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Well-posedness of logarithmic spiral vortex sheets

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vortr\"age aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field remains continuous across the spiral. Moreover, we give a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such a solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922. Another consequence of the main result is well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane.


Introduction
We are concerned with the two-dimensional incompressible Euler equations, ∂ t v + v · ∇v + ∇p = 0, div v = 0, (1.1) on R 2 × (0, ∞), where v denotes the velocity of an inviscid fluid and p denotes the pressure function.
We say that a vector field v ∈ L 2 loc (R 2 × (0, ∞)) is a weak solution of (1.1) if it is weakly divergencefree, i.e., for all ψ ∈ C ∞ 0 (R 2 ) and if it satisfies for all divergence-free ϕ ∈ C ∞ 0 (R 2 × (0, ∞); R 2 ). By considering the vorticity ω := ∂ 1 v 2 − ∂ 2 v 1 we see that, formally, ω satisfies the equation and v can be recovered from ω by the Biot-Savart law, v = K * ω, (1.4) where K(x) := 1 2π x ⊥ |x| 2 , (x 1 , x 2 ) ⊥ := (−x 2 , x 1 ). (1.5) This is often referred to as the vorticity formulation of the Euler equations (1.1). The vorticity formulation can be used to study more general notions of solutions, such as velocity fields v ∈ L 2 loc with vorticity in the form of a measure [7]. In fact the mathematical theory of vortex sheets, which are included in such family of solutions, is already well-established. In particular, global-in-time existence of weak solutions to (1.2)-(1.3) for every initial vorticity ω 0 ∈ H −1 loc that is compactly supported and has a distinguished sign has been established by Delort [5]; see also [12,24] for related results and simplifications.
A convenient way to describe a vortex sheet, with vorticity supported on a curve, was introduced by Birkhoff [2] and Rott [21] (see also [14]), and is based on a generalization of the Biot-Savart law (1.4) to the case of vorticities ω t in the form of measures, that is (1. 6) We note that the velocity field is well-defined only outside of Σ(t), but here we applied a common convention to define it on the sheet itself as the arithmetic mean of the limit velocities from both sides of the sheet (see [10, (2.15)-(2.16)]). We will show below (in Remark 5.1) that this assumption arises naturally in our approach to vortex sheets.
One can use methods of complex analysis to denote the velocity field v = (v 1 , v 2 ) in the complex notation as v 1 + iv 2 , and rewrite (1.6) as the Birkhoff-Rott equation, where the principal value comes from the arithmetic mean convention mentioned above, * denotes the complex conjugation, and Z(θ, t) ∈ C denotes the position of a point on the curve that is the support of vorticity, θ is the circulation parameter. Thus the evolution of the support of the sheet is described by the evolution of the curve Z(θ, t), and the vorticity distribution can be described by where Σ(t) := {Z(θ, t) : θ ∈ R} is the interface curve, and γ(Z(θ, t), t) is the difference of the tangential parts of limit velocities across Z(t).
In this work we are concerned with the vortex sheets of the form of logarithmic spirals (or a combination of coexisting spirals) given by      Z(θ, t) = t µ e a(θ−θ 0 ) e iθ , Γ(θ, t) = gt 2µ−1 e 2a(θ−θ 0 ) , r(θ, t) = |Z(θ, t)| = t µ e a(θ−θ 0 ) , θ ∈ R, t > 0, (1.8) where a > 0, µ ∈ R, g ∈ R \ {0} and θ 0 ∈ [0, 2π). Here Γ(θ, t) denotes the total circulation at radius r(θ, t), and Z(θ, t) denotes the parametrization of the spiral at time t with respect to the parameter θ; see Figure 1. We denote by Σ(t) the set of points on the spiral; i.e. parametrized by Z(θ, t). We often use the convention Σ ≡ Σ(1), and consider points z = (z 1 , z 2 ) located in a small neighbourhood close to Σ. In that case we assume that the spiral is oriented in the direction outward from 0, and so we refer to the points closer to 0 as the points on the left of a spiral Σ, and as the points on the right of Σ as those further away from 0. We will also denote by τ the unit vector tangent to the spiral that has the same direction as the direction of the parametrization, and we will denote by n = iτ the unit normal vector.
