Let \( \bar \xi\in\mathbb{R}\) and \( \bar b\in\mathbb{R}_{>0}\) represent a given potential vorticity and circulation, respectively. Let \( \rho_{\text{in}}\in \mathbb{R}_{\ge0} \) and \( \rho_{\text{out}}\in \mathbb{R}_{>0}\) denote the given inner and outer mass densities, and let \( R\in\mathbb{R}_{>0}\) be a given inner radius and \( \bar \epsilon\in\mathbb{R}_{>0}\) the radius of the cross-section. Suppose that the surface tension \( \bar \sigma\in\mathbb{R}_{\ge 0}\) is a \( C^1\) function of the aspect ratio \( \epsilon=\bar \epsilon/R\) that is either identically zero, \( \bar \sigma = 0\) , or it is strictly positive, \( \bar \sigma_{\epsilon}>0\) , with

Then there exists an \( \epsilon_0\in\mathbb{R}_{>0}\) such that for all \( \epsilon\in(0, \epsilon_0)\) , there is a unique axisymmetric traveling wave vortex ring solution with cross-sectional domain \( \Omega\) close to \( B_{\bar \epsilon}(R,0)\) . The speed of the ring is approximately given by

as \( \epsilon=\bar\epsilon/R\ll1\) .

Let \( \bar \xi, \bar b, \rho_{\text{in}},\rho_{\text{out}}\) , \( R\) , \( \bar \epsilon\) and \( \bar \sigma_{ \epsilon}\) be given as in Theorem 1. Then there exists an \( \epsilon_0\in\mathbb{R}_{>0}\) such that for all \( \epsilon \in (0, \epsilon_0)\) , there exists a unique solution \( (\Omega, \bar W, \bar \gamma, \bar \nu,\psi)\) to the overdetermined boundary problem with \( \Omega\) close to \( B_{\bar \epsilon}(0,R)\) . The speed \( \bar W\) is approximately given by formula (17). The leading order asymptotics of the flux constant and the Bernoulli constant are known.

Let \( \rho\in\mathbb{R}_{\ge0}\) denote the density ratio. Suppose that the surface tension \( \sigma\in\mathbb{R}_{\ge0}\) is a \( C^1\) function of the aspect ratio \( \epsilon\in\mathbb{R}_{>0}\) , that is either identically zero, \( \sigma=0\) , or it is strictly positive, \( \bar \sigma_{\epsilon}>0\) , with (49), (50), and (51). Then there exists an \( \epsilon_0\in\mathbb{R}_{>0}\) such that for any \( \epsilon\in(0,\epsilon_0)\) , there exists a solution \( (\theta,\varphi_{\text{in}},\varphi_{\text{out}},W,\gamma,\nu)\) to (39), (40), (41),(42), (43), (44), (45), (46), (47), and (48). The functions \( \theta,\varphi_{\text{in}}\) , and \( \varphi_{\text{out}}\) are all smooth and the shape function \( \theta\) satisfies the geometric conditions in (26).

Furthermore, the following holds:

1. The speed, flux and Bernoulli constants satisfy the following asymptotic expansions

   \[ \begin{align} W&=\frac{1}{4\pi}\left(\log 8-\frac{1}{2}+\log\frac{1}{\epsilon}\right)+\rho\pi+\frac{\epsilon\sigma_{\epsilon}\pi}{2} +o(1), \end{align} \]

   (52)

   \[ \begin{align} \gamma&=\frac{3}{8\pi}\log\left(\frac{8}{\epsilon}\right)-\frac{15}{16\pi}-\frac{\rho\pi}2-\frac{\epsilon \sigma_\epsilon\pi}{4}+o(1), \end{align} \]

   (53)

   \[ \begin{align} \nu & = 4\rho - \frac1{4\pi^2} +\epsilon\sigma_\epsilon+o(1), \\\end{align} \]

   (54)

   as \( \epsilon\to0\) .

2. The shape function \( \theta\) goes to \( 0\) in every \( H^k(\partial B)\) -space as \( \epsilon\rightarrow 0\) . In fact, \( \theta\) is a continuously Fréchet differentiable function of \( \epsilon\) and it holds that \( \partial_\epsilon\theta\rightarrow 0\) at \( \epsilon=0\) in every \( H^k(\partial B)\) space.

