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Vanishing viscosity limit for axisymmetric vortex rings

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Abstract

For the incompressible Navier-Stokes equations in \(\mathbb{R}^{3}\) with low viscosity \(\nu >0\), we consider the Cauchy problem with initial vorticity \(\omega _{0}\) that represents an infinitely thin vortex filament of arbitrary given strength \(\Gamma \) supported on a circle. The vorticity field \(\omega (x,t)\) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness \(\sqrt{\nu t}\) that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that \(\omega (x,t)\) is well-approximated on a large time interval by \(\omega _{\mathrm {lin}}(x-a(t),t)\), where \(\omega _{\mathrm {lin}}(\cdot ,t)=\exp (\nu t\Delta )\omega _{0}\) is the solution of the heat equation with initial data \(\omega _{0}\), and \(\dot{a}(t)\) is the instantaneous velocity given by Kelvin’s formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold’s geometric stability methods developed in the inviscid case \(\nu =0\) to the slightly viscous situation. It turns out that although the geometric structures behind Arnold’s approach are no longer preserved by the equation for \(\nu > 0\), the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold’s theory and interact favorably with the viscous terms.

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Notes

  1. Here the term current can be understood in its heuristic meaning but also in the technical meaning of the geometric measure theory, such as in [25].

  2. Fraenkel’s paper [27] contains formulae that can be used to obtain the same result. Tung and Ting in [57] also give a formula for \(C\) of a similar nature, which however needs a small correction.

  3. In the related situation of interacting vortices in \(\mathbb{R}^{2}\), this was already observed in [32].

  4. It is well-known that this is no longer the case in dimension four and higher [40].

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Acknowledgements

ThG is partially supported by the grants SingFlows ANR-18-CE40-0027 and Bourgeons ANR-23-CE40-0014-01 of the French National Research Agency (ANR). The research of VS has been supported in part by grants DMS 1956092 and DMS 2247027 from the National Science Foundation.

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Appendices

Appendix A: Appendix to Sect. 3

1.1 A.1 Inverting the operator \(\Lambda \)

Following [32], we give here a short proof of Proposition 3.8. Assume that \(n \ge 2\) and \(f \in \mathcal {Y}_{n} \cap \mathcal {Z}\), or that \(n = 1\) and \(f \in \mathcal {Y}'_{1} \cap \mathcal {Z}\). In both cases, we have \(f \in \mathop {\mathrm {Ker}}(\Lambda )^{\perp}\). We want to show that there exists a unique \(\eta \in \mathcal {Y}_{n} \cap \mathcal {Z}\) (respectively, \(\eta \in \mathcal {Y}_{1}' \cap \mathcal {Z}\) if \(n = 1\)) such that \(\Lambda \eta = f\).

To make things concrete, we suppose without loss of generality that \(f = a(\rho )\sin (n\vartheta )\), for some function \(a : \mathbb{R}_{+} \to \mathbb{R}\). Our hypotheses imply that \(a\) is smooth, that \(a(\rho ) = \mathcal {O}(\rho ^{n})\) as \(\rho \to 0\), and that \(e^{\rho ^{2}/4}a(\rho )\) grows at most polynomially as \(\rho \to \infty \). We look for a solution of the form \(\eta = \omega (\rho ) \cos (n\vartheta )\), where \(\omega : \mathbb{R}_{+} \to \mathbb{R}\) has to be determined. By (3.19), we have

$$ \Lambda \eta \,=\, \bigl\{ \phi _{0}\,,\eta \bigr\} + \{\Psi \,,\eta _{0} \bigr\} \,, \qquad \text{where}\quad \phi _{0} \,=\, \frac{1}{2\pi}\,L \eta _{0}\,, \quad \Psi \,=\, \frac{1}{2\pi}\,L\eta \,. $$
(A.1)

The function \(\phi _{0}\) is radially symmetric and satisfies \(\partial _{\rho }\phi _{0} = -\rho \varphi (\rho )\), see (3.24) and (A.12) below. It follows that

$$ \bigl\{ \phi _{0}\,,\eta \bigr\} \,=\, \partial _{\rho }\phi _{0}\, \frac{1}{\rho} \,\partial _{\vartheta }\eta \,=\, n \varphi (\rho )\omega ( \rho )\sin (n\vartheta )\,. $$
(A.2)

On the other hand, as \(-\Delta \Psi = \eta \), we have \(\Psi = \Omega (\rho ) \cos (n\vartheta )\), where \(\Omega \) is the unique regular solution of the differential equation

$$ -\Omega ''(\rho ) -\frac{1}{\rho}\,\Omega '(\rho ) + \frac{n^{2}}{\rho ^{2}}\, \Omega (\rho ) \,=\, \omega (\rho )~, \qquad \rho > 0~. $$
(A.3)

Since \(\eta _{0}\) is radially symmetric and \(\partial _{\rho }\eta _{0} = -(\rho /2)\eta _{0} = -\rho \varphi (\rho ) h( \rho )\), see (3.24), we deduce

$$ \{\Psi \,,\eta _{0}\bigr\} \,=\, -\partial _{\rho }\eta _{0}\, \frac{1}{\rho} \,\partial _{\vartheta }\Psi \,=\, -n\varphi (\rho )h(\rho ) \Omega (\rho )\sin (n\vartheta )\,. $$
(A.4)

In view of (A.1), (A.2), (A.4), the equation \(\Lambda \eta = f\) is equivalent to the relation (3.25), and using in addition (A.3) we obtain the differential equation (3.26) for the stream function \(\Omega \).

The main step in the proof is to show that (3.26) has a unique solution that is regular at the origin and decays to zero at infinity. Here we distinguish two cases according to the value of the angular Fourier mode \(n\).

1. If \(n \ge 2\), the homogeneous equation (3.26) with \(a \equiv 0\) has two linearly independent solutions \(\psi _{+}\), \(\psi _{-}\) which satisfy

$$ \psi _{-}(\rho ) \,\sim \, \textstyle\begin{cases} \rho ^{n} \hspace{6mm} \text{as }\rho \to 0\,, \\ \kappa \rho ^{n} \quad \text{as }\rho \to \infty \,,\end{cases}\displaystyle \qquad \psi _{+}(\rho ) \,\sim \, \textstyle\begin{cases} \kappa \rho ^{-n} \quad \text{as }\rho \to 0\,, \\ \rho ^{-n} \hspace{6mm} \text{as }\rho \to \infty \,,\end{cases} $$
(A.5)

for some \(\kappa > 0\), see [32]. Here we use the crucial observation that \((n^{2}/\rho ^{2}) - h(\rho ) > 0\) when \(n \ge 2\), so that the differential operator in the left-hand side of (3.26) satisfies the Maximum Principle. We deduce the following representation formula for the solution of the inhomogeneous equation

$$ \Omega (\rho ) \,=\, \psi _{+}(\rho )\int _{0}^{\rho }\frac{r}{w_{0}} \,\psi _{-}(r) \frac{a(r)}{n\varphi (r)}\,{\mathrm { d}}r + \psi _{-}(\rho )\int _{ \rho}^{\infty }\frac{r}{w_{0}} \,\psi _{+}(r)\frac{a(r)}{n\varphi (r)} \,{\mathrm { d}}r\,, $$
(A.6)

where \(w_{0} = 2n\kappa \). It is then straightforward to verify that \(\Omega (\rho ) = \mathcal {O}(\rho ^{n})\) as \(\rho \to 0\) and \(\Omega (\rho ) = \mathcal {O}(\rho ^{-n})\) as \(\rho \to \infty \). Moreover, if \(\omega \) is defined by (3.25), the function \(\eta = \omega (\rho )\cos (n\vartheta )\) lies in \(\mathcal {Y}_{n} \cap \mathcal {Z}\) and satisfies \(\Lambda \eta = f\) by construction. The details can be found in [32, Lemma 4].

