Existence of stationary vortex sheets for the 2D incompressible Euler equation

Abstract

We construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$ , $i=1,\ldots , m$ , be m arbitrarily fixed constants. For any given nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$ of the Kirchhoff–Routh function defined on $\Omega ^m$ corresponding to $(\kappa _1,\ldots , \kappa _m)$ , we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and concentrate on curves perturbed from small circles centered near $x_{0,i}$ , $i=1,\ldots ,m$ . The proof is accomplished via the implicit function theorem with suitable choice of function spaces.

1 Introduction

Let $\Omega \subset \mathbb {R}^2$ be a bounded or unbounded domain. We consider the stationary Euler equation

(1.1) $$ \begin{align} \begin{cases} \mathbf{u}\cdot \nabla \mathbf{u}=-\nabla P,&\quad \text{in}\, \Omega,\\ \nabla\cdot \mathbf{u}=0,&\quad \text{in}\, \Omega,\\ \mathbf{u}\cdot \nu=0, &\quad \text{on}\, \partial\Omega, \end{cases} \end{align} $$

where $\mathbf {u}=(u_1,u_2)$ is the velocity field, P is the scalar pressure, and $\nu $ is the outward unit normal of $\partial \Omega $ .

In a planar flow, the vorticity is defined as the third component of the curl of the velocity field $(u_1, u_2, 0)$ , namely, $\omega =\partial _1u_2-\partial _2u_1$ . Taking the curl of the first equation in (1.1), we find that $\omega $ satisfies the following vorticity equation:

(1.2) $$ \begin{align} \mathbf{u}\cdot \nabla \omega =0\quad\text{in}\,\Omega. \end{align} $$

The velocity is recovered by the Biot–Savart law

$$ \begin{align*}\ \mathbf{u}=\nabla^\perp(-\Delta)^{-1}\omega, \end{align*} $$

where $(x_1,x_2)^\perp =(x_2,-x_1)$ and the operator $(-\Delta )^{-1}$ is given by

$$ \begin{align*} (-\Delta)^{-1}\omega(x)=\int_{\Omega}G(x- y)\omega(y)dy. \end{align*} $$

Here, $G(x,y)$ is the Green function of $-\Delta $ in $\Omega $ with zero Dirichlet data. So $G(x,y)$ takes the form $G(x,y)=\frac {1}{2\pi }\ln \frac {1}{|x-y|}-H(x,y)$ with $H(x,y)$ the harmonic part of $G(x,y)$ . We denote $\psi =(-\Delta )^{-1}\omega $ to be the stream function, then the velocity field can be derived by $\mathbf {u}=\nabla ^\perp \psi $ .

In the last century, the two-dimensional Euler equation has been intensively studied, and the global well-posedness of the vorticity equation with initial data in $L^1\cap L^\infty $ was proved by Yudovich in the classical paper [45]. However, many physical phenomena possess strong and irregular fluctuations, such as fluids with small viscosity, where flows tend to separate from rigid walls and sharp corners [5, 37]. To model this phenomenon mathematically, the most natural way is to think of a solution to the Euler equation, in which the velocity changes sign discontinuously across a stream line. This discontinuity induces vorticity concentrated on a curve, which is only a measure rather than a bounded function.

A velocity discontinuity in an inviscid 2D flow is called a vortex sheet, whose vorticity concentrates as a measure along a curve. Suppose that $\omega $ is a weak solution to the Euler equation concentrated on a finite number of closed curves $\Gamma _i$ parameterized by $z_i(\theta )$ . Namely, for any test function $\phi \in C_c^\infty (\Omega )$ , $\omega $ is a measure such that

$$ \begin{align*}\int_\Omega \phi(x)d\omega(x)=\sum_i \int \gamma_i(\alpha)\phi(z_i(\alpha))|z_i'(\alpha)|d\alpha,\end{align*} $$

where $\gamma _i(\alpha )$ is the vorticity strength at $z_i(\alpha )$ . Then, the equation of the sheet can be derived by the Birkhoff–Rott operator in a domain [6, 23, 34, 37, 41]

(1.3) $$ \begin{align} BR(z,\gamma)(x):=\frac{1}{2\pi}P.V. \int \frac{(x-z(\alpha))^\perp}{|x-z(\alpha)|^2} \gamma(\alpha)|z'(\alpha)| d\alpha+\int \nabla^\perp H(x,z(\alpha))\gamma(\alpha)|z'(\alpha)| d\alpha, \end{align} $$

where $P.V.$ stands for Cauchy principal value of an integral. Equation (1.3) yields the motion of the sheet

(1.4) $$ \begin{align} \mathbf{u}(z_i(\theta))=-BR(z_i(\theta)) \end{align} $$

with $BR(z_i(\theta )):=-\sum _j BR(z_j,\gamma _j)(z_i(\theta ))$ .

Significant efforts have been made in mathematical study of the theory of vortex sheet. In the elegant paper [20], Delort proved global existence of weak solutions with an initial $L^2_{\text {loc}}$ velocity and a positive measure vorticity. Later, the proof was simplified by Majda [36]. Duchon and Robert [21] established global existence for a class of initial data concentrated closed to a line. Existence in different setting of vortex sheet with a distinguished sign was also obtained in [22, 42]. For vorticity without a definite sign, only partial results on the existence are known under some additional assumptions [35, 43, 44]. Note that uniqueness for such solutions still remains open.

On the other hand, blow up may occur in the motion of vortex sheet. Indeed, singular formulation was conjectured by Birkhoff [6], and by Birkhoff and Fisher [7]. In [39], Moore showed the possibility that the curvature blows up in finite time even though the initial data are analytic. Moore’s result was also supported by numerical study [30]. Ill-posedness for vortex sheet problem in the space $H^s$ with $s>\frac {3}{2}$ was obtained by Caflisch and Orellana [9]. These results demonstrate that the study of vortex sheet is extremely delicate, and hence exact solutions, in particular relative equilibria, are of great importance since their structures persist for long time.

Nevertheless, very few relative equilibria are known. For the vortex sheets in $\mathbb {R}^2$ , except for circles and lines, the only nontrivial examples include: uniformly rotating segment [4], in which the vorticity is supported on a segment of length $2a$ with density

$$ \begin{align*} \gamma(x)=\Omega_r\sqrt{a^2-x^2}, \ \ \ \ \ \ \text{for} \ \ x\in[-a,a] \end{align*} $$

and angular velocity $\Omega _r$ . A generalization of the rotating segment is the Protas–Sakajo class [40], which is made out of segments rotating about a common center of rotation with endpoints at the vertices of a regular polygon. Recently, a new class of vortex sheet was obtained in [24] via degenerate bifurcation from rotating circles. Note that the existence of nontrivial steady vortex sheet in $\mathbb R^2$ is not apparent in view of the rigidity results obtained in [23], where the authors showed for uniformly rotating vortex sheets with angular velocity $\Omega _r\leq 0$ and strength $\gamma>0$ , only trivial solutions exist.

In a domain $\Omega \subsetneq \mathbb {R}^2$ , as far as we know, there seems no nontrivial stationary vortex sheet is known so far. The purpose of the present paper is to construct a family of stationary vortex sheets for a domain (bounded or not), whenever the Kirchhoff–Routh function possesses nondegenerate critical points.

For any given integer $m>0$ , and m real numbers $\kappa _1,\kappa _2,\ldots ,\kappa _m$ , define the Kirchhoff–Routh function on $\Omega ^m =\{\mathbf {x}=(x_1,x_2,\ldots ,x_m)\,\mid \, x_i\in \Omega ,\text {for}\,i=1,\ldots ,m\}$ as follows:

(1.5) $$ \begin{align} {\mathcal{W}}_m(x_1,x_2,\ldots,x_m)=-\sum\limits_{i\neq j}^m\kappa_i\kappa_j G(x_i,x_j)+\sum\limits_{i=1}^m\kappa_i^2H(x_i,x_i). \end{align} $$

It is known that the location of m-point vortices with strength $\kappa _i$ ( $i=1,\ldots ,m$ ) in $\Omega $ must be a critical point of $\mathcal {W}_m$ (see, e.g., [31, 32]). Results on the existence and nondegeneracy of critical points for $\mathcal {W}_m$ can be found in [2, 3]. In [25], it was proved that if $\Omega $ is convex, then there is no critical point of $\mathcal {W}_m$ in $\Omega ^m$ with $m\geq 2$ and $\kappa _i>0$ for all $i = 1,\ldots , m$ . Let us point out that although the nondegeneracy of critical points for the Kirchhoff–Routh functions in a general domain is not an easy issue, it is true for most of the domains, as proved in [1, 38]. On the other hand, in [8], it was shown that in a convex domain, $\mathcal {W}_1$ has a unique critical point, which is also nondegenerate. In a recent paper [12], the first author, Yan, and Yu obtained some existence and results on the nondegeneracy of critical points of the Kirchhoff–Routh function for unbounded domains.