Logarithmic spirals (1.8) are often referred to as Prandtl spirals. In fact they were first introduced in 1922 by Ludwig Prandtl [18] as a model of vortices detaching from tips of wings, and were used to study the influence of the vortices on the lift of the wings as well as the influence of the vortices behind a given plane on a plane following it; see Fig. 4 in [16] and Fig. 2 and 4 in [18]. His contributions to the study of the spirals, long time before an introduction of the notion of weak solutions of the Euler equations or the Birkhoff-Rott [2,21] equations, were based on studying putative discontinuities of the velocity field, dimensional analysis as well as on physical experiments.
Since his pioneering work, significant effort has been dedicated towards mathematical understanding of the spirals. In fact, Prandtl himself has commented (on p. 704 in [18]) that his exposition has "touched on many unfinished things" and that its purpose was to "give suggestions for further research, rather than to report final results".
The introduction of Birkhoff-Rott equation (1.7) in [2,21] promised a more rigorous approach to Prandtl's spirals. Unfortunately, the right-hand side of the Birkhoff-Rott equation (1.7) does not converge at large scales (see [9] for instance). Saffman has introduced a way of dealing with this problem that is based on an unpublished note of Moore, see Section 8.3 in his extensive exposition [22] on vortex dynamics. However, this approach is far from mathematical rigor. Despite the introduction of the theory of vortex sheets as weak solutions of Euler equations by Di Perna and Majda [7], the analysis of the Prandtl spiral remained out of reach. In fact, the divergence of the right-hand side of the Birkhoff-Rott equation (1.7) is connected to the problem of defining the velocity field v via the Biot-Savart law (1.6), due to large contributions of vorticity from the large scales.
(1.9) Thus, if we fix t > 0 and we "cut" the spiral at some distance from the origin then, not only the velocity field is well-defined, but also ω ∈ H −1 loc (see [4]), which, due to compact support, implies that v ∈ L 2 loc , due to a result of Schochet [23]. Without cutting the spiral at large scales it has recently been proved in [3, Theorem 2.1] that the Biot-Savart law (1.6) defines an L 1 loc velocity field v if the vorticity ω is a nonnegative σ-finite measure on R 2 with which does not hold for Prandtl spirals due to (1.9) above.
Another approach of understanding vortex sheets is based on the weak vorticity formulation of the 2D Euler equations, i.e., where ω 0 denotes the initial vorticity and and K is the Biot-Savart kernel (1.5).
Note that this formulation does not involve the velocity function itself, but rather is concerned with the vorticity ω only. This formulation has been introduced by [24] (and appears implicitly already in [5]) in order to deal with the lack of continuity of v across a sheet, and [10] showed equivalence of the weak vorticity formulation with the the Birkhoff-Rott equation (1.7). However, the relation with the weak formulation (1.3) is still unclear. An equivalence between the two formulations was established by Delort [5] for vorticities that remain bounded in time with values in the space of bounded Radon measures and with values in H −1 loc , and such that the initial vorticity ω 0 has compact support. Unfortunately, the assumptions of the Delort's theorem do not hold for the Prandtl spiral (1.8), and they do not appear to be easily removable.
One more approach to understanding the logarithmic spirals, is to replace the single Prandtl spiral (1.8), with multiple coexisting logarithmic spirals that are obtained by rotating a single spiral around the origin. Namely, instead of (1.8), one can fix M ≥ 2 and consider a family of spirals parametrized by 11) which are defined by 2M + 2 parameters where m = 0, . . . , M − 1, and it is assumed (without loss of generality) that for simplicity. We will denote by Σ m (t) the spiral parametrized by Z m (θ, t), and we set Σ m := Σ m (1), Σ := M −1 m=0 Σ m and Z m (θ) := Z m (θ, 1) for brevity. For each m we denote by Ω m (t) the region between Σ m (t) and Σ m+1 (t) (with the convention that Σ M := Σ 0 ), and similarly Ω m := Ω m (1). below. This has been observed by Elling and Gnann [8], who applied the ansatz (1.11) in (1.7) to obtain