2. The situation is quite different when \(n = 1\), because the lower order term \(1/\rho ^{2} - h(\rho )\) in (3.26) is no longer positive. In that case, it happens that the homogeneous equation (3.26) with \(a \equiv 0\) has a solution \(\psi (\rho ) = \rho \varphi (\rho )\) which satisfies \(\psi (\rho ) \sim \rho /(8\pi )\) as \(\rho \to 0\) and \(\psi (\rho ) \sim 1/(2\pi \rho )\) as \(\rho \to \infty \). In other words, the linear operator in the left-hand side of (3.26) has a one-dimensional kernel, and for that reason we have to impose the solvability condition

$$ f \,\in \, \mathcal {Y}_{1}' \,\subset \, \mathop {\mathrm {Ker}}(\Lambda )^{\perp}\,, \quad \text{or equivalently} \quad \int _{0}^{\infty }a(\rho )\rho ^{2}\,{\mathrm { d}}\rho \,=\, 0\,. $$
(A.7)

To solve (3.26) for \(n = 1\), we look for a solution of the form \(\Omega (\rho ) = b(\rho )\psi (\rho )\), which leads to a first-order differential equation for \(b(\rho )\). In view of (A.7), we thus find

$$ b'(\rho ) \,=\, -\frac{1}{\rho \psi (\rho )^{2}}\int _{0}^{\rho }a(r)r^{2} \,{\mathrm { d}}r \,=\, \frac{1}{\rho \psi (\rho )^{2}}\int _{\rho}^{\infty }a(r)r^{2} \,{\mathrm { d}}r\,. $$
(A.8)

Integrating (A.8) gives the representation formula

$$ b(\rho ) \,=\, b_{0} - \int _{0}^{\rho }a(r) r^{2} \Bigl(\mathcal {F}(\rho ) - \mathcal {F}(r)\Bigr)\,{\mathrm { d}}r\,, \qquad \text{for some } b_{0} \in \mathbb{R}\,, $$

where

$$ \mathcal {F}(\rho ) \,=\, 8\pi ^{2} \left (\log \bigl(e^{\rho ^{2}/4}-1\bigr) - \frac{1}{ e^{\rho ^{2}/4}-1}\right )\,, \qquad \mathcal {F}'(\rho ) \,=\, \frac{1}{\rho \psi (\rho )^{2}}\,. $$

We now substitute \(\Omega (\rho ) = b(\rho )\psi (\rho )\) into (3.25) with \(n = 1\), and we choose the constant \(b_{0}\) so that \(\int _{0}^{\infty }\omega (\rho ) \rho ^{2} \,{\mathrm { d}}\rho = 0\). This is always possible in a unique way, since

$$ \int _{0}^{\infty }h(\rho ) \psi (\rho ) \rho ^{2} \,{\mathrm { d}}\rho \,=\, \int _{0}^{\infty }h(\rho ) \varphi (\rho ) \rho ^{3} \,{\mathrm { d}}\rho \,=\, \frac{1}{8\pi}\int _{0}^{\infty }e^{-\rho ^{2}/4}\rho ^{3}\,{\mathrm { d}}\rho \,= \, \frac{1}{\pi} \,\neq \, 0\,. $$

To conclude the proof, it remains to verify that the function \(\eta = \omega (\rho ) \cos (\vartheta )\) constructed above belongs to \(\mathcal {Y}_{1}' \cap \mathcal {Z}\) and satisfies \(\Lambda \eta = f\). These are straightforward calculations, which can be omitted. \(\Box \)

1.2 A.2 First order calculations

We first establish the relations (3.37). As \(\eta _{0} \in \mathcal {Y}_{0}\) has unit mass we find, using (3.11),

$$ \bigl(P_{1} \eta _{0}\bigr)(R,Z) \,=\, \int _{\mathbb{R}^{2}} \frac{R{+}R'}{2}\,\eta _{0}(R',Z') \,{\mathrm { d}}R' \,{\mathrm { d}}Z' \,=\, \frac{R}{2}\,, $$
(A.9)

hence \(\{P_{1} \eta _{0}\,,\eta _{0}\} = \frac{1}{2}\, \partial _{Z} \eta _{0}\). On the other hand, since \(\partial _{R} \eta _{0} = -(R/2)\eta _{0}\) and \(L\) is a convolution operator, which therefore commutes with derivatives, we have

$$\begin{aligned} \bigl(L P_{1} \eta _{0}\bigr)(R,Z) &=\, \frac{R}{2}\,(L\eta _{0})(R,Z) + L\Bigl(\frac{R}{2}\eta _{0}\Bigr)(R,Z) \\ &=\, \frac{R}{2}\,(L\eta _{0})(R,Z) - \partial _{R}\bigl(L\eta _{0}\bigr)(R,Z)\,. \end{aligned}$$

Recalling that \(L\eta _{0} = 2\pi \phi _{0}\), and that \(\{\phi _{0},\eta _{0}\} = 0\) because both \(\phi _{0}\), \(\eta _{0}\) are radially symmetric, we thus obtain

$$\begin{aligned} \frac{1}{2\pi}\bigl\{ L P_{1} \eta _{0}\,,\eta _{0}\bigr\} \,&=\, \Bigl\{ \frac{R}{2}\,\phi _{0}- \partial _{R}\phi _{0}\,,\eta _{0} \Bigr\} \,=\, \frac{1}{2}\,\phi _{0}\,\partial _{Z} \eta _{0} + \bigl\{ \phi _{0}\,, \partial _{R} \eta _{0} \bigr\} \\ \,&=\, \frac{1}{2}\,\phi _{0}\,\partial _{Z} \eta _{0} - \Bigl\{ \phi _{0} \,, \frac{R}{2}\,\eta _{0} \Bigr\} \,=\, \frac{1}{2}\,\phi _{0}\, \partial _{Z} \eta _{0} + \frac{1}{2}\,(\partial _{Z}\phi _{0})\eta _{0} \,, \end{aligned}$$

which concludes the proof of (3.37).