Giving a nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})\in \Omega ^m$ of $\mathcal {W}_m$ , for $\varepsilon $ small, we will construct a branch of vortex sheets concentrated on a finite number of closed curves $\Gamma _i$ . Moreover, each $\Gamma _i$ is the perturbation of a small circle with radius $\varepsilon $ centered at some point $x_{\varepsilon ,\tau , i}\in \Omega $ close to $x_{0,i}$ , and the vorticity $\omega \big |_{\Gamma _i}$ satisfies $\int _0^{2\pi } \gamma _i(\alpha )|z_i'(\alpha )| d\alpha \approx \kappa _i$ . This result shows the rich diversity of stationary vortex sheet solutions despite that the well-posedness is not fully understood.

Our main theorem is as follows.

Theorem 1.1 Let $\Omega \subset \mathbb {R}^2$ be a domain (may be unbounded), and let $\kappa _i\not =0\ (i=1,\ldots , m)$ be m given numbers. Suppose that $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})\in \Omega ^m$ with $x_{0,i}\not =x_{0,j}$ , for $i\not =j$ , is an isolated critical point of $\mathcal {W}_m$ defined by (1.5) satisfying the nondegeneracy condition: $\text {deg} \left (\nabla \mathcal {W}_m, \mathbf {x}_0\right )\not =0$ . Then, there are $\varepsilon _0>0$ and $\tau _0>0$ , such that for all $0<\varepsilon <\varepsilon _0$ and $-\tau _0<\tau <\tau _0$ , there exists a stationary vortex sheet $\omega _{\varepsilon ,\tau }$ possessing the following properties:

1. (i) $\omega _{\varepsilon ,\tau }=\sum \limits _{i=1}^m\gamma _i \delta _{\Gamma _i}$ concentrates on a finite number of closed curves $\Gamma _i$ with strength $\gamma _i$ . Moreover, it holds that $\gamma _i=\frac {\kappa _i+O(\varepsilon )}{2\pi \varepsilon }$ and each $\Gamma _i$ is a perturbation of a small circle with radius $\varepsilon $ and centered at some point $x_{\varepsilon ,\tau , i}\in \Omega $ satisfying $|x_{\varepsilon ,\tau ,i}-x_{0,i}|=O(\varepsilon )$ .

2. (ii) As $\varepsilon \to 0^+$ , one has in the sense of measure

   $$ \begin{align*} \omega_{\varepsilon,\tau}\to \sum\limits_{i=1}^{m}\kappa_i\delta(x-x_{0,i})\,\,\,\text{weakly}, \end{align*} $$
   where $\delta (x-x_{0,i})$ is the Dirac delta function concentrated at the point $x_{0,i}$ .

3. (iii) For any $i=1,\ldots ,m$ , the interior of $\Gamma _i$ is convex.

Remark 1.2 Our result does not rely on the sign of $\kappa _i$ , which is essential in the global existence of the initial problem as mentioned above. As we shall see in Section 5, the parameter $\tau $ stands for the projection on the kernel of the linearized operator, and it slightly affects the shape of $\Gamma _i$ . More precisely, $\Gamma _i$ in Theorem 1.1 takes the form $\Gamma _i=\{x_{\varepsilon ,\tau ,i}+\varepsilon (1+\varepsilon (f_{\varepsilon ,\tau ,i}(\theta )+\tau f_{0,i}(\theta ) ))(\cos \theta , \sin \theta )\mid \theta \in [0,2\pi )\}$ for some $f_{\varepsilon ,\tau ,i}$ depending on $\varepsilon $ and $\tau $ and some fixed $f_{0,i}$ in the kernel.

Remark 1.3 For simplicity, all the scales of $\Gamma _i$ ( $i=1,\ldots ,m$ ) are chosen to be of the same order. However, this is not necessary, and one may construct vortex sheet concentrated on $\Gamma _i$ with the scale of $\varepsilon _i$ ( $i=1,\ldots ,m$ ) different from each other.

Remark 1.4 Fixing $\tau \in (-\tau _0, \tau _0)$ in Theorem 1.1, say $\tau =0$ , we obtain a family of solutions with vortex sheet $\omega _\varepsilon $ parameterized only by $\varepsilon $ , which is of special interest since it is closely related to the classical problem: regularization of point vortices for the Euler equation. This means justifying the weak formulation for point vortex solutions of the incompressible Euler equations by approximating these solutions with more regular solutions. In fact, the vortex sheets obtained in Theorem 1.1 belong to the space $H^{-1}(\Omega)$ , whereas the point vortices solution belongs to $H^{-1-\sigma }(\Omega)$ for any $\sigma>0$ , which is more singular than a vortex sheet. Thus, our result can be regarded as a desingularization of point vortices by vortex sheet in some way. For more literature on the desingularization problem, we refer to [10, 11, 17, 28, 33] and the references therein.

Next, we shall sketch the basic ideas used to prove the main result. Thanks to Lemma 2.1 in [23], we are able to formulate the conditions that the Birkhoff–Rott integral (1.3) satisfies for a stationary vortex sheet into a system of $2m$ coupled integrodifferential equations $\mathcal {F}_{i,1}(\varepsilon , \mathbf {x}, \boldsymbol {f}, \boldsymbol {g})=0$ and $\mathcal {F}_{i,2}(\varepsilon , \mathbf {x}, \boldsymbol {f}, \boldsymbol {g})=0$ , $i=1,\ldots ,m$ . We expect that the case $(\varepsilon , \mathbf {x},\boldsymbol {f}, \boldsymbol {g})=(0,\mathbf {x}_0,0,0)$ corresponds to the point vortices and hence $(0, \mathbf {x}_0, 0, 0)$ is a solution to $\mathcal {F}_{i,1}=0$ and $\mathcal {F}_{i,2}=0$ provided that $\mathbf {x}_0$ is a critical point of $\mathcal {W}_m$ . Therefore, the first step is to extend $\mathcal {F}_{i,1}$ and $\mathcal {F}_{i,2}$ such that $\varepsilon \leq 0$ is allowed. Then, one can verify that $\mathcal {F}_{i,1} (0, \mathbf {x}_0, 0, 0)=\mathcal {F}_{i,2}(0, \mathbf {x}_0, 0, 0)=0$ does hold when $ \mathbf {x}_0$ is a critical point of $\mathcal {W}_m$ , and hence we obtain a trivial solution. To apply the implicit function theorem at the solution $(0, \mathbf {x}_0, 0, 0)$ , the Gateaux derivative of $\boldsymbol {\mathcal {F}}:=(\mathcal {F}_{1,1}, \mathcal {F}_{1,2}, \ldots , \mathcal {F}_{m,1}, \mathcal {F}_{m,2})$ should be an isomorphism, which unfortunately fails. Detailed calculations show that $D\boldsymbol {\mathcal F}$ has a $2m$ -dimensional kernel $\prod _{i=1}^m \{(a\cos (\theta )+b\sin (\theta ), \kappa _i(a\cos (\theta )+b\sin (\theta )))\mid (a,b)\in \mathbb {R}^2\}$ . Hence, we have to consider the equations in quotient spaces and impose the conditions $-\kappa _i \int \mathcal {F}_{i,1} \sin (\theta )d\theta =\int \mathcal {F}_{i,2} \cos (\theta )d\theta $ and $\kappa _i \int \mathcal {F}_{i,1}\cos (\theta )d\theta =\int \mathcal {F}_{i,2} \sin (\theta )d\theta $ for all $i=1,\ldots ,m$ . Although these conditions seem to be complicated, we successfully convert them into a concise equation $\nabla \mathcal { W}_m(\mathbf {x})=O(\varepsilon )$ , which is solvable near $\mathbf {x}_0$ due to the nondegeneracy of $\nabla \mathcal {W}_m$ at $\mathbf {x}_0$ . Finally, we can apply the implicit function theorem to obtain the existence. The convexity of the interior of $\Gamma _i$ follows from calculating the curvature directly. We point out that our procedure of proving Theorem 1.1 borrows the idea of Lyapunov–Schmidt reduction and local bifurcation theory.