Spirals
2ag k e 2aσ dσ 1 − e (a+i)σ+i(θ k −θm) * (1.13) where σ := θ − θ + θ m − θ k . It is clear from (1.13) that the right-hand side, while integrable for σ → −∞ for each pair k, m ∈ {0, . . . , M − 1}, is not integrable as σ → ∞, as expected from the above discussion of the Prandtl spiral. However, they observed the algebraic identity (1.14) for σ, ∆ ∈ R. By taking ∆ := θ k − θ m this gives that for all σ ∈ R (1.15) provided the compatibility conditions hold. Remarkably, assuming the compatibility conditions we can apply (1.15) to substitute the integrand in (1.13) for sufficiently large σ > 0 and hence obtain the missing integrability at σ → ∞. This illustrates the expected cancellation of multiple logarithmic spirals mentioned above. In other words, the compatibility conditions (1.16) express the desired cancellation of large contributions of vorticity at large scales.
Under the conditions (1.16), the equation (1.13) is equivalent to and for m = 0, . . . , M − 1, which is commonly referred to as Alexander's spirals, is more obvious choice, and has also been explored by [8]. In such case the left-hand side of (1.17) becomes coth(πA/M ) (see the Appendix), and so every a > 0, g ∈ R \ {0}, µ ∈ R such that give a solution to the Birkhoff-Rott equations. For M ≥ 3 the compatibility conditions (1.16) hold, which can be immediately seen by applying Viéte formulas to the equation z M − 1 = 0, and (1.21) gives 2 equations for three variables, which gives many generic solutions. For example, first choosing any a > 0 such that the direction of the term on the right-hand side of (1.21) is different from the direction of −1 + ai, we see that there exists a unique pair of µ ∈ R and g ∈ R \ {0} for which (1.21) holds. We note that the set of a's for which the two directions are the same is very small, see p. 30 in [8] for some numerical considerations.
Remarkably, this is false for M = 2 as then e −iθ 0 and e −iθ 1 are solutions to the quadratic equation z 2 − 1 = 0, and so Viéte formulas give which contradicts (1.16).
The above considerations give strong suggestions that both the Prandtl spiral, as well as the Alexander spirals with M = 2, cannot be given a rigorous mathematical meaning.
The main purpose of this work is to prove the contrary.
In fact, we find an explicit formula for the velocity field, where the profile function w is given in the polar coordinates by denotes the winding number of the spiral.
We note that here θ ∈ R, and we show (in Proposition 3.1(i)) that the definition (1.25) ensures that w is well-defined; namely that (1.24) is invariant with respect to adding 2πk, k ∈ Z, to θ. The explicit representation (1.23) allows us to consider v as a weak solution of the Euler equations (1.1), which turns out to offer a much more robust treatment of logarithmic spirals (1.11). In particular it shows that the compatibility conditions (1.16) can be neglected.
Our main result (Theorem 1.2) gives necessary and sufficient conditions for v to be a weak solution to the Euler equations (1.1). Before stating it, we discuss the main features of the velocity formula (1.23). The central idea of (1.24) is the introduction of the winding number J(r, θ, k), whose role is to give an efficient way of localizing a given point z = re iθ in relation to the M spirals Σ m (t). We discuss the role of the winding number J in more detail in Section 3.1, but we note that one of its key properties is that it is locally constant between any two neighbouring spirals, see Proposition 3.1(ii) for details. Thus, rewriting (1.24) in the form i.e. between the spirals, but also that the flow is described by the complex potential, that is This observation has a number of consequences. First, we note that (1.26) suggests that |w(z)| ∼ |z| Re (iA−1) = |z| (a 2 −1)/(a 2 +1) for z close to zero, which would indicate singular behaviour of v at the origin for small a > 0. We show (in Section 3.2) that this suggestion is false, as there is additional dependence on r hidden in D via the winding number J(r, θ, k). Note that J remains constant only when r and θ are coupled; in particular fixing θ ∈ [0, 2π) and taking r → 0 the definition (1.25) gives that J(r, θ, k) → +∞. We show below (in Section 3.2) that this implies that in fact and so in particular v ∈ L 2 loc (R 2 ×(0, ∞)) and v has locally finite kinetic energy, v(t) L 2 (B(x 0 ,1)) < ∞ for x 0 ∈ R 2 and each t > 0. Moreover, Furthermore the form (1.26) of w and the explicit formula (1.27) of the complex potential Φ imply that the pressure function p(z, t) can be determined explicitly using Bernoulli's law, as well as other properties of v which we summarize in the following statement.
, satisfies the pointwise estimate (1.28) and admits continuous extension to the closure of the region Ω m for any and v L (z, t) denote the limit velocities v(z , t) as z tends to z from the right and left sides of Σ(t), respectively. Then where n(z, t) denotes the unit normal vector to Σ(t) at z. In particular v is weakly divergencefree (1.2).
(iii) The vorticity curl v is a locally finite measure of the form of logarithmic spirals (1.11), i.e., (iv) v is a classical solution of the Euler equations in the region {(x, t) : t > 0, x ∈ Ω m (t)} between the spirals, with the pressure function p(z, t) defined by the Bernoulli law, We emphasize that the claims of the above theorem hold for all values of the parameters (1.12).
The relation of the logarithmic spirals (1.11) with the Euler equations is characterized by our main result.
and the pressure matching condition, , and p is given by the Bernoulli law (1.31).
Moreover the matching conditions (1.33) and (1.34) are equivalent to the imaginary part and the real part of (1.17), respectively.
We note that in the velocity matching condition (1.33) we take n · v := n · v R = n · v L , as the normal component of the velocity is continuous across each spiral due to Theorem 1.1(ii Theorem 1.5 can be proved using methods of contour integration, which gives further insight into the right-hand side of (1.24), particularly the meaning of the winding number J.
Observe that v from Theorem 1.5 equals, on each spiral, the average of its limit values from the left and right sides of the spiral, which often appears as an assumption of models of vortex sheets flow, see [2,13,14,22]. In the case of logarithmic spirals this is a consequence of the Biot-Savart law itself, since points on the sheet correspond to the contour crossing through a single pole, in which case contour integration gives the average of the integrals over the contours that avoid the pole from each side, see In Section 2 we introduce some basic tools and then, in Section 3, we discuss the properties of the winding number J (Section 3.1) and prove Theorem 1.1 (Section 3.2). In Section 4 we prove the main result, Theorem 1.2. Finally we prove Theorem 1.5 in Section 5.