We next prove formula (3.39) for the vertical velocity. Assuming that \(\dot{\bar{z}}_{0}\) is given by (3.39) for some \(v \in \mathbb{R}\), we see that the right-hand side of (3.38) belongs to \(\mathcal {Y}_{1}' = \mathcal {Y}\cap \mathop {\mathrm {Ker}}(\Lambda )^{\perp}\) if and only if

$$ \int _{\mathbb{R}^{2}} \Bigl(\frac{v}{2\pi}\,\partial _{Z} \eta _{0} - \frac{3}{2} (\partial _{Z} \phi _{0}) \eta _{0} -\frac{1}{2}\phi _{0} \partial _{Z} \eta _{0}\Bigr) Z\, \,{\mathrm { d}}R \,{\mathrm { d}}Z \,=\, 0\,. $$
(A.10)

Since \(\partial _{Z} \eta _{0} = -(Z/2)\eta _{0}\) and \(\int _{\mathbb{R}^{2}} Z^{2} \eta _{0} \,{\mathrm { d}}R\,{\mathrm { d}}Z = 2\), it is straightforward to verify that (A.10) is equivalent to

$$ v \,=\, \pi \int _{\mathbb{R}^{2}} \phi _{0} \eta _{0} \bigl(3 - Z^{2}\bigr) \,{\mathrm { d}}R \,{\mathrm { d}}Z \,=\, \frac{\pi}{2}\int _{\mathbb{R}^{2}} \phi _{0}\eta _{0} \bigl(6 - |X|^{2}\bigr)\,{\mathrm { d}}X\,, $$
(A.11)

where \(X = (R,Z)\) and \(|X|^{2} = R^{2} + Z^{2}\).

To evaluate the right-hand side of (A.11), we temporarily denote \(\psi _{0} = 2\pi \phi _{0} = L\eta _{0}\), namely

$$ \psi _{0}(X) \,=\, \frac{1}{4\pi} \int _{\mathbb{R}^{2}} \log \Bigl( \frac{8}{|X-Y|}\Bigr) \,e^{-|Y|^{2}/4}\,{\mathrm { d}}Y\,, \qquad X \in \mathbb{R}^{2}\,. $$

This function satisfies \(-\Delta \psi _{0} = 2\pi \eta _{0} = \frac{1}{2}\,e^{-|X|^{2}/4}\), so that

$$ \psi _{0}(X) \,=\, \psi _{0}(0) \,-\, \int _{0}^{|X|} \frac{1 - e^{-\rho ^{2}/4}}{\rho} \,{\mathrm { d}}\rho \,=:\, \tilde{\psi}_{0}(|X|) \,, \qquad X \in \mathbb{R}^{2}\,, $$
(A.12)

where

$$ \psi _{0}(0) \,=\, \log (8) - \frac{1}{4\pi} \int _{\mathbb{R}^{2}} \log (|Y|) \,e^{-|Y|^{2}/4}\,{\mathrm { d}}Y \,=\, 2\log (2) + \frac{\gamma _{E}}{2}\,. $$
(A.13)

Using (A.12), (A.13) and integrating by parts, we easily find

$$\begin{aligned} \int _{\mathbb{R}^{2}} \psi _{0} \eta _{0} \,{\mathrm { d}}X &=\, \frac{1}{2} \int _{0}^{ \infty }\tilde{\psi}_{0}(\rho ) e^{-\rho ^{2}/4} \rho \,{\mathrm { d}}\rho \\ &=\, \psi _{0}(0) + \int _{0}^{\infty }\tilde{\psi}_{0}'(\rho ) e^{-\rho ^{2}/4} \,{\mathrm { d}}\rho \,=\, \frac{3}{2}\log (2) + \frac{\gamma _{E}}{2}\,, \end{aligned}$$

and similarly

$$ \int _{\mathbb{R}^{2}} \psi _{0} \eta _{0} |X|^{2} \,{\mathrm { d}}X \,=\, 4\psi _{0}(0) + \int _{0}^{\infty }\tilde{\psi}_{0}'(\rho ) e^{-\rho ^{2}/4}(\rho ^{2} + 4)\,{\mathrm { d}}\rho \,=\, 6\log (2) + 2\gamma _{E} -1\,. $$

Returning to (A.11), we conclude that

$$ v \,=\, \frac{1}{4}\int _{\mathbb{R}^{2}} \psi _{0}\eta _{0} \bigl(6 - |X|^{2} \bigr)\,{\mathrm { d}}X \,=\, \frac{3}{4}\log (2) + \frac{1}{4}\gamma _{E} + \frac{1}{4}\,. $$
(A.14)

1.3 A.3 Second order calculations

Our goal here is to prove Lemma 3.12. To establish (3.48), we consider separately the various terms in (3.47). As \(\eta _{1} \in \mathcal {Y}_{1}\) has zero mean, we find as in (A.9) that \(P_{1} \eta _{1}\) is a constant, which can be disregarded. Moreover \(L P_{1} \eta _{1} = \frac{R}{2}L\eta _{1} + L\bigl(\frac{R}{2}\eta _{1} \bigr)\), hence using the expression (3.41) of \(\eta _{1}\) we find that

$$ L P_{1} \eta _{1} \,=\, (R^{2}-Z^{2}) \chi _{1}(\rho ) + \delta RZ \chi _{2}(\rho ) + \chi _{3}(\rho )\,, $$

where \(\chi _{1}, \chi _{2}, \dots \) are functions of the radial variable \(\rho = (R^{2} + Z^{2})^{1/2}\). As \(\eta _{0}\) itself is radially symmetric, we deduce that

$$ \bigl\{ (\beta _{\epsilon }- 1)P_{1} \eta _{1} + LP_{1} \eta _{1}\,, \eta _{0}\bigr\} \,=\, RZ \chi _{4}(\rho ) + \delta (R^{2}-Z^{2}) \chi _{5}(\rho )\,. $$
(A.15)

Next, using the expression (3.11) of \(P_{2}\), we see that

$$\begin{aligned} \bigl(P_{2} \eta _{0}\bigr)(R,Z) &=\, \frac{1}{16}\int _{\mathbb{R}^{2}} \Bigl( (R{-}R')^{2} + 3 (Z{-}Z')^{2}\Bigr)\,\eta _{0}(R',Z')\,{\mathrm { d}}R' \,{\mathrm { d}}Z' \\ &=\, \frac{R^{2}}{16} + \frac{3 Z^{2}}{16} + \frac{1}{2}\,, \end{aligned}$$

and a similar calculation gives \(Q_{2} \eta _{0} = \frac{3R^{2}}{16} - \frac{Z^{2}}{16} + \frac{1}{4}\). Moreover,

$$\begin{aligned} &\bigl(L P_{2} \eta _{0}\bigr)(R,Z) \\ &\quad=\, \frac{1}{16} \int _{\mathbb{R}^{2}} \log \Bigl(\frac{8}{D}\Bigr)\Bigl(2D^{2} + (Z{-}Z')^{2} - (R{-}R')^{2} \Bigr) \,\eta _{0}(R',Z')\,{\mathrm { d}}R' \,{\mathrm { d}}Z'\,, \end{aligned}$$

where \(D^{2} = (R{-}R')^{2} + (Z{-}Z')^{2}\). Using the fact that \(\eta _{0}\) given by (3.32) is radially symmetric, we easily obtain

$$ \frac{1}{2\pi}\bigl(L P_{2} \eta _{0}\bigr)(R,Z) \,=\, \chi _{6}( \rho ) + (R^{2} - Z^{2})\chi _{7}(\rho )\,. $$