The ideas and methods introduced in the present paper may be widely applied to a variety of situations and other models. For example, one may consider an ideal fluid with an irrotational background flow $u_0=\nabla ^\perp \psi _0$ , where $\psi _0$ is a given harmonic function. In this case, the Kirchhoff–Routh function is given by (see [12])

$$ \begin{align*} \mathcal{W}_{m, \psi_0}(x_1,x_2,\ldots,x_m)=-\sum\limits_{i\neq j}^m\kappa_i\kappa_j G(x_i,x_j)+\sum\limits_{i=1}^m\kappa_i^2H(x_i,x_i)+2\sum\limits_{i=1}^m\kappa_i \psi_0(x_i). \end{align*} $$

Although $\mathcal {W}_{m, \psi _0}$ is slightly different from $\mathcal {W}_{m}$ given by (1.5), we believe that our method can be modified to construct vortex sheets near critical points of $\mathcal {W}_{m, \psi _0}$ in this situation. In addition, for domains with some symmetry properties, such as the unit disk or the half-space, one may modify our method by considering in function spaces with certain symmetries to construct solutions with vortex sheets near degenerate critical points of the Kirchhoff–Routh function.

We would like to make a brief remark on the approach of constructing vortex patches via the contour dynamics equation, which shares a similar spirit as the construction of vortex sheet we consider in the present paper. Many celebrated contributions have been made with the contour dynamics equation method in recent years (see, e.g., [13–16, 18, 19, 26–29] and the references therein). However, since a vortex patch is actually a bounded function, the contour dynamics equation is more regular than the equations of a vortex sheet. Hence, more effort is needed in the process of our proof.

This paper is organized as follows. In Section 2, we derive the equations that the Birkhoff–Rott integral satisfies for a stationary vortex sheet and define the function spaces which will be used later. In Section 3, we extend the functionals and show their $C^1$ regularity. Section 4 is devoted to study the linearization operators, where we prove that the derivative is an isomorphism in quotient spaces. In Section 5, we choose $\mathbf {x}$ properly such that the range of our functional belongs to the quotient spaces and apply the implicit function theorem to prove Theorem 1.1.

2 Formulation and functional setting

Since $\omega $ is a stationary sheet, using Lemma 2.1 in [23], we derive the following equations that the BR equation (1.4) and $\gamma _i$ satisfy.

(2.1) $$ \begin{align} BR(z_i(\theta))\cdot \mathbf{n}(z_i(\theta))=0, \end{align} $$

where $\mathbf {n}(z_i(\theta ))$ is the unit normal vector of $\Gamma _i$ at $z_i(\theta )$ , and

(2.2) $$ \begin{align} BR(z_i(\theta))\cdot \mathbf{s}(z_i(\theta))\frac{\gamma_i(\theta)}{|z_i'(\theta)|}=C, \end{align} $$

where $\mathbf {s}(z_i(\theta ))$ is the unit tangential vector. Note that (2.2) can be rewritten as

(2.3) $$ \begin{align} (I-P_0)\left[BR(z_i(\theta))\cdot \mathbf{s}(z_i(\theta))\frac{\gamma_i(\theta)}{|z_i'(\theta)|}\right]=0, \end{align} $$

where $P_0$ is the projection to the mean value defined by $P_0 f:=\frac {1}{2\pi }\int _0^{2\pi } f(\theta )d\theta $ . Since $\mathbf {x}_0\in \Omega ^m$ , we can take $r_0>0$ sufficiently small such that $B_{r_0}(x_{0,i})\subset \Omega $ for all $i=1,\ldots ,m$ , where $B_{r_0}(x_{0,i})$ is the ball with radius $r_0$ and centered at $x_{0,i}$ . We aim to construct vortex sheets localized near $\mathbf {x}_0$ . Thus, for $\varepsilon>0$ small, we assume that $z_i$ ( $i=1,\ldots , m$ ) are of the following form:

$$ \begin{align*}z_i(\theta)=x_{i}+\varepsilon R_i(\theta)(\cos\theta, \sin\theta)\end{align*} $$

with $R_i(\theta )=1+\varepsilon f_i(\theta ),$ and $x_i\in B_{r_0}(x_{0,i})$ to be chosen later. We also assume

$$ \begin{align*}\gamma_i(\alpha)=\frac{\kappa_i+\varepsilon g_i(\alpha)}{2\pi |z^{\prime}_i(\alpha)|}.\end{align*} $$

We end this section by introducing some notations and definitions that will be used in this paper and reformulating equations (2.1) and (2.3). Denote the mean value of integral of g on the unit circle by

and set

$$ \begin{align*} &A(\theta, \alpha):=4\sin^2\left(\frac{\theta-\alpha}{2}\right),\,\,\,A_{ij}=|x_{i}-x_{j}|^2,\\[6pt] &B(f,\theta, \alpha):=4(f(\theta)+f(\alpha))\sin^2\left(\frac{\theta-\alpha}{2}\right)+\varepsilon\left((f(\theta)-f(\alpha))^2+4 f(\theta)f(\alpha)\sin^2\left(\frac{\theta-\alpha}{2}\right)\right),\\[6pt] &B_{ij}(\theta, \alpha)=2(x_{i}-x_{j})\cdot((\cos\theta, \sin\theta)-(\cos\alpha, \sin\alpha))+2\varepsilon(x_{i}-x_{j})\cdot\left(f_i(\theta)(\cos\theta, \sin\theta) \right.\\[6pt] &\qquad\quad\qquad\left.-f_j(\theta)(\cos\alpha, \sin\alpha)\right)+\varepsilon((1+\varepsilon f_i(\theta))(\cos\theta, \sin\theta)-(1+\varepsilon f_j(\alpha))(\cos\alpha, \sin\alpha))^2. \end{align*} $$

For $k\geq 3$ , we will also frequently use the function spaces given in the following, whose norms are naturally defined as norms of product spaces.

For given $ \boldsymbol {f}=(f_1,\ldots , f_m)$ and $ \boldsymbol {g}=(g_1,\ldots , g_m)$ , denote $\tilde {g}_{i,\varepsilon }(t)=\kappa _i+\varepsilon g_i(t)$ . Then, we can reduce equations (2.1) and (2.3) to

(2.4)

and

(2.5) $$ \begin{align} 0=\mathcal{F}_{i,2}:=(I-P_0)\tilde{\mathcal{F}}_{i,2}, \end{align} $$

where $\tilde {\mathcal {F}}_{i,2}$ is given by

(2.6)

3 Extension and regularity of functionals

To apply the implicit function theorem at $\varepsilon =0$ , we need to extend the functions $\mathcal {F}_{i,1}$ and $\mathcal {F}_{i, 2}$ defined in Section 2 to $\varepsilon \leq 0$ and check the $C^1$ regularity.

Let us first show the continuity of these functionals. Letting V be the unit ball centered at origin in $\left (X^{k+1}\times X^k\right )^m$ and $B_{r_0}(\mathbf {x}_0)$ be the ball centered at $\mathbf {x_0}$ in $\Omega ^m$ with radius $r_0$ , we have the following proposition.

Proposition 3.1 The functionals $\mathcal {F}_{i,1}$ and $\mathcal {F}_{i,2}$ can be extended from $(-\varepsilon _0, \varepsilon _0)\times B_{r_0}(\mathbf {x}_0) \times V$ to $X^{k}\times X^k$ as continuous functionals.

Proof Throughout the proof, we will frequently use the following Taylor’s formula:

(3.1) $$ \begin{align} \frac{1}{(A+B)^\lambda}=\frac{1}{A^\lambda}-\lambda\int_0^1\frac{B}{(A+tB)^{1+\lambda}}dt. \end{align} $$

Let us consider $\mathcal {F}_{i,1}$ first. We need to prove that $\partial ^l \mathcal {F}_{i,1}\in L^2$ for $l=0,1, \ldots , k$ . For the first term

since $R_i(x)=1+\varepsilon f_i(x)$ , the possible singularity caused by $\varepsilon =0$ may occur only when we take zeroth-order derivative of $\mathcal {F}_{i,11}$ . Thus, we first show that $\mathcal {F}_{i, 11}\in L^2$ . We decompose the kernel into two parts