Preliminaries
We denote the complement of A ⊂ R 2 by A c , and we denote the complex conjugate of z = x + iy by z * = x − iy. We say that a holomorphic function where f denotes the complex derivative of a complex function f . In such case In other words, in real variables, the velocity field (v 1 , v 2 ) satisfies . Note that existence of the complex potential implies that the flow (v 1 , v 2 ) is irrotational, due to the Cauchy-Riemann equations, see [14, Appendix 1.3] for details.
Assuming that both v 1 , v 2 , ψ, χ are also functions of time, then the unsteady Bernoulli law, In particular the velocity field (v 1 , v 2 )(x, y, t) satisfies the 2D incompressible Euler equations (1.1) with pressure function p, which follows by taking the gradient of (2.1). 1). Then w * = Φ and we the complex Bernoulli law (2.2) takes the form

Now suppose that the flow is self-similar
In particular, taking "∇" gives the self-similar form of the Euler equations (1.1), We conclude this section with a technical lemma regarding separation of exp(z) from 0.
We first choose M > 0 such that e M ≥ 3r/2 and e −M ≤ r/2.
Since |e z − r| = 0 on the compact set B ∩ {−M ≤ Re z ≤ M }, there exists c ∈ (0, r/2) such that For Rez ≥ M we can use the triangle inequality to obtain and similarly

Velocity field of logarithmic spirals
In this section we prove Theorem 1.1, that is we prove a number of properties of velocity field v(z, t) recall (1.24).
3.1. The winding number J. Here we comment on the notion of the winding number J(r, θ, k).
First we note that J(r, θ, k) decreases by 1 whenever z crosses the k-th spiral from left to right. Indeed, the inequality in (1.25) becomes an equality when r = e a((θ−2πj)−θ k ) , i.e. when r = |Z k (θ )| for θ := θ − 2πj, where j ∈ Z. Since the exponential representation z = re iθ is invariant with respect to adding 2πj to θ, the winding number J(r, θ, k) can be thought of as "the number of the loop of the the spiral Σ k ". For example, if θ ∈ [0, 2π) and r ∈ [e a(θ−θ k ) , e a(θ+2π−θ k ) ) then the part of the spiral Σ k given by {Z k (θ ) : θ ∈ [0, 2π)} is the closest segment of Σ k to z = re iθ in the direction towards the origin.
(1) During such procedure J(r, θ, m) would remain constant, but all the other J(r, θ, k), for k = m, would increase by 1 as during the procedure we must have crossed each of Σ k , right to left, see Figure 2 for a sketch. Thus, for k = m, J(r, θ, k) has increased by 1, and so the k-th ingredient of the sum (1.24) got multiplied by e 2πA * . Moreover all ingredients got multiplied by e −2πA * , from the term "e A(θ k −θ) ", as θ increased. Thus the overall effect, i.e. multiplication of the m-th term by e −2πA * , is the same as before.
Re z Im z Here the two circles at the darkest spiral represent a point on the spiral considered as a limit point from both sides of the spiral, the dashed line illustrates the procedure described below Proposition 3.1, and the three black dots represent the crossing points with the other spirals that must occur during the procedure.