3. For small enough \( \epsilon\) , the solution is unique among all \( (\theta,\varphi_{\text{in}},\varphi_{\text{out}},W,\gamma,\nu)\) such that \( \theta\) is in the class (26) with \( \left\lVert \theta \right\rVert_{H^5}\leq\epsilon^{\ell}\) for some \( \ell\in (1/2,1)\) , and \( W\) satisfies the bound

   \[ \begin{equation} |W||\log \epsilon|\left(\epsilon^2 +\|\theta\|_{H^k}^2\right)=o(1). \end{equation} \]

   (55)

• a)

  Let \( U\) and \( X\hookrightarrow Y\) be Banach spaces and let \( V\subset U\) be open. Suppose \( G:V\rightarrow X\) is twice Gateux-differentiable as a map to \( Y\) . If its second derivative is (locally) bounded as a map from \( U^2\) to \( X\) , then \( G\) is (locally) continuously Fréchet differentiable.
• b)

  Let \( \Omega'\subset \mathbb{R}^d\) be a smooth and bounded domain. Let \( j_1,j_2,j_3,l\in\mathbb{N}_0\) be given with \( j_1=j_2(1+d+\dots+d^l)\) and let \( F:\mathbb{R}^{j_3+j_1}\rightarrow \mathbb{R}^m\) be a smooth vector field. If \( s>l+\frac{d}{2}\) , then the map

  \[ \begin{align} (t,u)\rightarrow F(t,u, \nabla u, \dots ,\nabla^l u) \end{align} \]

  (72)

  is Fréchet smooth from \( \mathbb{R}^{j_3}\times H^s(\Omega';\mathbb{R}^{j_2})\) to \( H^{s-l}(\Omega';\mathbb{R}^m)\) and the Fréchet derivatives agree with the pointwise derivatives.

The function \( (\theta,\epsilon)\mapsto h_{\theta,\epsilon}\in H^{k-2}(\partial B)\) is well-defined on a small open neighborhood of \( (\theta,\epsilon)=(0,0)\in H^{k}(\partial B)\times \mathbb{R}\) and Fréchet smooth in the joint variable. It holds that

The functions \( \theta\mapsto n_\theta\circ\chi_\theta\in H^{k-1}(\partial B)\) and \( \theta\mapsto m_\theta\in H^{k-1}(\partial B)\) are both Fréchet smooth near \( \theta=0\in H^{k}(\partial B)\) , and it holds that

where \( \tau_0=y^{\perp}\) is the tangent on \( \partial B\) .

The manifold \( V^k\) is smooth in a neighborhood of \( \theta=0\in H^k(\partial B)\) . In particular \( \mathcal{M}\) is smooth around \( (\theta,\epsilon)=(0,0)\in H^k(\partial B)\times \mathbb{R}\) with a \( C^{1,\frac{1}{\ell}-1}\) -boundary. The tangent space of \( V^k\) at \( \theta=0\) is given through

For \( \theta\in V^k\) and \( \delta\theta\in T_\theta V^k\) it holds that

The mapping \( (\theta,\epsilon)\mapsto \lambda_{\theta,\epsilon}\) is Fréchet smooth from \( H^k(\partial B)\) to \( H^{k-1}(\partial B)\) near \( (\theta,\epsilon)=(0,0)\) . It holds that \( \lambda_{0,0}=-2\) and

where \( \mathcal{L}\) is the Dirichlet-to-Neumann operator associated with the Laplacian on the unit ball.

The Dirichlet-to-Neumann operator \( \mathcal{L}\) is defined on \( H^{\frac{1}{2}}(\partial B)\) by taking some boundary datum \( g\in H^{\frac{1}{2}}(\partial B)\) , solving the equation

and setting \( \mathcal{L} g=\partial_n f|_{\partial B}\) . On the Fourier level, it takes a particularly simple form. Indeed, Fourier transforming the Laplace equation in angular direction, we obtain for any wave number \( l\in \mathbb{Z}\) and radius \( s\in(0,1)\) that

This ODE has the solution \( \hat f_l(s) = s^{|l|}\hat g_l\) , and thus

From this Fourier representation, we draw three immediate conclusions:

1. The Dirichlet-to-Neumann operator is bounded from \( H^s(\partial B)\) to \( H^{s-1}(\partial B)\) .

2. The operator maps constants to zero.

3. The operator maps \( y_1\) to \( y_1\) .

The last observation implies that the derivative with respect to \( \theta\) in the previous lemma is vanishing on \( x_1\) . This degeneracy reflects the translation invariance of the limiting elliptic problem in (86). Moreover, the above observations yield

The function \( (\epsilon,\theta)\mapsto (\partial_n\varphi_{\text{in}})\circ \chi_{\theta}\) is well-defined and Fréchet-smooth on a small open neighborhood of \( (\epsilon,\theta)=(0,0)\in \mathbb{R}\times H^{k}(\partial B)\) and takes values in \( H^{k-1}(\partial B)\) . It holds that

for any \( \delta \theta\in H^k\) .

This function has an expansion

for positive \( s\) near \( 0\) , where \( f_1\) and \( f_2\) are smooth functions with \( f_1(0)=f_2(0)=0\) .

The convolution kernel \( \log|x-y|\) acts on \( L^2(\partial B)\) as the Fourier multiplier \( e^{ik\alpha}\rightarrow \frac{-\pi}{|k|}e^{ik\alpha}\) for \( k\in \mathbb{Z}_{\neq 0}\) and maps \( 1\) to \( 0\) , where we used the identification with \( \mathbb{C}\) . In particular, for \( y\in \partial B\) it holds that

and the associated linear map maps \( H^{k-1}(\partial B)\) to \( H^{k}(\partial B)\) boundedly.

For any \( y,\tilde y\in \partial B\) , it holds that

Let \( k\ge 5\) and \( \delta\in(0,1)\) be given, let \( d \geq 1\) be an integer and let

be smooth. Assume that \( \tilde\theta\in H^k(\partial B,\mathbb{R}^d)\) is such that \( \left\lVert \tilde\theta \right\rVert_{H^k(\partial B, \mathbb{R}^d)}\leq \delta'\) with \( \delta'\ll\delta\) . Consider the kernels

and denote by \( \mathcal{R}_1\) and \( \mathcal{R}_2\) the associated linear maps given by

These maps are bounded from \( H^{k-1}(\partial B)\) to \( H^k(\partial B)\) , and it holds

for any \( f\in H^{k-1}(\partial B)\) , uniformly in \( r_{1/2}\) .

Let \( \mathcal{R}_1\) and \( \mathcal{R}_2\) be as above, then \( (\tilde\theta,\epsilon)\rightarrow \mathcal{R}_1\) and \( (\tilde\theta,\epsilon) \rightarrow \mathcal{R}_2\) are Fréchet smooth as maps from \( H^{k}(\partial B)\times \mathbb{R}\) to \( C(H^{k-1}(\partial B),H^k(\partial B))\) , locally around \( 0\) and its Fréchet derivatives are given through the pointwise derivatives of the kernel.

The linear map \( \widetilde{\mathcal{K}}_{\theta,0}: H^{k-1}(\partial B)\to H^k(\partial B)\) is Fréchet smooth in \( \theta\in H^k(\partial B)\) , provided that \( \|\theta\|_{H^k}\) is sufficiently small. Moreover for all \( f\in H^{k-1}(\partial B)\) , it holds that

For any \( g\in H^k(\partial B)\) , the problem

has a unique solution \( f\in H^{k-1}(\partial B)\) . Moreover, it holds that

and

For any sufficiently small \( \theta\in H^k(\partial B)\) and any \( g\in H^k(\partial B)\) , the problem

has a unique solution \( f\in H^{k-1}(\partial B)\) . It satisfies the estimate

Moreover, for any fixed \( g\in H^k(\partial B)\) , the map \( \theta\rightarrow f\) is Fréchet-smooth near \( \theta=0\) as a function from \( H^k(\partial B)\) to \( H^{k-1}(\partial B)\) .

The function \( \tilde{\mu}_{\theta,0}\) is well-defined, at least for small enough \( \theta\) , and it is Fréchet smooth in \( \theta\) as a map from \( H^{k}(\partial B)\) to \( H^{k-1}(\partial B)\) . It holds \( \tilde \mu_{0,0}=\frac1{2\pi}\) , and its first derivative at \( \theta=0\) is

provided that \( \delta\theta\in H^k(\partial B)\) is a mean-zero function.