(3.2) $$ \begin{align} \frac{1}{A(\theta, \alpha)+\varepsilon B(f_i,\theta,\alpha)} =\frac{1}{4\sin^2\left(\frac{\theta-\alpha}{2}\right)}\cdot\frac{1}{1+2\varepsilon f_i(\theta)+\varepsilon^2(f_i(\theta)^2+f^{\prime}_i(\theta)^2)} +\mathcal{K}_R, \end{align} $$

where $\mathcal {K}_R:=\frac {1}{A(\theta , \alpha )+\varepsilon B(f_i,\theta ,\alpha )}-\frac {1}{4\sin ^2\left (\frac {\theta -\alpha }{2}\right )}\cdot \frac {1}{1+2\varepsilon f_i(\theta )+\varepsilon ^2(f_i(\theta )^2+f^{\prime }_i(\theta )^2)}$ is more regular than $\frac {1}{4\sin ^2\left (\frac {\theta -\alpha }{2}\right )}$ . Indeed, by using (3.1), we calculate

$$ \begin{align*} &\sin\left(\theta-\alpha\right)\mathcal{K}_R\\[6pt] &=\frac{\sin\left(\theta-\alpha\right)}{A(\theta, \alpha)+\varepsilon B(f_i,\theta,\alpha)}-\frac{\sin\left(\theta-\alpha\right)}{4\sin^2\left(\frac{\theta-\alpha}{2}\right)}\cdot\frac{1}{1+2\varepsilon f_i(\theta)+\varepsilon^2(f_i(\theta)^2+f^{\prime}_i(\theta)^2)}\\[6pt] &=\frac{\sin\left(\theta-\alpha\right)}{1+2\varepsilon f_i(\theta)+\varepsilon^2(f_i(\theta)^2+f^{\prime}_i(\theta)^2)} \nonumber \\&\quad \times\frac{\left(1+2\varepsilon f_i(\theta)+\varepsilon^2(f_i(\theta)^2+f^{\prime}_i(\theta)^2)\right)4\sin^2\left(\frac{\theta-\alpha}{2}\right)-A(\theta, \alpha)-\varepsilon B(f_i,\theta,\alpha)}{4\sin^2\left(\frac{\theta-\alpha}{2}\right)(A(\theta, \alpha)+\varepsilon B(f_i,\theta,\alpha))}\\[6pt] &=\frac{\varepsilon \sin\left(\theta-\alpha\right)}{1+2\varepsilon f_i(\theta)+\varepsilon^2(f_i(\theta)^2+f^{\prime}_i(\theta)^2)}\\[6pt] &\quad \times\frac{ f_i(\theta)-f_i(\alpha)+\varepsilon f_i(\theta)(f_i(\theta)-f_i(\alpha))+\varepsilon\frac{4f^{\prime}_i(\theta)^2\sin^2\left(\frac{\theta-\alpha}{2}\right)-(f_i(\theta)-f_i(\alpha))^2}{4\sin^2\left(\frac{\theta-\alpha}{2}\right)}}{A(\theta, \alpha)+\varepsilon B(f_i,\theta,\alpha)}\\[6pt] &=\varepsilon\left(\frac{(f_i(\theta)-f_i(\alpha))\sin\left(\theta-\alpha\right)}{4\sin^2\left(\frac{\theta-\alpha}{2}\right)}+O(\varepsilon)\right), \end{align*} $$

where the constant in $O(\varepsilon )$ depends on $\|f\|_{W^{2,\infty }}\leq C\|f\|_{H^3}$ . This implies

$$ \begin{align*}|\sin\left(\theta-\alpha\right)\mathcal{K}_R|\leq C\varepsilon.\end{align*} $$

Then, it is easy to see that

where $\mathcal {R}_{111}$ is regular and bounded. Hence, to prove $\mathcal {F}_{i, 11}\in L^2$ , we only need to estimate the rest term

. By the odd symmetry and (3.1), we have

Using the expansion $f_i(\alpha )=f_i(\theta )+O(|\sin \left (\frac {\theta -\alpha }{2}\right )|)$ and $g_i(\alpha )=g_i(\theta )+O(|\sin \left (\frac {\theta -\alpha }{2}\right )|)$ , then we find

where we have used the fact $\left |\frac {\sin (\theta -\alpha )}{4\sin ^2\left (\frac {\theta -\alpha }{2}\right )} O(\sin \left (\frac {\theta -\alpha }{2}\right ))\right |\leq C$ . Therefore, it holds that

belongs to $L^\infty $ and hence belongs to $L^2$ . Moreover, $\mathcal {R}_{112}$ is regular and bounded. We conclude that $\mathcal {F}_{i,11}\in L^\infty $ . Furthermore, it holds

(3.3)

where $\mathcal {R}_{11}=\mathcal {R}_{111}+\mathcal {R}_{112}$ is regular and bounded.

Next, we prove that $\partial ^k \mathcal {F}_{i,11}\in L^2$ . To simplify notation, we rewrite $\mathcal {F}_{i,11}$ as follows by changing the variable $\alpha $ to $\theta -\alpha $ :

Taking kth derivatives of $\mathcal {F}_{i,11}$ , we see that the most singular term is

We first deal with $I_1$ . By the splitting of the kernel (3.2), we derive

Noting that $|\mathcal {K}_R \sin (\theta -\alpha )|\leq C\varepsilon $ , we have

is bounded. Since

is the Hilbert transformation of the function $\partial ^k f_i(\alpha )(\kappa _i+\varepsilon g_i(\alpha ))$ , we have

Similarly, one can check that $||I_2||_{L^2}\leq C(1+||f||_{L^2})||g||_{H^k}$ .

To estimate $I_3$ , we split the kernel as follows:

$$ \begin{align*} \frac{4\sin^2\left(\frac{\alpha}{2}\right)}{(A(\theta, \theta-\alpha)+\varepsilon B(f_i,\theta,\theta-\alpha))^2} =\frac{1}{4\sin^2\left(\frac{\alpha}{2}\right)} \cdot\frac{1}{(1+2\varepsilon f_i(\theta)+\varepsilon^2(f_i(\theta)^2+f^{\prime}_i(\theta)^2))^2} +\tilde{\mathcal{K}}_R, \end{align*} $$

where $\tilde {\mathcal {K}}_R$ satisfies $| \tilde {\mathcal {K}}_R\sin \alpha |\leq C$ . Since convolution with the kernel $\frac {\sin \alpha }{4\sin ^2\left (\frac {\alpha }{2}\right )}$ defines the Hilbert transformation, we find that

belongs to $L^2$ due to the $L^2$ boundedness of Hilbert transformation and the regularity of $\tilde {\mathcal {K}}_R$ . For the remaining term in $I_3$

we decompose the kernel

$$ \begin{align*} \frac{(1+\varepsilon f_i(\theta-\alpha))(\kappa_i+\varepsilon g_i(\theta-\alpha))\sin(\alpha)}{(A(\theta, \theta-\alpha)+\varepsilon B(f_i,\theta,\theta-\alpha))^2}(f_i(\theta)-f_i(\theta-\alpha)) =\frac{(1+\varepsilon f_i(\theta))(\kappa_i+\varepsilon g_i(\theta))f^{\prime}_i(\theta)}{4\sin^2\left(\frac{\alpha}{2}\right)}+\bar{\mathcal{K}}_R, \end{align*} $$

where $\bar {\mathcal {K}}_R$ satisfies $|\bar {\mathcal {K}}_R\sin \alpha |\leq C$ . Then, we deduce

By Fourier transformation and the Hardy inequality, we obtain

$$ \begin{align*}||(-\Delta)^{\frac{1}{2}}(\partial^kf_i)||_{L^2}\leq ||\nabla \partial^kf_i||_{L^2}\leq ||f_i||_{H^{k+1}}\end{align*} $$

and

Consequently, we have $\partial ^k \mathcal {F}_{i,11}\in L^2$ and hence $\mathcal {F}_{i,11}\in H^k$ .

Now, we turn to the second term

Since $|1-\cos (\theta -\alpha )|=\sin ^2\left (\frac {\theta -\alpha }{2}\right )$ , the kernel of this term is actually regular and bounded. Therefore, it is easy to see that $\mathcal {F}_{i,12}\in H^k$ . Moreover, by (3.1), we find

(3.4)

where $\mathcal {R}_{12}$ is smooth and we have used the identity $1-\cos (\theta -\alpha )=2\sin ^2\left (\frac {\theta -\alpha }{2}\right )=\frac {A(\theta , \alpha )}{2}$ .