3.2.
Proof of Theorem 1.1. That w * is holomorphic in Ω m and admits continuous extension to the closure of Ω m for each m follows as observed in (1.26), by recalling the well-definiteness (3.1) and Proposition 3.1(ii). In particular this gives that v is irrotational and divergence-free in the regions between the spirals, as these properties are equivalent to the Cauchy-Riemann equations. As for the pointwise estimate (1.28), note that Proposition 3.1(iii) gives Indeed, since v R and v L can be expressed using (1.23) by applying the values J R and J L (respectively) of the winding number J from Proposition 3.1(iv), we obtain which has the same direction as the tangent to Σ m (t) at Z m (θ, t), as Since the jump across each spiral occurs in the direction tangent to the spiral, we see that v · n is continuous across each spiral, where n denotes a unit vector orthogonal to the spiral. This together with the decay (1.28) of v at the origin, and the fact that div v = 0 pointwise in the regions between the spirals shows that v is weakly divergence free. To be more precise, for every ψ ∈ C ∞ 0 (R 2 ) and every t > 0, where we used (1.28) in the last step. As for claim (iii) of Theorem 1.1 we note that, since w * is holomorphic in Ω m for every m we see that v is irrotational in the regions between the spirals. Hence the measure curl v(t) is supported on Σ(t) at each time. In order to show the explicit formula (1.30), where γ(Z k (θ, t), t) := ∂ θ Γ k (t, θ) |∂ θ Z k (θ, t)| −1 , we note that (3.2) and (3.3) give |∂ θ Z k (θ, t)| denotes the unit tangent vector to Σ k (t) in the same direction as the orientation of the spiral at Z(θ, t) ∈ Σ k (t). Thus Theorem 1.1(iii) follows by writing for any ψ ∈ C ∞ 0 (R 2 ), where we used (1.28) in the last step. The claim of Theorem 1.1(iv) follows directly from the Bernoulli law, recall the comment below (2.2).

Logarithmic spirals as weak solutions of the Euler equations
In this section we prove Theorem 1.2, that is we show that the velocity field (1.23) is a weak solution of the Euler equations on R 2 × (0, ∞) if and only if the matching conditions (1.33), (1.34) hold, i.e. that n(z, t) · (v(z, t) − µz) = 0, and p is continuous at z for z ∈ Σ(t), t > 0, and that these conditions are equivalent to the constraint (1.17), being valid for all m ∈ {0, . . . , M − 1}, where A mk is defined in (1.19).
Using the decay (1.28) of w at the origin we obtain where, in the second step, we have applied integration by parts in space and noted that w is independent of time. Here we have also used the notation with the obvious generalization to the set Σ RL ∩ B(η) c , where f R and f L denote the limit values of f at Σ ≡ M −1 m=0 Σ m from the right and left sides, respectively. Noting that φ has compact support in time we can integrate the third term on the right-hand side by parts in time to get and so we see that, since the Euler equations are satisfied in the classical sense in the region between the spirals (see (2.4)), the sum of the integrals over (0, ∞) × B(η) c on the right-hand side of the above calculation becomes where q(z) := p(z, 1), and we used Theorem 1.1(iv) in the first equality, the fact that div φ = 0 and (1.32) in the second equality.
Thus the weak form of the Euler equations holds if and only if , where we used that fact that n · w R = n · w L =: n · w (recall Theorem 1.1(ii)) in the second line. Since the vectors w R − w L and n are orthogonal (by the same fact), and w R − w L = 0 (by (3.2)), this holds if and only if both matching conditions (1.33), (1.34) hold.
The claim of Theorem 1.2 about the equivalence of the weak formulation (1.3) of the Euler equations and the distributional formulation can be proved in an analogous way, simply by not restricting ourselves to the divergence-free test functions ψ.