Written as a Fourier multiplier, this derivative at \( 0\) is

by Lemma 5. In particular, the derivative is orthogonal to \( x_1\) , which reflects the translation invariance of the problem.

We have that \( \widetilde{\mathcal{K}}_{\theta,0}\tilde{\mu}_{\theta,0}\) is smooth in \( \theta\in H^k\) in a neighborhood of \( 0\) and it holds that

for all small enough \( \theta\in V^k\) .

The kernel \( K_{\theta,\epsilon}\) enjoys the asymptotic expansion

and the linear operator \( \mathcal{R_{\theta,\epsilon}}\) , associated to the remainder \( R_{\theta,\epsilon}\) is an integral kernel mapping \( H^{k-1}(\partial B)\) to \( H^{k}(\partial B)\) with estimate

if \( \left\lVert \theta \right\rVert_{H^k}\) and \( \epsilon\) are small enough.

Moreover \( \mathcal{R_{\theta,\epsilon}}\) is continuously Fréchet differentiable in \( (\theta,\epsilon)\) (away from \( 0\) ) as an operator from \( H^{k-1}(\partial B)\) to \( H^k(\partial B)\) with estimates

if \( \left\lVert \theta \right\rVert_{H^k}\) and \( \epsilon\) are small enough.

The map \( \mathcal{K}_{\theta,\epsilon}\) is bounded from \( H^{k-1}(\partial B)\) to \( H^k(\partial B)\) with estimate

for small enough \( \theta\) and \( \epsilon\) and it is continuously Fréchet differentiable in \( (\theta,\epsilon)\) in these spaces away from \( 0\) with estimate

The operator \( \mathcal{K}_{\theta,\epsilon}\) is invertible modulo constants for \( (\|\theta\|_{H^k},\epsilon)\in \mathcal{M}\) sufficiently small, in the sense that for any \( g\in H^k(\partial B)\) and every \( c\in \mathbb{R} \) , there exists a unique function \( f\in H^{k-1}(\partial B)\) such that

Moreover, there holds the continuity estimate

The unknown constant in the first equation in (143) is at most of the order \( |\log\epsilon|(\left\lVert g \right\rVert_{H^k}+|c|)\) .

Furthermore, for a fixed \( c\) and any smooth family \( g=g_{\theta,\epsilon}\in H^k(\partial B)\) , the function \( f=f_{\theta,\epsilon}\) is continuously Fréchet differentable in \( (\theta,\epsilon)\) for small \( (\theta,\epsilon)\neq 0\) and it holds that

Let \( (\theta,\epsilon)\in \mathcal{M}\) be small. Then, for any \( S\in \mathbb{R}\) , there exists a constant \( W\in \mathbb{R} \) such that

where \( \widetilde{\mathcal{E}}_{\theta,\epsilon}(S)\) is a smooth error term satisfying

This construction does indeed yield a well-defined \( \mu_{\theta,\epsilon}(S)\in H^k(\partial B)\) for \( (\theta,\epsilon)\in \mathcal{M}\) sufficiently small. The smallness condition is uniform in \( S\) .

Moreover, \( \mu =\mu_{\theta,\epsilon}(S)\) is continuously Fréchet differentiable in the joint variable \( (\theta,\epsilon,S)\in \mathcal{M}\times \mathbb{R}\) in a neighborhood of \( (0,0)\times \mathbb{R}\) , and it has the same derivative in \( \theta\) at \( (\theta,\epsilon)=(0,0)\) as \( \tilde{\mu}_{\theta,0}\) .

Finally, the error functional satisfies the following estimates

We have the following estimates for small enough \( (\theta,\epsilon)\in \mathcal{M}\) :

Here, all derivatives are Fréchet derivatives.

It holds that

Suppose \( (\epsilon,\theta)\in \mathcal{M}\) are small and let \( \tilde \gamma\) and \( \tilde W\) be given such that there is some \( \tilde \mu\in H^{k-1}\) with

then there is a unique \( \tilde S\) such that \( \tilde\mu=\mu_{\theta,\epsilon}(\tilde S)\) . Furthermore \( |\tilde S|\lesssim \tilde W+|\log\epsilon|\) .