For $\mathcal {F}_{i,13}$ , taking kth derivatives of $\mathcal {F}_{i,13}$ , we see that the most singular terms are

Since

, by Taylor’s expansion of $f_i(\alpha )$ at $\alpha =\theta $ , we know that the first term $J_1$ is bounded in $L^2$ . To deal with $J_2$ , we split the kernel

$$ \begin{align*} \frac{\kappa_i+\varepsilon g_i(\theta-\alpha)}{A(\theta, \theta-\alpha)+\varepsilon B(f_i,\theta,\theta-\alpha)} =\frac{\kappa_i+\varepsilon g_i(\theta)}{4\sin^2\left(\frac{\alpha}{2}\right)}+\hat{\mathcal{K}}_R, \end{align*} $$

where $|\hat {\mathcal {K}}_R\sin \frac {\alpha }{2}|\leq C$ . Therefore, we conclude

By Fourier transformation and Hardy inequality, we obtain

$$ \begin{align*}||(-\Delta)^{\frac{1}{2}}(\partial^kf_i)||_{L^2}\leq ||\nabla \partial^kf_i||_{L^2}\leq ||f_i||_{H^{k+1}}\end{align*} $$

and

We can show that the remaining terms $J_3$ and $J_4$ are bounded in $L^2$ similarly. Moreover, it can be seen that

(3.5)

where $\mathcal {R}_{13}$ is regular.

Since $H(x,y)$ is smooth in $\Omega $ , the terms $\mathcal {F}_{i,14}$ , $\mathcal {F}_{i,15}$ , and $\mathcal {F}_{i,16}$ are apparently smooth and belong to $H^k$ . Furthermore, we have

(3.6)

where $\mathcal {R}_{14}$ is bounded and smooth.

By (3.3)–(3.6), we conclude

(3.7)

where $\mathcal {R}_{1}:=\mathcal {R}_{11}+\mathcal {R}_{12}+\mathcal {R}_{13}+\mathcal {R}_{14}$ is regular. Hence, we can define

(3.8)

Next, we prove the continuity of $\mathcal {F}_{i,1}$ . By (3.7) and the definition of $\mathcal {F}_{i,1}( 0,\mathbf {x}, \boldsymbol {f},\boldsymbol {g})$ , one can easily check that $\mathcal {F}_{i,1}$ is continuous with respect to $\varepsilon $ at $\varepsilon =0$ . Thus, we only need to prove that $\mathcal {F}_{i,1}$ is continuous with respect to $\varepsilon $ for $\varepsilon \not =0$ . However, it is easy to see that the continuity of $ \mathcal {F}_{i,1}$ with respect to $\varepsilon $ is a consequence of its continuity with respect to $\boldsymbol {f}$ and $\boldsymbol {g}$ when $\varepsilon \not =0$ , on which we will focus below.

We only prove the continuity of $\mathcal {F}_{i,11}$ with respect to $f_i$ and $g_i$ , and the continuity of other terms in $\mathcal {F}_{i, 1}$ can be proved by a similar or even easier way. We will use the following notations: for a general function h, we denote $\Delta h=h(\theta )-h(\alpha ), \ \ \ h=h(\theta ), \ \ \ \tilde h=h(\alpha ),$ and

$$ \begin{align*}D(h)=\varepsilon^2(\Delta h)^2+4(1+\varepsilon h)(1+\varepsilon\tilde h)\sin^2\left(\frac{\theta-\alpha}{2}\right).\end{align*} $$

To show the continuity of $\mathcal {F}_{i,11}$ with respect to $f_i$ , let $(f_1,g),(f_2,g)\in X_i^k$ . Then, we can calculate the difference

For the first term $K_1$ , since $\frac {1}{D(f_2)}$ has the same singularity as $\frac {1}{\sin ^2\left (\frac {\theta -\alpha }{2}\right )}$ , it is easy to prove $\|K_1\|_{H^k}\leq C\|f_1-f_2\|_{H^{k+1}}$ by the technique we have used before. For the second term $K_2$ , since

$$ \begin{align*} &\frac{1}{D(f_2)}-\frac{1}{D(f_1)}\\[6pt] &=\frac{\varepsilon^2((\Delta f_1)^2-(\Delta f_2)^2)+4\varepsilon\big((f_1-f_2)(1+\varepsilon\tilde f_2)+(\tilde f_1-\tilde f_2)(1+\varepsilon f_1)\big)\sin^2(\frac{x-y}{2})}{D(f_1)D(f_2)}\\[6pt] &=\varepsilon\frac{\varepsilon(\Delta f_1+\Delta f_2)(\Delta f_1-\Delta f_2)+4\big((f_1-f_2)(1+\varepsilon\tilde f_2)+(\tilde f_1-\tilde f_2)(1+\varepsilon f_1)\big)\sin^2(\frac{x-y}{2})}{D(f_1)D(f_2)}, \end{align*} $$

it holds that the singularity of $\frac {1}{D(f_2)}-\frac {1}{D(f_1)}$ is also of the order $O\left (\frac {1}{4\sin ^2\left (\frac {\theta -\alpha }{2}\right )} \right )$ , the same as the kernel in $\mathcal {F}_{i,11}$ itself. Therefore, using argument similar to the above, we can prove that $\|K_2\|_{H^k}\leq C\|f_2-f_1\|_{H^{k+1}}$ , which shows the continuity of $\mathcal {F}_{i,11}$ with respect to $f_i$ . Notice that $\mathcal {F}_{i,11}$ is linear with respect to $g_i$ . Then, the continuity of $\mathcal {F}_{i,11}$ with respect to g can be obtained by argument similar to the proof of boundedness of $\mathcal {F}_{i,11}$ in $H^k$ .

We have shown that the conclusion of Proposition 3.1 holds true for $\mathcal {F}_{i,1}$ . The fact that $\mathcal {F}_{i,2}$ is well defined and continuous can be verified in a similar way. Attention should be paid to the fact that the projection operator $I- P_0$ eliminates all constant terms in $\tilde {\mathcal {F}}_{i,2}$ , which also removes singularity in $\mathcal {F}_{i,2}$ . By (3.1), we obtain

(3.9)

where $\mathcal {R}_{2}$ is smooth. Thus, we define

(3.10)

▪

Our next proposition concerns the $C^1$ regularity.

Proposition 3.2 The Gateaux derivatives $\partial _{(\boldsymbol {f}, \boldsymbol {g})} \mathcal {F}_{i, 1}$ and $\partial _{(\boldsymbol {f}, \boldsymbol {g})} \mathcal {F}_{i, 2}$ exist and are continuous.

Proof We first prove that the derivative of $\mathcal {F}_{i,11}$ with respect to $f_i$ exists and is as follows:

(3.11) $$ \begin{align} \partial_{f_i} \mathcal{F}_{i,11} h=Fh, \quad \forall \,h\in X^{k+1}, \end{align} $$

where $F h$ is given by

(3.12)

To prove (3.11), one needs to verify

(3.13) $$ \begin{align} \lim\limits_{t\to0}\left\|\frac{\mathcal{F}_{i,11}(\varepsilon, f_i+th,g_i)-\mathcal{F}_{i,11}(\varepsilon, f_i,g_i)}{t}-F h\right\|_{H^{k}}= 0. \end{align} $$

Using the notations given in the proof of Proposition 3.1, we deduce

By the mean value theorem, we find

$$ \begin{align*} &\frac{1}{D(f_i+th)}-\frac{1}{D(f_i)}=O\left( \frac{t\varepsilon}{4\sin^2\left(\frac{x-y}{2}\right)}\right),\\[6pt] &\frac{1}{D(f_i+th)}-\frac{1}{D(f_i)}+t\frac{2\varepsilon^2\Delta f_i\Delta h +4\big(\varepsilon\tilde R h +\varepsilon \tilde h R\big)\sin^2(\frac{x-y}{2})}{D(f_i)^2}=O\left( \frac{t^2\varepsilon^2}{4\sin^2\left(\frac{x-y}{2}\right)}\right), \end{align*} $$

which means that the kernels in $F_{1}$ and $F_{2}$ are of the same order as the kernel in $\mathcal {F}_{i,11}$ . Therefore, by argument similar to Proposition 3.1, we have

$$ \begin{align*} \|F_{1}\|_{H^{k}}+\|F_{2}\|_{H^{k}}\le Ct\|h\|_{X^{k+1}}. \end{align*} $$

Letting $t\rightarrow 0$ , we obtain (3.13) and hence obtain the existence of Gateaux derivative of $\mathcal {F}_{i,11}$ . To prove the continuity of $\partial _{f_i} \mathcal {F}_{i,11}(\varepsilon , f_i, g_i)h$ , one just needs to verify by definition. Since there is no other new idea than the proof of continuity for $\mathcal {F}_{i,11}$ , we omit it therefore. The existence and continuity of Gateaux derivatives of other terms in $\mathcal {F}_{i,1}$ and $\mathcal {F}_{i,2}$ can be obtained via similar argument, which we leave out here. Noting that $\mathcal {F}_{i,1}$ is linearly dependent on g and $\mathcal {F}_{i,2}$ is quadratically dependent on g, it is much easier to compute their Gateaux derivatives with respect to g, so we leave them to our reader. For readers’ convenience, we also write down the derivatives of $\mathcal {F}_{i,1}$ and $\mathcal {F}_{i,2}$ in the following form directly without proof here.