Altogether, we arrive at

$$ \frac{1}{2\pi}\bigl\{ \beta _{\epsilon }P_{2}\eta _{0} + LP_{2} \eta _{0} + Q_{2} \eta _{0}\,, \eta _{0}\bigr\} \,=\, \frac{\beta _{\epsilon}}{16\pi}\, RZ \eta _{0} + RZ \chi _{8}(\rho ) \,. $$
(A.16)

The remaining terms in (3.47) are easier to treat. In view of (3.39), (3.41), (3.42), we have

$$\begin{aligned} &\bigl\{ \phi _{1}\,,\eta _{1}\bigr\} - \frac{r_{0} \dot{\bar{z}}_{0}}{\Gamma} \,\partial _{Z} \eta _{1} \,=\, \Bigl\{ \phi _{1} - \frac{\beta _{\epsilon }-1}{4\pi}\,R \,,\eta _{1} \Bigr\} - \frac{v}{2\pi}\,\partial _{Z} \eta _{1} \\ &\quad=\, \Bigl\{ \frac{R}{2}\,\phi _{0} - \partial _{R} \phi _{0} + R\, \phi _{10}(\rho ) + \delta Z\,\phi _{11}(\rho )\,, R\,\eta _{10}( \rho ) + \delta Z\,\eta _{11}(\rho )\Bigr\} - \frac{v}{2\pi}\, \partial _{Z} \eta _{1} \\ &\quad=\, RZ\,\chi _{9}(\rho ) + \delta \Bigl(\chi _{10}(\rho ) + (R^{2}-Z^{2}) \chi _{11}(\rho )\Bigr) + \delta ^{2}RZ\,\chi _{12}(\rho )\,. \end{aligned}$$
(A.17)

It is also easy to verify that the terms \((\partial _{Z} \phi _{1})\eta _{0} + (\partial _{Z} \phi _{0})\eta _{1} - 2R(\partial _{Z}\phi _{0})\eta _{0} + \delta \partial _{R}(R\eta _{0})\) are exactly of the same form. Finally, using again (3.41), (3.42), we obtain

$$ R\Bigl(\bigl\{ \phi _{1}\,,\eta _{0}\bigr\} + \bigl\{ \phi _{0}\,,\eta _{1} \bigr\} \Bigr) \,=\, R\Bigl(\frac{\beta _{\epsilon}-1}{4\pi}\, \partial _{Z} \eta _{0} + Z \chi _{13}(\rho ) + \delta R \chi _{14}( \rho )\Bigr)\,. $$
(A.18)

If we now combine (A.15), (A.16), (A.17), (A.18), we arrive at (3.48). \(\Box \)

1.4 A.4 Higher order calculations

The calculations carried out in Sects. 3.5 and 3.6 do not require new ideas, but a more compact notation is often helpful. To prove Lemma 3.13 and similar statements, it is important to understand how the decomposition (3.21) of the function space \(\mathcal {Y}\) behaves under the Poisson bracket. If we use polar coordinates \(R = \rho \cos \vartheta \), \(Z = \rho \sin \vartheta \), we recall that \(\mathcal {Y}_{n}\) is the subspace of \(\mathcal {Y}\) spanned by functions of the form \(a(\rho )\cos (n\vartheta )\) and \(b(\rho ) \sin (n\vartheta )\). Since

$$ \bigl\{ f\,,g\bigr\} \,=\, \partial _{R} f \partial _{Z} g - \partial _{Z} f \partial _{R} g \,=\, \frac{1}{\rho}\Bigl(\partial _{\rho }f \partial _{\vartheta }g - \partial _{\vartheta }f \partial _{\rho }g\Bigr)\,, $$

we easily obtain the following result

Lemma A.1

If \(a,b : \mathbb{R}_{+} \to \mathbb{R}\) are smooth functions and \(n,m \in \mathbb{N}\), then

$$\begin{aligned} \bigl\{ a(\rho )\cos (n\vartheta )\,,\,b(\rho )\cos (m\vartheta )\bigr\} \,&=\, c_{11}( \rho )\sin ((n{-}m)\vartheta ) + c_{12}(\rho )\sin ((n{+}m)\vartheta )\,, \\ \bigl\{ a(\rho )\sin (n\vartheta )\,,\,b(\rho )\sin (m\vartheta )\bigr\} \,&=\, c_{21}( \rho )\sin ((n{-}m)\vartheta ) + c_{22}(\rho )\sin ((n{+}m)\vartheta )\,, \\ \bigl\{ a(\rho )\sin (n\vartheta )\,,\,b(\rho )\cos (m\vartheta )\bigr\} \,&=\, c_{31}( \rho )\cos ((n{-}m)\vartheta ) + c_{32}(\rho )\cos ((n{+}m)\vartheta )\,, \end{aligned}$$

where \(c_{ij} : \mathbb{R}_{+} \to \mathbb{R}\) are smooth functions. In particular \(\{\mathcal {Y}_{n},\mathcal {Y}_{m}\} \subset \mathcal {Y}_{n-m} + \mathcal {Y}_{n+m}\) if \(m \le n\).

It is also necessary to compute the homogeneous polynomials \(P_{j}, Q_{j}\) in (3.10) for higher values of \(j\) than in Lemma 3.3. This is a cumbersome calculation that can be done for instance using computer algebra. For \(j = 3\) we find

$$ \begin{aligned} P_{3} \,&=\, -\frac{1}{32}(R+R')\Bigl((R-R')^{2} + 3 (Z-Z')^{2} \Bigr)\,, \\ Q_{3} \,&=\, -\frac{1}{48}(R+R')\Bigl((R+R')^{2} - 6 (Z-Z')^{2}\Bigr) \,, \end{aligned} $$
(A.19)

and the calculation for \(j = 4\) yields the more complicated expressions

$$ \begin{aligned} P_{4} \,=\, &-\frac{15}{1024}\,(Z{-}Z')^{4} + \frac{21}{512}\,(R{-}R')^{2} (Z{-}Z')^{2} + \frac{3}{16}\,R R'\,(Z{-}Z')^{2} \\ & + \frac{17}{1024}\, (R^{2}{-}R^{\prime \,2})^{2} - \frac{1}{256}\,R R'\,(R{-}R')^{2} \,, \\ Q_{4} \,=\, &\frac{31}{2048}\,(Z{-}Z')^{4} - \frac{89}{1024}\,(R{+}R')^{2}(Z{-}Z')^{2} + \frac{1}{256}\,R R'\,(Z{-}Z')^{2} \\ &- \frac{19}{6144}\,(R^{2}{-}R^{\prime \,2})^{2} + \frac{35}{1536}\,R R'(R{+}R')^{2} - \frac{1}{128}\,R^{2} R^{\prime \,2}\,. \end{aligned} $$
(A.20)

