_\varepsilon$ is determined by

which gives an expansion for $s_\varepsilon$ as follows

For any $x \in \Omega$, define $U_{\varepsilon,x,a}(y) = U_{\varepsilon,a}(y-x)$. Because $U_{\varepsilon,x,a}$ does not vanish on $\partial\Omega$, we need to make a projection. Let $PU_{\varepsilon,x,a}$ be the solution of

Then

where $g(y,x) = \ln R +2\pi H(y,x)$ and $H(y,x)$ is the regular part of the Green's function $G$ defined by (6).

Take

where $0<r_1<r_2$ are two fixed constants, which will be determined later, and set

where $(\partial O_0)_{d_i} = \{y\in\Omega|\, dist(y,\partial O_0)<d_i\}$ is the neighborhood of $\partial O_0$. Denote $d(x) = dist(x,\partial O_0)$ and take $\hat{x}\in \partial O_0$ such that $d(x) = |x-\hat{x}|$. Define

Then we can rewrite $S$ as follows

In the rest of this paper, we assume that $x\in S$. We want to construct solutions for (5) of the form

where $\omega_\varepsilon$ is a perturbation term. To make $PU_{\varepsilon,x,a_\varepsilon}$ a good approximate solution, we need to require that $a_\varepsilon$ and $s_\varepsilon$ satisfy

Let us now show that (18) is solvable for $\varepsilon>0$ small. From the Taylor expansion we have

where $\nu$ is the outward unit normal of $\partial\Omega$. On the other hand, the following expansion for Robin's function is proved in [15] Appendix B:

Then by our assumption on $\lambda_\varepsilon$, as in Lemma 2.1 in [6], we can solve (18) to obtain $a_\varepsilon(x)$ and $s_\varepsilon(x)$. For simplicity, we use $a_\varepsilon$ and $s_\varepsilon$ instead of $a_\varepsilon(x)$ and $s_\varepsilon(x)$, respectively. Then for $y\in B_{Ls_\varepsilon}(x)$, where $L>0$ is any fixed constant, we have

where $0<\sigma<1$ is a small constant.

Remark 1. It is not difficult to get the following expansions:

where $\sigma>0$ is a small constant and

Moreover, in view of $\frac{\partial \varphi(x)}{\partial x_i} = O(\frac{1}{d(x)})$, we can prove that

By (9) and (21), we have the following expansion, which will be used in the rest of this paper,

3.   The reduction

In this section, we reduce the problem of finding a solution for (5) of the form (17) to a finite dimension problem.

Define

and

For any $u\in L^p(\Omega)$, define the following projection

where $b_1$ and $b_2$ are the constants such that $Q_\varepsilon u\in F_{\varepsilon,x}$. Thus $b_1$ and $b_2$ should satisfy

The existence of $b_1$ and $b_2$ can be obtained by the following estimate

where $c>0$ is a constant, $\delta_{ij} = 1$, if $i = j$ and $\delta_{ij} = 0$, if $i\neq j$.

Set

We have the following result for the operator $\mathbb{Q}_\varepsilon\mathbb{L}_\varepsilon$.

Proposition 1. There are constants $\varepsilon_0>0$ and $\sigma_0>0$, such that for any $\varepsilon\in (0,\varepsilon_0)$, $x\in S$, $u\in E_{\varepsilon,x}$ with $\mathbb{Q}_\varepsilon\mathbb{L}_\varepsilon u = 0$ in $\Omega\setminus B_{Ls_{\varepsilon}}(x)$ for some large $L>0$, then

As a consequence, $\mathbb{Q}_\varepsilon\mathbb{L}_\varepsilon$ is one to one and onto from $E_{\varepsilon,x}$ to $F_{\varepsilon,x}$.

Proof. Suppose to the contrary that there are $\varepsilon_n\to0$, $x_n\in S_n$, $u_n\in E_{\varepsilon_n,x_n}$ with $\mathbb{Q}_{\varepsilon_n}\mathbb{L}_{\varepsilon_n} u_n = 0$ in $\Omega\setminus B_{Ls_{\varepsilon_n}}(x_n)$, and $\|u_n\|_{L^\infty(\Omega)} = 1$, such that

First of all, we estimate $b_{1,n}$ and $b_{2,n}$ in the following formula:

where $b_{1,n}$ and $b_{2,n}$ satisfy

From (19), (21) and Lemma A.1, we have

Then by (24), (28) and (29), we obtain

Write

where

Define

Then we have

From (26), (30) and Lemma A.1, we find

Since the right hand side of (32) is bounded in $L_{loc}^p(\mathbb{R}^2)$, $\tilde{u}_n$ is bounded in $W_{loc}^{2,p}(\mathbb{R}^2)$. By the Sobolev embedding, $\tilde{u}_n$ is bounded in $C_{loc}^\alpha(\mathbb{R}^2)$ for some $\alpha>0$. So, we can assume that $\tilde{u}_n$ converges uniformly in any compact set of $\mathbb{R}^2$ to $\tilde{u}\in L^\infty(\mathbb{R}^2)\cap C(\mathbb{R}^2)$. It is easy to check that $\tilde{u}$ satisfies

where

So there exist constants $b_1$ and $b_2$ (see [10]), such that

From $u_n\in E_{\varepsilon_n,x_n}$, we see that

So we get $b_1 = b_2 = 0$. That is, $\tilde{u}\equiv0$. Then we find

By our assumption,

We find that

However, $u_n = 0$ on $\partial\Omega$ and $u_n = o_n(1)$ on $\partial B_{Ls_{\varepsilon_n}}(x_n)$. By the maximum principle,

So, we have proved that

which contradicts $\|u_n\|_{L^\infty(\Omega)} = 1$.