Recall the definitions $\tilde {g}_{i,\varepsilon }(t)=\kappa _i+\varepsilon g_i(t)$ , $R_i(t)=1+\varepsilon f_i(t)$ . For any $h_1\in X^{k+1}$ and $h_2\in X^k$ , we have

(3.14)

(3.15) $$ \begin{align} \partial_{f_j}\mathcal{F}_{i,1}(\varepsilon, \mathbf{x}, \boldsymbol{f}, \boldsymbol{g}) h_1=O(\varepsilon), \end{align} $$

(3.16)

(3.17)

(3.18)

(3.19) $$ \begin{align} \partial_{f_j}\mathcal{F}_{i, 2}(\varepsilon,\mathbf{x}, \boldsymbol{f}, \boldsymbol{g}) h_1=O(\varepsilon), \end{align} $$

(3.20)

and

(3.21)

▪

4 Linearization and isomorphism

In this section, we study the linearization of the functionals defined in Section 2. Denote $\boldsymbol {\mathcal {F}}_i:=(\mathcal {F}_{i,1}, \mathcal {F}_{i,2})$ and $\boldsymbol {\mathcal {F}}:=(\boldsymbol {\mathcal {F}}_1,\ldots , \boldsymbol {\mathcal {F}}_m)$ .

By (3.8) and (3.10), one can check that $(0,\mathbf {x}, 0,0)$ is a solution to $\boldsymbol {\mathcal {F}}=0$ if and only if $\mathbf {x}$ is a critical point of $\mathcal {W}_m$ . Now, we take $\mathbf {x}_0$ to be a critical point of $\mathcal {W}_m$ , and hence $(0,\mathbf {x}_0, 0,0)$ is a solution to $\boldsymbol {\mathcal {F}}=0$ . We study the linearization of $\boldsymbol {\mathcal {F}}$ at $(0,\mathbf {x}_0, 0,0)$ .

According to (3.14)–(3.21) at the end of the proof of Proposition 3.2, when $\varepsilon =0$ and $\boldsymbol {f}, \boldsymbol {g}\equiv 0$ , for all $i=1,\ldots ,m$ , the Gateaux derivatives are

(4.1)

Taking $(h_1,h_2)\in X^{k+1}\times X^k$ , where

(4.2) $$ \begin{align} h_1(\theta)=\sum_{j=1}^\infty( a_j \cos(j\theta)+b_j\sin(j\theta)) \ \ \ \text{and} \ \ \ h_2(\theta)=\sum_{j=1}^\infty( c_j \cos(j\theta)+d_j\sin(j\theta)), \end{align} $$

we will prove that the linearization of $\boldsymbol {\mathcal {F}}_i$ at $(0,\mathbf {x}_0,0,0)$ has the following Fourier series form:

(4.3) $$ \begin{align} D_{(f_i,g_i)} \boldsymbol{\mathcal{F}}_i(0,\mathbf{x}_0,0,0) (h_1, h_2):=&\begin{pmatrix} \partial_{f_i}\mathcal{F}_{i,1}(0,\mathbf{x}_0, 0,0) h_1 + \partial_{g_i}\mathcal{F}_{i,1}(0,\mathbf{x}_0, 0,0) h_2 \\[6pt] \partial_{f_i}\mathcal{F}_{i,2}(0,\mathbf{x}_0, 0,0) h_1 + \partial_{g_i}\mathcal{F}_{i,2}(0,\mathbf{x}_0, 0,0) h_2 \end{pmatrix} \nonumber \\ &=\sum_{j=1}^\infty\begin{pmatrix} \hat a_j \sin(j\theta) +\hat b_j \cos(j\theta)\\[6pt] \hat c_j \cos(j\theta)+\hat d_j \sin(j\theta)\end{pmatrix}, \end{align} $$

where

$$ \begin{align*} \begin{pmatrix} \hat a_j \\[6pt] \hat c_j \end{pmatrix}=M_j \begin{pmatrix} a_j \\[6pt] c_j \end{pmatrix} \quad\text{and}\,\,\begin{pmatrix} \hat b_j \\[6pt] \hat d_j \end{pmatrix}=N_j \begin{pmatrix} b_j \\[6pt] d_j \end{pmatrix} \end{align*} $$

with $M_j$ and $N_j$ two $2\times 2$ matrices given in Lemma 4.2.

To compute $M_j$ and $N_j$ , we need the following identities.

Lemma 4.1 For all $j\ge 1$ and $j\in \mathbb N^*$ , there hold

(4.4)

(4.5)

(4.6)

(4.7)

Proof Identities (4.4) and (4.6) were proved in Lemma A.8 [24]. Indeed, (4.4) can be deduced from the identity

where $\mathcal H(\cdot )$ is the Hilbert transform on torus and hence $H(\cos (j\theta ))=\sin ({j\theta })$ . Identity (4.6) can be obtained by computing the fractional Laplacians

Finally, we point out that the identities (4.5) and (4.7) can be derived by calculating derivatives of (4.4) and (4.6), respectively.▪

Now, we can prove (4.3) and find the explicit formula for $M_j$ and $N_j$ .

Lemma 4.2 The derivative of $\boldsymbol {\mathcal {F}}_i$ at $(0,\mathbf {x}_0,0,0)$ is given by (4.3) with

(4.8) $$ \begin{align} M_j=\begin{pmatrix} - \frac{\kappa_i j}{2} & \frac{1}{2} \\[6pt] \frac{(2-j)\kappa_i^2}{2} & -\frac{\kappa_i}{2} \end{pmatrix}, \quad N_j=\begin{pmatrix} \frac{\kappa_i j}{2} & -\frac{1}{2} \\[6pt] \frac{(2-j)\kappa_i^2}{2} & -\frac{\kappa_i}{2} \end{pmatrix}, \end{align} $$

for any $j\geq 1$ .

Moreover, $D_{(f_i, g_i)}\boldsymbol {\mathcal {F}}_i(0,\mathbf {x}_0,0,0)$ is an isomorphism from $X^k_i$ to $Y^k_i$ and $D_{(\boldsymbol {f}, \boldsymbol {g})}\boldsymbol {\mathcal {F}}(0,\mathbf {x}_0,0,0)$ is an isomorphism from $\mathcal X^k$ to $\mathcal Y^k$ .

Proof Using (4.1), (4.2), and Lemma 4.1, we obtain by direct calculations

$$ \begin{align*} \partial_{f_i}\mathcal{F}_{i,1}(0,\mathbf{x}_0, 0,0) h_1=\sum_{j=1}^\infty\left(\frac{-\kappa_ij}{2} a_j \sin(j\theta)+\frac{\kappa_ij}{2} b_j \cos(j\theta)\right), \end{align*} $$

$$ \begin{align*} \partial_{g_i}\mathcal{F}_{i,1}(0,\mathbf{x}_0, 0,0) h_2=\frac{1}{2}\sum_{j=1}^\infty (c_j\sin(j\theta)-d_j\cos(j\theta)), \end{align*} $$

$$ \begin{align*} \begin{array}{ll} \partial_{f_i}\mathcal{F}_{i,2}(0,\mathbf{x}_0, 0,0)h_1&\\[6pt] \quad\qquad\,=\kappa_i^2\sum_{j=1}^\infty( a_j \cos(j\theta)+b_j\sin(j\theta))-\kappa_i^2\sum_{j=1}^\infty\left( \frac{j}{2} a_j \cos(j\theta)+\frac{j}{2}b_j\sin(j\theta)\right)&\\[6pt] \quad\qquad\,=\sum_{j=1}^\infty\left( \frac{\kappa_i^2(2-j)}{2} a_j \cos(j\theta)+\frac{\kappa_i^2(2-j)}{2}b_j\sin(j\theta)\right),&\\ \end{array} \end{align*} $$

and

$$ \begin{align*} \partial_{g_i}\mathcal{F}_{i,2}(0,\mathbf{x}_0, 0,0)h_2=-\frac{\kappa_i}{2}\sum_{j=1}^\infty( c_j \cos(j\theta)+d_j\sin(j\theta)). \end{align*} $$