The proof of Lemma 3.13 is similar to that of Lemma 3.12, and the details can be omitted. We use the expressions (3.41), (3.50) of the vorticities \(\eta _{1}, \eta _{2}\), the formulas (3.42), (3.51) for the stream functions \(\phi _{1}, \phi _{2}\), and the definition (3.15) of the Biot-Savart operators, which involve the polynomials (3.11) and (A.19). Using Lemma A.1, it is straightforward to verify that the quantity defined in (3.55) satisfies \(\mathcal {R}_{3} \in \mathcal {Y}_{1} + \mathcal {Y}_{3}\) and takes the form

$$ \mathcal {R}_{3} \,=\, \chi _{1}(\rho )\sin (\vartheta ) + \chi _{2}(\rho )\sin (3 \vartheta ) + \delta \Bigl(\chi _{3}(\rho )\cos (\vartheta ) + \chi _{4}(\rho ) \cos (3\vartheta )\Bigr) + \mathcal {O}(\delta ^{2})\,, $$

where \(\chi _{1},\chi _{2},\chi _{3},\chi _{4}\) are radially symmetric functions which may depend linearly on \(\beta _{\epsilon}\). To arrive at (3.56), it remains to verify that \(\mathcal {R}_{3}\) does not contain any term involving \(\beta _{\epsilon}^{2}\). Indeed, according to (3.11), (3.50), we have

$$ \frac{\beta _{\epsilon}}{2\pi}\,P_{1} \eta _{2} \,=\, \frac{\beta _{\epsilon}}{4\pi} \int _{\mathbb{R}^{2}}(R+R')\,\eta _{2}(R',Z') \,{\mathrm { d}}R'\,{\mathrm { d}}Z' \,=\, \frac{\beta _{\epsilon }R}{4\pi} \int _{\mathbb{R}^{2}}\, \eta _{24}(R',Z')\,{\mathrm { d}}R'\,{\mathrm { d}}Z'\,, $$

so that the first term in (3.55) does not contain \(\beta _{\epsilon}^{2}\). The only other terms that we have to check are

$$ \bigl\{ \phi _{1}\,,\eta _{2}\bigr\} - \frac{r_{0}}{\Gamma}\, \dot{\bar{z}}_{0} \partial _{Z} \eta _{2} \,=\, \Bigl\{ \phi _{1} - \frac{\beta _{\epsilon }-1 + 2v}{4\pi}\,R \,,\eta _{2}\Bigr\} \,, $$

but using the expressions (3.42), (3.50) we immediately see that the right-hand side does not contain any factor \(\beta _{\epsilon}^{2}\). Altogether we arrive at (3.56). \(\Box \)

Appendix B: Appendix to Sect. 4

2.1 B.1 Properties of the energy functional

Proof of Lemma 4.4

We use the first expression of \(E_{\epsilon}^{\mathrm {kin}}[\eta ]\) in (4.22) and the representation formula (2.20) for the stream function \(\phi \). Since \(\mathop {\mathrm {supp}}(\eta ) \subset B_{\epsilon}\) by assumption, we have

$$ E_{\epsilon}^{\mathrm {kin}}[\eta ] \,=\, \frac{1}{4\pi} \int _{B_{\epsilon}} \int _{B_{\epsilon}} K_{\epsilon}(R,Z;R',Z')\,\eta (R,Z)\,\eta (R',Z') \,{\mathrm { d}}X\,{\mathrm { d}}X'\,, $$
(B.1)

where the integral kernel \(K_{\epsilon}\) is defined in (3.8). As \(R^{2}+Z^{2} \le \epsilon ^{-2\sigma _{1}}\) and \({R'}^{2}+{Z'}^{2} \le \epsilon ^{-2\sigma _{1}}\), the argument of \(F\) in (3.8) is not larger than \(C \epsilon ^{2-2\sigma _{1}}\) for some \(C > 0\). Using the asymptotic expansion of \(F(s)\) as \(s \to 0\) and proceeding as in Sect. 3.1, we easily obtain the decomposition

$$ K_{\epsilon}(R,Z;R',Z') \,=\, \beta _{\epsilon }- 2 + \log \frac{8}{D} + \tilde{K}_{\epsilon}(R,Z;R',Z')\,, $$
(B.2)

where \(\beta _{\epsilon }= \log (1/\epsilon )\) and \(D^{2} = (R{-}R')^{2} + (Z{-}Z')^{2}\). The remainder \(\tilde{K}_{\epsilon}\) satisfies the estimate

$$ |\tilde{K}_{\epsilon}(R,Z;R',Z')| \,\le \, C\epsilon \bigl(|R| + |R'| \bigr) \Bigl(\beta _{\epsilon }+ 1 + \log \frac{8}{D}\Bigr) + \mathcal {O}\bigl(\beta _{\epsilon }\epsilon ^{2-2\sigma _{1}}\bigr)\,. $$
(B.3)

If we insert the decomposition (B.2) into (B.1), the contributions of \(\beta _{\epsilon }- 2\) and \(\log (8/D)\) give exactly the first two terms in the right-hand side of (4.24), in view of (4.23). Moreover, taking into account estimate (B.3) where \(\epsilon ^{2-2\sigma _{1}} \le \epsilon \), we see that the contributions of \(\tilde{K}_{\epsilon}\) to the kinetic energy (B.1) are of order \(\mathcal {O}\bigl(\epsilon \beta _{\epsilon}\|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \bigr)\), as stated in (4.24). □

Proof of Proposition 4.6

Given \(\eta \in \mathcal {X}_{\epsilon}\), we decompose \(\eta = \eta _{1} + \eta _{2}\) where \(\eta _{1} = \eta \mathbf {1}_{B_{\epsilon}}\) and \(\mathbf {1}_{B_{\epsilon}}\) is the indicator function of the ball \(B_{\epsilon }= \{(R,Z) \in \Omega _{\epsilon}\,;\, R^{2} + Z^{2} \le \epsilon ^{-2\sigma _{1}}\}\). We thus have

$$ E_{\epsilon}[\eta ] \,=\, \frac{1}{2} \int _{\Omega _{\epsilon}} W_{ \epsilon}\,\eta _{1}^{2}\,{\mathrm { d}}X + \frac{1}{2} \int _{\Omega _{\epsilon}} W_{\epsilon}\,\eta _{2}^{2}\,{\mathrm { d}}X - \frac{1}{2} \int _{\Omega _{ \epsilon}} \bigl(\phi _{1} + \phi _{2}\bigr)\bigl(\eta _{1} + \eta _{2} \bigr)\,{\mathrm { d}}X\,, $$
(B.4)

where \(\phi _{j} = \mathrm {BS}^{\epsilon}[\eta _{j}]\) for \(j = 1,2\). We claim that

$$ \frac{1}{2} \int _{\Omega _{\epsilon}}\bigl(\phi _{1} + \phi _{2} \bigr)\bigl(\eta _{1} + \eta _{2} \bigr)\,{\mathrm { d}}X \,=\, E^{\mathrm {kin}}_{ \epsilon}[\eta _{1}] + \mathcal {O}\bigl(\epsilon ^{\infty }\|\eta \|_{\mathcal {X}_{ \epsilon}}^{2}\bigr)\,, $$
(B.5)

so that

$$ E_{\epsilon}[\eta ] \,=\, E_{\epsilon}[\eta _{1}] + \frac{1}{2} \| \eta _{2}\|_{\mathcal {X}_{\epsilon}}^{2} + \mathcal {O}\bigl(\epsilon ^{\infty}\| \eta \|_{\mathcal {X}_{\epsilon}}^{2}\bigr)\,. $$
(B.6)

