Circular flows for the Euler equations in two-dimensional annular domains

In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs are based on the study of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function, which is shown to satisfy a semilinear elliptic equation in the whole domain. In exterior or punctured domains, the method of moving planes is applied to some almost circular domains located between some streamlines of the flow, and the radial symmetry of the stream function is shown by a limiting argument. The paper also contains two Serrin-type results in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary and the domains are then proved to be disks or annuli.


Introduction and main results
Throughout this paper, | | denotes the Euclidean norm in R 2 and, for 0 ≤ a < b ≤ ∞, Ω a,b denotes the two-dimensional domain defined by Ω a,b = x ∈ R 2 : a < |x| < b .
When a < b are two positive real numbers, Ω a,b is a bounded smooth annulus. When 0 < a < b = ∞, Ω a,∞ is an exterior domain which is the complement of a closed disk. When 0 = a < b < ∞, Ω 0,b is a punctured disk. When 0 = a < b = ∞, Ω 0,∞ is the punctured plane R 2 \{0}, where we denote 0 = (0, 0) with a slight abuse of notation.
In Ω a,b , we consider steady flows v = (v 1 , v 2 ) of an inviscid fluid, solving the system of the Euler equations: where the solutions v and p are always understood in the classical sense, that is, they are (at least) of class C 1 in Ω a,b and therefore satisfy (1.1) everywhere in Ω a,b . We always assume rigid wall boundary conditions, that is, v is (at least) continuous up to the regular parts of ∂Ω a,b and tangential there: v · e r = 0 on C a if a > 0, v · e r = 0 on C b if b < ∞. (1.2) A flow v in Ω a,b is called a circular flow if v(x) is parallel to the vector e θ (x) at every point x ∈ Ω a,b , that is, v · e r = 0 in Ω a,b . The main goal in this paper is to show that, under some conditions, the flow is a circular flow and thus inherits the radial symmetry of the domain Ω a,b . We obtain such results in the four cases 0 < a < b < ∞, 0 < a < b = ∞, 0 = a < b < ∞, and 0 = a < b = ∞.
We also consider Euler flows in simply or doubly connected bounded domains whose boundaries are free: under the additional condition that the flow has constant norm on the boundary, we show that both the domain and the flow are then circular.
The case of bounded smooth annuli Ω a,b with 0 < a < b < ∞ The first result is concerned with flows having no stagnation point in the closed annulus Ω a,b . Throughout the paper, the stagnation points of a flow v are the points x for which |v(x)| = 0. Theorem 1.1 Assume 0 < a < b < ∞. Let v be a C 2 (Ω a,b ) flow solving (1.1)-(1.2) and such that |v| > 0 in Ω a,b . Then v is a circular flow, and there is a C 2 ([a, b]) function V with constant strict sign such that v(x) = V (|x|) e θ (x) for all x ∈ Ω a,b .
It actually turns out that the assumption |v| > 0 in Ω a,b can be slightly relaxed. Namely, if |v| > 0 in the open annulus Ω a,b and if the set of stagnation points is assumed to be properly included in one of the connected components of ∂Ω a,b , then the same conclusion holds, and then in fact |v| > 0 in Ω a,b . This is the purpose of the following result. Theorem 1.2 Assume 0 < a < b < ∞. Let v be a C 2 (Ω a,b ) flow solving (1.1)-(1.2) and such that x ∈ Ω a,b : |v(x)| = 0 C a or x ∈ Ω a,b : |v(x)| = 0 Then |v| > 0 in Ω a,b and the conclusion of Theorem 1.1 holds.
Theorem 1.2 is clearly stronger than Theorem 1.1, but we preferred to state Theorem 1.1 separately since the assumption is simpler.
Several further comments are in order. First of all, despite the fact that Ω a,b is not simply connected, the flow v has a stream function u : Ω a,b → R of class C 3 (Ω a,b ) defined by ∇ ⊥ u = v, that is, ∂u ∂x 1 = v 2 and ∂u ∂x 2 = −v 1 (1.4) in Ω a,b , since v is divergence free and tangential on C a . Notice that the stream function u is uniquely defined in Ω a,b up to an additive constant. Theorems 1.1 and 1.2 can then be viewed as Liouville-type symmetry results since their conclusion means that the stream function u is radially symmetric (and strictly monotone with respect to |x| in Ω a,b ). Furthermore, if for x in Ω a,b one calls ξ x the solution of ξ x (t) = v(ξ x (t)), the conclusion of Theorems 1.1 and 1.2 then implies that each function ξ x is defined in R and periodic, and that the streamlines Ξ x = ξ x (R) of the flow are concentric circles. Theorems 1.1 and 1.2 also mean equivalently that any C 2 (Ω a,b ) non-circular flow for (1.1)-(1.2) must either have a stagnation point in the open annulus Ω a,b , or must have stagnation points in both circles C a and C b , or in the whole circle C a , or in the whole circle C b .