Then, one can easily check that the derivative of $\boldsymbol {\mathcal {F}}_i$ at $(0,\mathbf {x},0,0)$ is given by (4.3) with

$$ \begin{align*}M_j=\begin{pmatrix} - \frac{\kappa_i j}{2} & \frac{1}{2} \\[6pt] \frac{(2-j)\kappa_i^2}{2} & -\frac{\kappa_i}{2} \end{pmatrix},\,\,N_j=\begin{pmatrix} \frac{\kappa_i j}{2} & -\frac{1}{2} \\[6pt] \frac{(2-j)\kappa_i^2}{2} & -\frac{\kappa_i}{2} \end{pmatrix}.\end{align*} $$

Now we are going to prove that $D_{(f_i,g_i)} \boldsymbol {\mathcal {F}}_i(0,\mathbf {x}_0,0,0)$ is an isomorphism from $X^k_i$ to $Y^k_i$ . Recall the definition of $X^k_i$ and $Y^k_i$ given at the end of Section 2. From the above calculations, one has $M_1=\begin {pmatrix} -\kappa _i/2 & 1/2 \\ \kappa _i^2/2 & -\kappa _i/2 \end {pmatrix}$ and $N_1=\begin {pmatrix} \kappa _i/2 & -1/2 \\ \kappa _i^2/2 & -\kappa _i/2 \end {pmatrix}$ , then it is obvious that $D_{(f_i,g_i)} \boldsymbol {\mathcal {F}}_i(0,\mathbf {x}_0,0,0)$ maps $X^k_i$ to $Y^k_i$ . Hence, only the invertibility needs to be considered.

For $j\ge 2$ , $\det (M_j)=-\det (N_j)=\frac {\kappa _i^2(j-1)}{2}>0$ which implies that $M_j$ and $N_j$ are invertible, and their inverse are given by

(4.9) $$ \begin{align} M_j^{-1}=\begin{pmatrix} \frac{-1}{\kappa_i(j-1)} & \frac{-1}{\kappa_i^2(j-1)} \\[6pt] \frac{j-2}{j-1} & \frac{-j}{\kappa_i(j-1)} \end{pmatrix}, \ \ \ \forall \, j\ge 2, \end{align} $$

and

(4.10) $$ \begin{align} N_j^{-1}=\begin{pmatrix} \frac{1}{\kappa_i(j-1)} & \frac{-1}{\kappa_i^2(j-1)} \\[6pt] \frac{2-j}{j-1} & \frac{-j}{\kappa_i(j-1)} \end{pmatrix}, \ \ \ \forall \, j\ge 2. \end{align} $$

Thus, for any $(u,v)\in Y_i^k$ with

$$ \begin{align*} u&=\sum_{j=1}^\infty p_j\sin(j\theta) +q_j\cos(j\theta)\ \ \ \text{and} \\ v&=-\kappa_i p_1\cos(\theta)+\kappa_iq_1\sin(\theta)+\sum_{j=2}^\infty r_j\cos(j\theta)+s_j\sin(j\theta), \end{align*} $$

we can write $D_{(f_i,g_i)}\boldsymbol {\mathcal {F}}_i(0,\mathbf {x}_0,0,0)^{-1}(u,v)$ as

$$ \begin{align*} &D_{(f_i,g_i)}\boldsymbol{\mathcal{F}}_i(0,\mathbf{x}_0,0,0)^{-1}(u,v)\\[6pt] &=\begin{pmatrix} -\frac{p_1}{\kappa_i} \cos(\theta)+\frac{q_1}{\kappa_i} \sin(\theta)\\[6pt] p_1 \cos(\theta)-q_1\sin(\theta)\end{pmatrix}+\sum_{j=2}^\infty M_j^{-1} \begin{pmatrix} p_j \\[6pt] r_j \end{pmatrix}\cos(j\theta)+ N_j^{-1} \begin{pmatrix} q_j \\[6pt] s_j \end{pmatrix}\sin(j\theta). \end{align*} $$

Denote

$$ \begin{align*} \begin{pmatrix} \tilde p_j \\[6pt] \tilde r_j \end{pmatrix}=M_j^{-1} \begin{pmatrix} p_j \\[6pt] r_j \end{pmatrix}, \ \ \ \begin{pmatrix} \tilde q_j \\[6pt] \tilde s_j \end{pmatrix}=N_j^{-1} \begin{pmatrix} q_j \\[6pt] s_j \end{pmatrix},\ \ \ \forall \, j\ge 2. \end{align*} $$

From (4.9) and (4.10), we have the asymptotic behavior: $\tilde p_j=O(j^{-1}(|p_j|+|r_j|))$ , $\tilde r_j=O( |p_j|+|r_j|)$ , $\tilde q_j=O(j^{-1}(|q_j|+|s_j|))$ and $\tilde s_j=O( |q_j|+|s_j|)$ as $j\to +\infty $ , which implies that $D_{(f_i,g_i)}\boldsymbol {\mathcal {F}}_i(0,\mathbf {x},0,0)^{-1}(u,v)$ does belong to $ X^k_i$ .

Noticing that by (4.1), we have $\partial _{f_j}\mathcal {F}_{i,1}(0,\mathbf {x}_0, 0,0) h_1$ , $\partial _{g_j}\mathcal {F}_{i,1}(0,\mathbf {x}_0, 0,0) h_2$ , $\partial _{f_j}\mathcal {F}_{i,2}(0,\mathbf {x}_0, 0,0) h_1$ , and $\partial _{g_j}\mathcal {F}_{i,2}(0,\mathbf {x}_0, 0,0) h_2=0,\,j\not =i$ . Therefore, we find

$$ \begin{align*}D_{(\boldsymbol{f},\boldsymbol{g})}\boldsymbol{\mathcal{F}}(0,\mathbf{x}_0, 0,0)=\text{diag}\left(D_{(f_1,g_1)}\boldsymbol{\mathcal{F}}_1(0,\mathbf{x}_0, 0,0),\ldots, D_{(f_m,g_m)}\boldsymbol{\mathcal{F}}_m(0,\mathbf{x}_0, 0,0)\right),\end{align*} $$

and hence $D_{(\boldsymbol {f},\boldsymbol {g})}\boldsymbol {\mathcal {F}}(0,\mathbf {x}_0, 0,0)$ is an isomorphism from $\mathcal X^k$ to $\mathcal Y^k$ .

The proof of is thus completed.▪

5 Existence of vortex sheets

In this section, inspired by the classical Crandall–Rabinowitz theorem on bifurcation theory, we use the implicit function theorem to obtain a branch of solutions for arbitrarily fixed small $\varepsilon $ .

From the previous sections, we know that $(0,\mathbf {x}_0, 0,0)$ is a solution to $\boldsymbol {\mathcal {F}}=0$ if and only if $\mathbf {x}_0$ is a critical point of $\mathcal {W}_m$ . Moreover, $D_{(\boldsymbol {f},\boldsymbol {g})}\boldsymbol {\mathcal {F}}(0,\mathbf {x}_0, 0,0)$ is an isomorphism from $\mathcal X^k$ to $\mathcal Y^k$ . It can be seen from Lemma 4.2 that the kernel of $D_{(\boldsymbol {f}, \boldsymbol {g})}\boldsymbol {\mathcal {F}}(0,\mathbf {x}_0,0,0)$ in $\left (X^{k+1}\times X^k\right )^m$ is

$$ \begin{align*}\prod_{i=1}^m \{(a\cos(\theta)+b\sin(\theta), \kappa_i(a\cos(\theta)+b\sin(\theta)))\mid (a,b)\in \mathbb{R}^2\}.\end{align*} $$

We take arbitrary nontrivial $(\boldsymbol {f}_0, \boldsymbol {g}_0)\in \mathcal {X}^k_0$ and define the following new functional:

(5.1) $$ \begin{align} \overline{\boldsymbol{\mathcal{F}}}(\varepsilon, \tau, \mathbf{x}, \boldsymbol{f}, \boldsymbol{g}):= \boldsymbol{\mathcal{F}}(\varepsilon, \mathbf{x}, \boldsymbol{f} +\tau\boldsymbol{f}_0, \boldsymbol{g}+ \tau\boldsymbol{g}_0). \end{align} $$

To apply the implicit function theorem, we need to make sure that $\overline {\boldsymbol {\mathcal {F}}}$ maps a suitable subset of $\mathcal X^k$ into $\mathcal Y^k$ . This aim will be achieved by choosing $\mathbf {x}$ properly. Indeed, letting $V_1:=\{(\boldsymbol {f}, \boldsymbol {g})\in \mathcal X^k \mid \sum _{j=1}^m( ||f_i||_{H^{k+1}}+||g_i||_{H^{k}})<1\} \subset \mathcal X^k$ be the unit ball, we have the following key proposition.