To prove (B.5), we recall that \(\phi _{j}(R,Z) = \frac{1}{2\pi}\int _{\Omega _{\epsilon}} K_{ \epsilon}(R,Z;R',Z')\eta _{j}(R',Z')\,{\mathrm { d}}X'\), where the kernel \(K_{\epsilon}\) is given by (3.8). Using the crude estimate \(|F(s)| \le C\bigl(|\log s| + 1\bigr)\), we easily obtain

$$ \bigl|K_{\epsilon}(R,Z;R',Z')\bigr| \,\le \, C\bigl(1{+}\epsilon |R| \bigr)^{a} \bigl(1{+}\epsilon |R'|\bigr)^{a} \bigl(\beta _{\epsilon }+ \bigl|\log D\bigr| + 1\bigr)\,, $$
(B.7)

for some \(a > 1/2\). It follows in particular that

$$ |\phi (R,Z)| \,\le \, C\bigl(\beta _{\epsilon }+ 1\bigr)(1 + \rho )^{b} \|\eta \|_{\mathcal {X}_{\epsilon}}\,, \qquad \rho \,=\, \sqrt{R^{2}+Z^{2}}\,, $$

for some \(b > 1/2\), and using Hölder’s inequality we deduce

$$\begin{aligned} &\int _{\Omega _{\epsilon}} |\phi (R,Z)|\,|\eta _{2}(R,Z)|\,{\mathrm { d}}X \\ &\quad\le \, C\bigl(\beta _{\epsilon }+ 1\bigr) \|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \biggl(\int _{B_{\epsilon}^{c}}(1+\rho )^{2b} \,W_{\epsilon}(R,Z)^{-1} \,{\mathrm { d}}X\biggr)^{1/2}\,, \end{aligned}$$

where the last integral is \(\mathcal {O}(\epsilon ^{\infty})\) in view of (4.17). In a similar way we have

$$\begin{aligned} |\phi _{2}(R,Z)| &\le \, C\bigl(\beta _{\epsilon }+ 1\bigr)(1+\rho )^{b} \biggl(\int _{B_{\epsilon}^{c}}(1+\rho ')^{2b} |\eta (R',Z')|^{2}\,{\mathrm { d}}X' \biggr)^{1/2} \\&=\, \mathcal {O}\bigl(\epsilon ^{\infty}\|\eta \|_{\mathcal {X}_{ \epsilon}}\bigr)(1+\rho )^{b}\,, \end{aligned}$$

so that \(\int _{\Omega _{\epsilon}}\!\phi _{2} \eta _{1}\,{\mathrm { d}}x = \mathcal {O}\bigl( \epsilon ^{\infty }\|\eta \|_{\mathcal {X}_{\epsilon}}^{2}\bigr)\). Altogether we arrive at (B.5).

Now, since \(\eta _{1}\) is supported in the ball \(B_{\epsilon}\), it follows from (4.18) and Lemma 4.4 that

$$ \begin{aligned} &\|\eta _{1}\|_{\mathcal {X}_{\epsilon}}^{2} \,=\, \|\eta _{1}\|_{\mathcal {X}_{0}}^{2} + \mathcal {O}\bigl(\epsilon ^{\gamma _{1}} \|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \bigr)\,, \\ & E_{\epsilon}^{\mathrm {kin}}[\eta _{1}] \,=\, \frac{\beta _{\epsilon}{-}2}{4\pi}\,\tilde{\mu}_{0}^{2} + E_{0}^{\mathrm {kin}}[ \eta _{1}] + \mathcal {O}\bigl(\epsilon \beta _{\epsilon}\|\eta \|_{\mathcal {X}_{ \epsilon}}^{2}\bigr)\,. \end{aligned}$$
(B.8)

Moreover we know from Proposition 4.5 that

$$ \|\eta _{1}\|_{\mathcal {X}_{0}}^{2} \,\le \, C_{4} E_{0}[\eta _{1}] + C_{5} \bigl(\tilde{\mu}_{0}^{2} + \tilde{\mu}_{1}^{2} + \tilde{\mu}_{2}^{2} \bigr)\,, $$
(B.9)

where \(\tilde{\mu}_{0}, \tilde{\mu}_{1}, \tilde{\mu}_{2}\) are the moments of \(\eta _{1}\), which satisfy \(\tilde{\mu}_{j} = \mu _{j} + \mathcal {O}\bigl(\epsilon ^{\infty}\|\eta \|_{ \mathcal {X}_{\epsilon}}\bigr)\). Combining both estimates in (B.8) we obtain

$$ E_{0}[\eta _{1}] \,=\, \frac{1}{2} \|\eta _{1}\|_{\mathcal {X}_{0}}^{2} - E^{ \mathrm {kin}}_{0}[\eta _{1}] \,\le \, \frac{1}{2} \|\eta _{1}\|_{\mathcal {X}_{ \epsilon}}^{2} - E^{\mathrm {kin}}_{\epsilon}[\eta _{1}] + \frac{\beta _{\epsilon}{-}2}{4\pi} \,\tilde{\mu}_{0}^{2} + \mathcal {O}\bigl( \epsilon ^{\gamma _{1}}\|\eta \|_{\mathcal {X}_{\epsilon}}^{2}\bigr)\,, $$

namely \(E_{0}[\eta _{1}] \le E_{\epsilon}[\eta _{1}] + \frac{\beta _{\epsilon}-2}{4\pi}\tilde{\mu}_{0}^{2} + \mathcal {O}\bigl( \epsilon ^{\gamma _{1}}\|\eta \|_{\mathcal {X}_{\epsilon}}^{2}\bigr)\). Using in addition (B.9) we deduce

$$\begin{aligned} \|\eta _{1}\|_{\mathcal {X}_{\epsilon}}^{2} &\le \, \|\eta _{1}\|_{\mathcal {X}_{0}}^{2} + \mathcal {O}\bigl(\epsilon ^{\gamma _{1}} \|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \bigr) \\ &\le \, C_{4} E_{\epsilon}[\eta _{1}] + C\bigl(\beta _{ \epsilon}\tilde{\mu}_{0}^{2} + \tilde{\mu}_{1}^{2}+ \tilde{\mu}_{2}^{2} \bigr) + \mathcal {O}\bigl(\epsilon ^{\gamma _{1}}\|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \bigr)\,. \end{aligned}$$