The velocity matching.
Here we show that the velocity matching condition (1.33), n(z) · (w(z) − µz) = 0, for z ∈ Σ, is equivalent to the imaginary part of (1.17), where A mk is defined by (1.19).
Indeed noting noting that, by Theorem 1.1(ii), on Σ we have n · w = n · (w R + w L )/2, where n(z) denotes the unit normal vector to Σ at z ∈ Σ we consider Since n has the same direction as for all θ ∈ R and m ∈ {0, . . . , M − 1}, which is equivalent to (4.1).

4.3.
The pressure matching condition. Here we show that the pressure function obtained via the Bernoulli law 3)) is continuous across each spiral Σ m if and only if the real part of (1.17) holds, i.e.

Re 1 sinh(πA)
We will show below that Then the claim follows by noting that, by Proposition 3.1, taking z = Z m (θ) in Φ(z) and w * (z) (recall (1.27) and (1.24)), makes the "left and right differences" of the terms in the sum M −1 k=0 vanish, except for k = m, and the winding number J(e a(θ−θm) , θ, k) becomes either 1 k<m or 1 k≤m , depending on the side of the spiral. To be more precise Proposition 3.1(iv) gives for all θ ∈ R. Thus, since w * = iA z Φ, (4.2) gives that Since the right-hand side vanishes for all θ ∈ R if and only if Re K = (2µ − a 2 − 1)/2a 2 , this proves the claim.

The Biot-Savart integral
In this section we prove Theorem 1.5, that we show that if the vorticity is given in terms of circulation Γ m in (1.11) on the m-th spiral, m ∈ {0, . . . , M −1}, the compatibility conditions (1.16) are satisfied, and v is given by the Biot-Savart law (1.6), i.e. v(z, t) : Applying the change of variable θ → θ − θ k =: σ for each k ∈ {0, . . . , M − 1}, we obtain where, for brevity, we used the notation We fix r > 0 and θ ∈ R such that a(2πj + ∆ k ) + ln r = 0 Note that (5.2) is equivalent to re iθ ∈ Σ ∪ {0} and also equivalent to the denominator in (5.4) not vanishing for any σ. Then multiplying (5.1) by e iθ gives w(z) = w(z) for z ∈ Σ ∪ {0}, as required.
In the remainder of this section we prove (5.3).
for k ∈ {0, . . . , M − 1}, r > 0 and j ∈ Z. Observe that the l'Hôpital rule gives that This shows that each σ j (j ∈ Z) is a simple pole of f and We aim to find the integral on the left-hand side of (5.3) by contour integration as sketched in Figure 3 below. We set ϕ 0 := π + arctan (a) + ε, (5.8) where ε > 0 is sufficiently small so that ϕ 0 ∈ (π, 3π/2) and we also set and , 2π]}, with the anticlockwise orientation (see Figure 3), for j ≥ J. In fact j ≥ J if and only if Im σ j < 0, which gives another intuition regarding the meaning of the winding index J = J(r, θ, k), introduced in (1.25). This implies that, for large j, the closed contour . . , σ j }, see Figure 3.
If fact, we can say more: the contour Γ j − ∪ Γ j + is separated from the poles σ l 's by a positive distance. Namely, there exists j 0 ∈ Z such that for all ϕ ∈ [π, 2π], l ≥ J and j ≥ j 0 , since for sufficiently large j and every l ≥ J as j → ∞, where the last convergence can be verified by a direct calculation using (5.9) and the fact that |σ j | 2 = ln 2 r + (2jπ + ∆ k ) 2 1 + a 2 , recall (5.6). This shows (5.10), see also Figure 3. In Step 1 below we deduce from (5.10) that We also show, in Step 2 below, that our choice of ϕ 0 (5.8) and the equivalent form (5.5) of f give that where we used (5.12), (5.11) in the third line, (5.7) in the fourth line, and noted that (a+i)σ l +i∆ k = ln r − 2πil in the fifth line.
It therefore remains to verify (5.11), (5.12), which we discuss in Steps 1 and 2 below.
The claim (5.15) now follows by writing Remark 5.1. We note that for z ∈ Σ m (for some m) the only difference in the above contour integration is the fact that the contour in (5.3) passes through the pole. (Recall (5.6) that Im σ j = 0 if and only if (5.2) fails.) However, since the pole is simple, such integral equals the average of the integrals along contours that "avoid" the pole from each side. This shows that the velocity field v on Σ m (t) equals the average of the limit velocities from the two sides of Σ m (t), as mentioned below Theorem 1.1.