Using the same argument as in Proposition 3.2 in [6], it is not difficult to prove that $\mathbb{Q}_{\varepsilon}\mathbb{L}_{\varepsilon}$ is one to one and onto from $E_{\varepsilon,x}$ to $F_{\varepsilon,x}$. Therefore, we complete the proof.

We now want to find a solution for (5) of the form $PU_{\varepsilon,x,a_\varepsilon}+\omega$. Then $\omega$ should satisfy

where

and

From (37), we see

Fot $\omega\in E_{\varepsilon,x}$, using Proposition 1, (40) is equivalent to

We have the following estimates for $l_\varepsilon$ and $R_\varepsilon(\omega)$.

Lemma 3.1. It holds

and if $\|\omega\|_{L^\infty(\Omega)} = O(s_\varepsilon)$, then

Proof. For any $y\in B_{Ls_\varepsilon}(x)$, from (19), we have

So we get

Similarly, using (19) and Lemma A.1, it ie east to see that

So we complete the proof.

Using Lemma 3.1, we can solve (41) in a standard way and obtain the following proposition.

Proposition 2. There is $\varepsilon_0>0$, such that for any $\varepsilon\in (0,\varepsilon_0)$ and $x\in S$, (40) has a unique solution $\omega_{\varepsilon,x}\in E_{\varepsilon,x}$, with

Furthermore, $\omega_{\varepsilon,x}$ is a $C^1$ map from $x\in S$ to $E_{\varepsilon,x}$.

4.   Proof of main results

In this section, we will prove our main results.

Define

and set

where $x\in S$ and $\omega_{\varepsilon,x}$ is found in Proposition 2. It is well-known that if $x$ is a critical point of $F$, then $PU_{\varepsilon,x,a_\varepsilon}+\omega_{\varepsilon,x}$ is a solution of (5).

Lemma 4.1. There holds:

Proof. Since $\omega_{\varepsilon,x}\in E_{\varepsilon,x}$, then we have the following energy expansion

where

One sees

Then by Lemma 3.1, we can easily check that

Lemma 4.2. We have

where $\sigma>0$ is a small constant, $A_\varepsilon$ is given in Remark 1, and $C_\varepsilon$ is a constant, depending on $\varepsilon$.

Proof. We have

On the other hand, by (19)

Then we get that

Therefore, since $\lambda_\varepsilon$ satisfies $(H_\varepsilon)$, it is not difficult from Remark 1 to see that

where $\sigma>0$ is a small constant, $A_\varepsilon$ is given in Remark 1, and

Proof of Theorem 1.3. By Lemma 4.1 and Lemma 4.2, we have that for $x\in S$,

Consider the following minimization problem

Let

The Hopf lemma shows that for any $y\in \partial O_0$, $\frac{\partial \eta(y)}{ \partial \nu}>0$. Define

and choose $r_1,r_2\in \mathbb{R}$, such that $0<r_1<m_1\leq m_2<r_2$. Then there exists $x_0\in S$, such that

where $r(x)$ is given in (15). So

For $x\in \partial S$, $r(x) = r_1$, we have

if $0<r_1<m_1$ small enough.

Similarly, for $x\in \partial S$, $r(x) = r$, we have

if $r_2\geq m_2$ large enough.

Therefore, from (42), (46) and (47), for any $x\in \partial S$, we have

Thus there is a minimum point $x_\varepsilon\in S$ of $F(x)$ in $S$, which is a critical point. As a result, $u_\varepsilon = PU_{\varepsilon,x_\varepsilon,a_\varepsilon}+\omega_{\varepsilon,x_\varepsilon}$ is a solution of (5).

By our construction, we find

and as $\varepsilon\to0$, $dist(x_\varepsilon,\partial O_0)\to0$.

We remark that if there is a minimum interval $I$ of $\frac{\partial \eta(x(s))}{\partial \nu}$ for $x(s)\in \partial O_0$, then we can find a critical point $x_\varepsilon$ of $F$, which nears $x(I)$.

Appendix A.   The estimate for the radius of vortex core

In this appendix, we give the estimates for the radius of vortex core. The proofs of such results can be found in [6] Lemma A.1.

In the following, we assume that $x\in S$, where $S$ is given in (14).

Lemma A.1. Let $0<\sigma<1$ be a constant. Then there are $\varepsilon_0>0$ and $L>0$ large enough, such that for any $\varepsilon\in (0,\varepsilon_0)$,

while

Lemma A.2. Suppose that $\omega$ satisfies

Then there is $L>0$ large enough, such that

while