Finally, invoking (B.6) and recalling that \(C_{4} > 2\), we find

$$ \|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \,\le \, \|\eta _{1}\|_{\mathcal {X}_{ \epsilon}}^{2} + \frac{C_{4}}{2} \|\eta _{2} \|_{\mathcal {X}_{\epsilon}}^{2} \,\le \, C_{4} E_{\epsilon}[\eta ] + C\bigl(\beta _{\epsilon} \tilde{\mu}_{0}^{2} + \tilde{\mu}_{1}^{2}+ \tilde{\mu}_{2}^{2}\bigr) + \mathcal {O}\bigl(\epsilon ^{\gamma _{1}}\|\eta \|_{\mathcal {X}_{\epsilon}}^{2} \bigr)\,, $$

and estimate (4.27) follows, since \(\tilde{\mu}_{j} = \mu _{j} + \mathcal {O}\bigl(\epsilon ^{\infty }\|\eta \|_{ \mathcal {X}_{\epsilon}}\bigr)\) for \(j = 0,1,2\). □

2.2 B.2 Diffusive terms in the energy functional

We justify here the expression (4.32) of the quantity \(I_{4}\). Integrating by parts as in [35], we find

$$ \int _{\Omega _{\epsilon}} W_{\epsilon }\tilde{\eta}\,\mathcal {L}\tilde{\eta} \,{\mathrm { d}}X \,=\, -\int _{\Omega _{\epsilon}} W_{\epsilon }|\nabla \tilde{\eta}|^{2}\,{\mathrm { d}}X - \int _{\Omega _{\epsilon}} (\nabla W_{ \epsilon }\cdot \nabla \tilde{\eta})\tilde{\eta} \,{\mathrm { d}}X - \int _{\Omega _{ \epsilon}} \tilde{V}_{\epsilon }\tilde{\eta}^{2}\,{\mathrm { d}}X\,, $$

where \(\tilde{V}_{\epsilon }= \frac{1}{4}(R\partial _{R} + Z\partial _{Z})W_{ \epsilon }- \frac{1}{2} W_{\epsilon}\). Similarly,

$$ \epsilon \int _{\Omega _{\epsilon}} W_{\epsilon}\tilde{\eta}\, \partial _{R}\tilde{\zeta} \,{\mathrm { d}}X \,=\, \epsilon \int _{\Omega _{ \epsilon}} W_{\epsilon}(1+\epsilon R)\tilde{\zeta}\,\partial _{R} \tilde{\zeta} \,{\mathrm { d}}X \,=\, -\frac{\epsilon}{2}\int _{\Omega _{\epsilon}} \partial _{R} \bigl(W_{\epsilon}(1+\epsilon R)\bigr)\tilde{\zeta}^{2} \,{\mathrm { d}}X\,. $$

On the other hand, integrating by parts and using the relation (2.19) between \(\tilde{\phi}\) and \(\tilde{\eta}\), we obtain

$$\begin{aligned} \int _{\Omega _{\epsilon}} \tilde{\phi}\Bigl(\mathcal {L}\tilde{\eta}+ \epsilon \partial _{R} \tilde{\zeta}\Bigr)\,{\mathrm { d}}X \,&=\, \int _{\Omega _{ \epsilon}} \tilde{\eta}\Bigl(\Delta \tilde{\phi}- \frac{\epsilon \partial _{R} \tilde{\phi}}{1+\epsilon R}\Bigr)\,{\mathrm { d}}X -\frac{1}{2}\int _{\Omega _{ \epsilon}} \tilde{\eta}\bigl(R\partial _{R} + Z\partial _{Z}\bigr) \tilde{\phi} \,{\mathrm { d}}X \\ \,&=\, -\int _{\Omega _{\epsilon}} \tilde{\eta}^{2}(1+\epsilon R)\,{\mathrm { d}}X - \frac{1}{2}\int _{\Omega _{\epsilon}} \tilde{\eta}\bigl(R\partial _{R} + Z\partial _{Z}\bigr)\tilde{\phi} \,{\mathrm { d}}X\,. \end{aligned}$$

It remains to treat the last term in the right-hand side. Here again, we use the relation (2.19) and integrate by parts to obtain

$$ \frac{1}{2}\int _{\Omega _{\epsilon}}\tilde{\eta}\bigl(R\partial _{R} + Z\partial _{Z}\bigr) \tilde{\phi} \,{\mathrm { d}}X \,=\, \frac{\epsilon}{4}\int _{ \Omega _{\epsilon}} \frac{R|\nabla \tilde{\phi}|^{2}}{(1+\epsilon R)^{2}}\,{\mathrm { d}}X\,. $$

Altogether we arrive at (4.32), with \(V_{\epsilon }= \tilde{V}_{\epsilon }- (1+\epsilon R)\).

2.3 B.3 Coercivity of the diffusive quadratic form

This section is devoted to the proof of Proposition 4.15. Given \(\epsilon > 0\) sufficiently small, we take a smooth partition of unity of the form \(1 = \chi _{3}^{2} + \chi _{4}^{2}\), where \(\chi _{3}, \chi _{4}\) are radially symmetric and \(\chi _{3} = 1\) when \(\rho \le \frac{1}{2} \epsilon ^{-\sigma _{1}}\), \(\chi _{3} = 0\) when \(\rho \ge \epsilon ^{-\sigma _{1}}\). We can also assume that \(|\nabla \chi _{3}| + |\nabla \chi _{4}| \le C \epsilon ^{\sigma _{1}}\). Given \(\eta \) as in the statement of Proposition 4.15, we define \(\eta _{3} = \chi _{3}\eta \), \(\eta _{4} = \chi _{4}\eta \). We thus have the decompositions \(\eta ^{2} = \eta _{3}^{2} + \eta _{4}^{2}\), \(\eta \nabla \eta = \eta _{3}\nabla \eta _{3} + \eta _{4}\nabla \eta _{4}\), and

$$ |\nabla \eta |^{2} \,=\, |\nabla \eta _{3}|^{2} + |\nabla \eta _{4}|^{2} - \bigl(|\nabla \chi _{3}|^{2} + |\nabla \chi _{4}|^{2} \bigr)\eta ^{2} \,. $$
(B.10)

As a consequence, the quadratic form \(Q_{\epsilon}[\eta ]\) can be decomposed as

$$ Q_{\epsilon}[\eta ] \,=\, Q_{\epsilon}[\eta _{3}] + Q_{\epsilon}[ \eta _{4}] - \int _{\Omega _{\epsilon}} W_{\epsilon }\bigl(|\nabla \chi _{3}|^{2} + |\nabla \chi _{4}|^{2} \bigr)\eta ^{2}\,{\mathrm { d}}X\,. $$
(B.11)

The last term in (B.11) is bounded by \(C \epsilon ^{2\sigma _{1}}\|\eta \|_{\mathcal {X}_{\epsilon}}^{2}\) and is thus negligible when \(\epsilon \ll 1\). So our main task is to estimate from below the terms \(Q_{\epsilon}[\eta _{3}]\) and \(Q_{\epsilon}[\eta _{4}]\).