Without the assumption |v| > 0 in Ω a,b or the weaker one (1.3), the conclusion of Theorems 1.1 and 1.2 obviously does not hold in general, in the sense that there are non-circular flows which do not fulfill (1.3). To construct such flows explicitly, we first point out that, for any continuous function f : R → R and any non-radial C 2 (Ω a,b ) solution u of ∆u + f (u) = 0 (1.6) in Ω a,b which is constant on C a and on C b and which has a critical point in Ω a,b , the C 1 (Ω a,b ) field v = ∇ ⊥ u is a non-circular solution of (1.1)-(1.2) with a stagnation point in Ω a,b : notice indeed that v = ∇ ⊥ u satisfies the boundary condition v · e r = −∇u · e θ = 0 on ∂Ω a,b since u is constant on C a and on C b , and v solves (1.1) with pressure where F = f . As an example, let λ ∈ R and ϕ ∈ C ∞ ([a, b]) be the principal eigenvalue and the principal eigenfunction of the eigenvalue problem −ϕ (r) − r −1 ϕ (r) + r −2 ϕ(r) = λ ϕ(r) in [a, b] with ϕ > 0 in (a, b) and Dirichlet boundary condition ϕ(a) = ϕ(b) = 0 (the principal eigenvalue λ is unique and the principal eigenfunction ϕ is unique up to multiplication by positive constants). The C ∞ (Ω a,b ) function u defined by u(x) = ϕ(|x|) x 1 /|x| (that is, u(x) = ϕ(r) cos(θ) in the usual polar coordinates) satisfies ∆u + λu = 0 in Ω a,b and it has some critical points in Ω a,b (since min Ω a,b u < 0 < max Ω a,b u and u = 0 on ∂Ω a,b ). Actually, it can easily be seen that ϕ has only one critical point in [a, b] and that u has exactly 6 critical points in Ω a,b (2 in Ω a,b , 2 on C a , and 2 on C b ). Then the C ∞ (Ω a,b ) flow v = ∇ ⊥ u is a non-circular flow solving (1.1)-(1.2) and having 2 stagnation points in Ω a,b and 4 on ∂Ω a,b . However, we do not know whether the hypothesis (1.3) could be more relaxed for the conclusion of Theorems 1.1 and 1.2 to still hold. For instance, would it be sufficient to assume that v has no stagnation point in Ω a,b ? We refer to Remark 2.5 in Section 2 below for further comments on this question.
On the other hand, we point out that the sufficient conditions |v| > 0 in Ω a,b or the more general one (1.3) are obviously not equivalent to being a circular flow, in the sense that there are circular flows for (1.1)-(1.2) which do not fulfill (1.3) (besides the trivial flow v = (0, 0)!). Actually, any C 1 (Ω a,b ) circular flow v(x) = V (|x|) e θ (x) solving (1.1)-(1.2) and for which V ∈ C 1 ([a, b]) does not have a constant strict sign, has a set of stagnation points containing at least a circle. For instance, let µ ∈ R and φ ∈ C ∞ ([a, b]) be the principal eigenvalue and the principal eigenfunction of the eigenvalue problem with φ > 0 in (a, b) and Dirichlet boundary condition φ(a) = φ(b) = 0, and let u = φ(| · |).
Then v = ∇ ⊥ u = φ (| · |) e θ is a C ∞ (Ω a,b ) non-trivial circular flow solving (1.1)-(1.2) with pressure p(x) = −φ (|x|) 2 /2 − µφ(|x|) 2 /2 and with a circle of stagnation points in Ω a,b : more precisely, if r * ∈ (a, b) denotes a real number such that φ(r * ) = max [a,b] φ (it is easy to see that r * is the only critical point of φ in [a, b]), then the set of stagnation points of the flow v is equal to the whole circle C r * .