Proposition 5.1 The condition that $\overline {\boldsymbol {\mathcal {F}}}$ maps $(-\varepsilon _0, \varepsilon _0)\times (-\tau _1, \tau _1)\times B_{r_0}(\mathbf {x}_0)\times V_1$ into $\mathcal Y^k$ is equivalent to a system of $2m$ equations of the form

(5.2) $$ \begin{align} \nabla \mathcal{ W}_m(\mathbf{x})=O_{\tau_1}(\varepsilon), \end{align} $$

where $\tau _1$ is any fixed small positive number and $O_{\tau _1}(\varepsilon )$ means a vector that is of the order $\varepsilon $ up to a constant depending on $\tau _1$ .

Proof For arbitrary $i=1,\ldots , m$ , we take $(\boldsymbol {f}, \boldsymbol {g})\in V_1$ with

$$ \begin{align*} &f_i(\theta)=\sum_{j=1}^\infty( a_j \cos(j\theta)+b_j\sin(j\theta)), \\[6pt] &g_i(\theta)=-\kappa_i a_1\cos(\theta)-\kappa_i b_1\sin(\theta)+\sum_{j=2}^\infty( c_j \cos(j\theta)+d_j\sin(j\theta)). \end{align*} $$

By the definition of $\mathcal Y^k$ , in order to make $\overline {\boldsymbol {\mathcal {F}}}(\varepsilon , \tau , \mathbf {x}, \boldsymbol {f}, \boldsymbol {g})= \boldsymbol {\mathcal {F}}(\varepsilon , \mathbf {x}, \boldsymbol {f} +\tau \boldsymbol {f}_0, \boldsymbol {g}+ \tau \boldsymbol {g}_0)\in \mathcal {Y}^k$ , we need to ensure that the following equations hold true:

(5.3)

where $i=1,\ldots ,m$ . By (3.7), (3.9), and calculations in Lemmas 4.1 and 4.2, we obtain

(5.4)

(5.5)

(5.6)

and

(5.7)

Then, by the above equations (5.4)–(5.7), we conclude that (5.3) is equivalent to the following equations:

(5.8)

and

(5.9)

Since (5.8) and (5.9) hold for all $i=1,\ldots ,m$ , we arrive at (5.2) and complete the proof of Proposition 5.1.▪

Now, we are ready to prove Theorem 1.1.

Proof of Theorem 1.1 Since we have the nondegeneracy condition $\text {deg} \left (\nabla \mathcal {W}_m, \mathbf {x}_0\right )\not =0$ , equation (5.2) is solvable near $\mathbf {x}_0$ whenever $\varepsilon $ is small. We solve (5.2) and write the solution $\mathbf {x}_{\varepsilon ,\tau }$ in the form $\mathbf {x}_{\varepsilon ,\tau }=\mathbf {x}_0+\varepsilon \bar {\mathcal {R}}_{\mathbf {x}}(\varepsilon , \tau , \boldsymbol {f}, \boldsymbol {g})$ . Then, we know that $\bar {\mathcal R}_{\mathbf {x}}$ defined on $ (-\varepsilon _0, \varepsilon _0) \times (-\tau _1, \tau _1) \times V_1$ is at least of $C^1$ due to the regularity of $\boldsymbol {\mathcal {F}}$ .

Now, set

$$ \begin{align*}\overline{\boldsymbol{\mathcal{F}}}^*(\varepsilon,\tau, \boldsymbol{f}, \boldsymbol{g}):= \overline{\boldsymbol{\mathcal{F}}}(\varepsilon,\tau, \mathbf{x}_0+\varepsilon \bar{\mathcal R}_{\mathbf{x}}(\varepsilon, \tau, \boldsymbol{f}, \boldsymbol{g}), \boldsymbol{f}, \boldsymbol{g}).\end{align*} $$

Then, we conclude from Proposition 5.1 that $\overline {\boldsymbol {\mathcal {F}}}^*$ maps $(-\varepsilon _0, \varepsilon _0)\times (-\tau _1, \tau _1) \times V_1$ into $\mathcal Y^k$ . Moreover, $\overline {\boldsymbol {\mathcal {F}}}^*$ is $C^1$ continuous with respect to $\boldsymbol {f}$ and $\boldsymbol {g}$ . Next, we need to verify that $D_{(\boldsymbol {f}, \boldsymbol {g})}\boldsymbol {\mathcal {F}}^*(0,0,0,0)$ is an isomorphism from $\mathcal X^k$ to $\mathcal Y^k$ . In fact, by the chain rule, we get

$$ \begin{align*}D_{(\boldsymbol{f}, \boldsymbol{g})} \overline{\boldsymbol{\mathcal{F}}}^*=D_{(\boldsymbol{f}, \boldsymbol{g})} \overline{\boldsymbol{\mathcal{F}}}+ D_{\mathbf{x}}\overline{\boldsymbol{\mathcal{F}}}\cdot D_{(\boldsymbol{f}, \boldsymbol{g})}\left(\mathbf{x}_0+\varepsilon \bar{\mathcal R}_{\mathbf{x}}(\varepsilon, \tau, \boldsymbol{f}, \boldsymbol{g}) \right),\end{align*} $$

which implies

$$ \begin{align*}D_{(\boldsymbol{f}, \boldsymbol{g})} \overline{\boldsymbol{\mathcal{F}}}^*(0,0,0,0)=D_{(\boldsymbol{f}, \boldsymbol{g})} \boldsymbol{\mathcal{F}}(0, \mathbf{x}_0, 0,0).\end{align*} $$

Therefore, $D_{(\boldsymbol {f}, \boldsymbol {g})} \overline {\boldsymbol {\mathcal {F}}}^*(0,0,0,0)$ is an isomorphism from $\mathcal X^k$ to $\mathcal Y^k$ by Lemma 4.2.

Now, applying implicit function theorem to $\overline {\boldsymbol {\mathcal {F}}}^*$ at the point $(0,0,0,0)$ , we obtain that there exist $\varepsilon _0>0$ and $0<\tau _0\leq \tau _1$ such that the solutions set

$$ \begin{align*} \left\{(\varepsilon, \tau, \boldsymbol{f}, \boldsymbol{g})\in (-\varepsilon_0,\varepsilon_0)\times (-\tau_0,\tau_0)\times V_1 \ : \ \overline{\boldsymbol{\mathcal{F}}}^*(\varepsilon,\tau,\boldsymbol{f}, \boldsymbol{g})=0\right\} \end{align*} $$

is not empty and can be parameterized by a two-dimensional surface $(\varepsilon ,\tau )\in (-\varepsilon _0,\varepsilon _0)\times (-\tau _0,\tau _0)\to (\varepsilon , ,\tau , \boldsymbol {f}_{\varepsilon ,\tau }, \boldsymbol {g}_{\varepsilon ,\tau })$ . So we obtain a family of nontrivial vortex sheet solutions and finish the proof of (i) in Theorem 1.1.

Since (ii) of Theorem 1.1 is obvious, to end our proof, we only need to show the convexity of the interior of $\Gamma _i$ for $i=1,\ldots , m$ . This can be done by computing the sign of the curvature. Recall that $z_i(\theta )=x_{\varepsilon ,\tau ,i}+\varepsilon R_i(\theta )(\cos \theta ,\sin \theta )$ with $R_i(\theta )=1+\varepsilon (f_{\varepsilon ,\tau ,i}(\theta )+\tau f_{0,i})$ . Given $\theta \in [0,2\pi )$ , the signed curvature of $\Gamma _i$ at $z_i(\theta )$ is

$$ \begin{align*} \varepsilon \kappa(\theta)=\frac{R_i(\theta)^2+2R^{\prime}_i(\theta)^2-R_i(\theta)R_i''(\theta)}{\left(R_i(\theta)^2+R_i'(\theta)^2\right)^{\frac{3}{2}}}=\frac{1+O(\varepsilon)}{1+O(\varepsilon)}>0, \end{align*} $$

for $\varepsilon $ and $\tau $ small, which implies the convexity and thus completes the proof of Theorem 1.1.▪

We point out that for fixed $\varepsilon $ , if $\tau _1\not =\tau _2$ with $0<\tau _1, \tau _2<\tau _0$ , then obviously one has $\omega _{\varepsilon ,\tau _1}\not \equiv \omega _{\varepsilon ,\tau _2}$ . Thus, we have obtained a large family of stationary solutions with vortex sheet for every $\varepsilon>0$ small.