We first consider the function \(\eta _{3}\) which is supported in the region where \(\rho \le \epsilon ^{-\sigma _{1}}\). We recall that the weight \(W_{\epsilon}\) in (4.16) satisfies the estimates (4.18), which read

$$\begin{aligned} &|\nabla W_{\epsilon}(R,Z) - \nabla A(\rho )| + |W_{\epsilon}(R,Z) - A( \rho )| \,\le \, C \epsilon ^{\gamma _{1}} A(\rho )\,, \\ &\quad \text{when}~ \,\rho \le \epsilon ^{-\sigma _{1}}\,, \end{aligned}$$
(B.12)

where \(\gamma _{1} > 0\). We easily deduce that

$$ Q_{\epsilon}[\eta _{3}] \,\ge \, Q_{0}[\eta _{3}] - C \epsilon ^{ \gamma _{1}} \bigl(\|\nabla \eta _{3}\|_{\mathcal {X}_{0}}^{2} + \|\rho \eta _{3} \|_{\mathcal {X}_{0}}^{2} + \|\eta _{3}\|_{\mathcal {X}_{0}}^{2}\bigr)\,, $$
(B.13)

where \(Q_{0}\) is the limiting quadratic form (4.62). On the other hand, we know from Proposition 4.14 that

$$ C_{8} Q_{0}[\eta _{3}] \,\ge \, \|\nabla \eta _{3}\|_{\mathcal {X}_{0}}^{2} + \|\rho \eta _{3}\|_{\mathcal {X}_{0}}^{2} + \|\eta _{3}\|_{\mathcal {X}_{0}}^{2} - C_{9} \bigl(\tilde{\mu}_{0}^{2} + \tilde{\mu}_{1}^{2} + \tilde{\mu}_{2}^{2} \bigr)\,, $$
(B.14)

where \(\tilde{\mu}_{0}, \tilde{\mu}_{1}, \tilde{\mu}_{2}\) are the moments of \(\eta _{3}\), which satisfy \(\tilde{\mu}_{j} = \mu _{j} + \mathcal {O}\bigl(\epsilon ^{\infty}\|\eta \|_{ \mathcal {X}_{\epsilon}}\bigr)\). Combining (B.13), (B.14) and using (B.12) once again, we arrive at

$$ \|\nabla \eta _{3}\|_{\mathcal {X}_{\epsilon}}^{2} + \|\rho \eta _{3}\|_{\mathcal {X}_{ \epsilon}}^{2} + \|\eta _{3}\|_{\mathcal {X}_{\epsilon}}^{2} \,\le \, 2C_{8} Q_{ \epsilon}[\eta _{3}] + C\bigl(\tilde{\mu}_{0}^{2} + \tilde{\mu}_{1}^{2} + \tilde{\mu}_{2}^{2}\bigr)\,. $$
(B.15)

We next consider the function \(\eta _{4}\), which is nonzero only if \(\rho \ge \frac{1}{2} \epsilon ^{-\sigma _{1}}\). Our starting point is the lower bound

$$ Q_{\epsilon}[\eta _{4}] \,\ge \, \frac{1}{4}\int _{\Omega _{\epsilon}} W_{\epsilon }|\nabla \eta _{4}|^{2}\,{\mathrm { d}}X + \int _{\Omega _{\epsilon}} \Bigl(V_{\epsilon }- \frac{|\nabla W_{\epsilon}|^{2}}{3W_{\epsilon}} \Bigr)\eta _{4}^{2}\,{\mathrm { d}}X\,, $$

which is obtained from (4.61) by applying Young’s inequality to the middle term in the right-hand side. Using the expression (4.16) of the weight function, as well as the estimates (B.12) in the inner region \(\Omega _{\epsilon}'\), it is not difficult to verify that

$$ \frac{V_{\epsilon}}{W_{\epsilon}} - \frac{|\nabla W_{\epsilon}|^{2}}{3W_{\epsilon}^{2}} ~\ge ~ \textstyle\begin{cases} C \rho ^{2} - \tilde{C} & \text{in}~\,\Omega _{\epsilon}'\,, \\ - \tilde{C} & \text{in}~\,\Omega _{\epsilon}''\,, \\ C \rho ^{2\gamma} & \text{in}~\,\Omega _{\epsilon}'''\,, \end{cases} $$

for some positive constants \(C,\tilde{C}\). It follows that

$$ Q_{\epsilon}[\eta _{4}] \,\ge \, \frac{1}{4} \|\nabla \eta _{4}\|_{ \mathcal {X}_{\epsilon}}^{2} + C \int _{\Omega _{\epsilon}' \cup \Omega _{ \epsilon}'''} W_{\epsilon}\,\rho _{\gamma}^{2}\eta _{4}^{2}\,{\mathrm { d}}X - \tilde{C} \int _{\Omega _{\epsilon}''} W_{\epsilon}\,\eta _{4}^{2}\,{\mathrm { d}}X \,. $$
(B.16)

If we now combine (B.15) and (B.16), we obtain

$$\begin{aligned} & \|\nabla \eta _{3}\|_{\mathcal {X}_{\epsilon}}^{2} + \|\nabla \eta _{4}\|_{\mathcal {X}_{\epsilon}}^{2} + \|\eta \|_{\mathcal {X}_{\epsilon}}^{2} + \int _{\Omega _{\epsilon}'\cup \Omega _{\epsilon}'''} W_{\epsilon}\, \rho _{\gamma}^{2} \eta ^{2}\,{\mathrm { d}}X \\ &\quad\qquad\le \, C_{10}\bigl(Q_{\epsilon}[\eta _{3}] + Q_{\epsilon}[\eta _{4}] \bigr) + C_{11}\Bigl(\tilde{\mu}^{2} + \int _{\Omega _{\epsilon}''} W_{ \epsilon }\eta ^{2}\,{\mathrm { d}}X\Bigr)\,, \end{aligned}$$
(B.17)

for some positive constants \(C_{10}, C_{11}\), where \(\tilde{\mu}^{2} = \tilde{\mu}_{0}^{2} + \tilde{\mu}_{1}^{2} + \tilde{\mu}_{2}^{2}\). Finally, using again (B.10) as well as (B.11), and recalling that \(\tilde{\mu}_{j} = \mu _{j} + \mathcal {O}\bigl(\epsilon ^{\infty}\|\eta \|_{ \mathcal {X}_{\epsilon}}\bigr)\), we deduce (4.65) from (B.17). \(\Box \)

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Gallay, T., Šverák, V. Vanishing viscosity limit for axisymmetric vortex rings. Invent. math. 237, 275–348 (2024). https://doi.org/10.1007/s00222-024-01261-5

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