Lastly, the assumption on the C 2 (Ω a,b ) smoothness of v is a technical assumption which is used in the proof. It warrants the C 1 smoothness of the vorticity ω = ∂v 2 ∂x 1 − ∂v 1 ∂x 2 , satisfying v · ∇ω = 0 in Ω a,b , and the C 1 smoothness of the vorticity function f arising in the semilinear elliptic equation of the type (1.6) satisfied by the stream function u. We refer to the proofs in Section 2 and especially Lemma 2.4 below for further details.
As for Theorems 1.1 and 1.2, the conclusion of Theorem 1.3 says that the stream function u is radially symmetric and strictly monotone with respect to |x| in Ω a,∞ , and that the streamlines of the flow v are concentric circles.
As far as the behavior of v at infinity is concerned, we do not know what could be the critical behavior of v(x) · e r (x) as |x| → +∞, or another type of asymptotic condition at infinity, for the conclusion of Theorem 1.3 to hold. However, we can say that without the condition (1.8) the conclusion of Theorem 1.3 does not hold in general. For instance, consider the C ∞ (Ω a,∞ ) function u defined by u(x) = 2(|x| 2 /a 2 − 1) + (|x|/a − a/|x|)x 1 /|x|, that is, in the usual polar coordinates. The function u satisfies ∆u − 8/a 2 = 0 in Ω a,∞ with Dirichlet boundary condition u = 0 on C a , and the C ∞ (Ω a,∞ ) field v = ∇ ⊥ u satisfies (1.1)-(1.2) with pressure p = −|v| 2 /2 + 8u/a 2 . In the usual polar coordinates, the field v is given by v = 4r (1.9) It satisfies condition (1.7) (and even inf Ωa, and v is not a circular flow. However, since u(x) → +∞ as |x| → +∞ and u = 0 on C a and since u has no critical point, it is easily seen that all solutions ξ x of (1.5) are defined in R and periodic and that all streamlines Ξ x = ξ x (R) (which are level sets of u) surround the origin. 2 Nevertheless, the streamlines do not converge to any family of circles at infinity since a calculation yields max y∈Ξx |y| − min y∈Ξx |y| = max R |ξ x (·)| − min R |ξ x (·)| → a/2 > 0 as |x| → +∞. We point out that, in Theorem 1.3, the flow v is not assumed to be bounded. Actually, there are unbounded circular flows satisfying all assumptions of Theorem 1.3: consider for instance the C ∞ (Ω a,∞ ) unbounded circular flow v defined by v(x) = |x| e θ (x), solving (1.1)-(1.2) with stream function u(x) = |x| 2 /2 and pressure p(x) = |x| 2 /2, and satisfying inf Ωa,∞ |v| = a > 0.
Notice lastly that the condition (1.7) is fulfilled in particular when inf Ωa,∞ |v| > 0. Furthermore, as soon as |v| > 0 on C a (that holds if inf Ωa,∞ |v| > 0), the boundary condition (1.2) and the continuity of v imply in particular that v · e θ has a constant strict sign on C a . Under the condition inf Ωa,∞ |v| > 0, the following result then provides some estimates on the infimum or the supremum of the vorticity ∂v 2 ∂x 1 − ∂v 1 ∂x 2 in Ω a,∞ , in terms of the sign of v · e θ on C a .
The flow v given by (1.9) is an example of a flow satisfying the assumptions of Theorem 1.4, with v · e θ > 0 on C a , and for which the vorticity (namely ∆u) is actually equal to the positive constant 8/a 2 everywhere in Ω a,∞ . Theorem 1.4 can also be viewed as a Liouville-type result. Namely, we show in its proof that, if inf Ωa,∞ |v| > 0, if v ·e θ > 0 on C a , and if the vorticity is nonpositive everywhere in Ω a,∞ , then v is a circular flow of the type v = V (| · |) e θ with V : [a, +∞) → [η, +∞) for some η > 0. Therefore, the vorticity ∂v 2 ∂x 1 (x)− ∂v 1 ∂x 2 (x) = V (|x|)+V (|x|)/|x| can not be nonpositive everywhere (since otherwise the function r → r V (r) (≥ ηr) would be nonincreasing in [a, +∞), leading to a contradiction). 3 Notice that Theorem 1.4 does not hold good if the assumption inf Ωa,∞ |v| > 0 is dropped. There are actually some circular flows v satisfying (1.1)-(1.2) such that |v| > 0 in Ω a,∞ and v ·e θ > 0 on C a , but inf Ωa,∞ |v| = 0 and for which the vorticity is negative everywhere. Consider for instance the C ∞ (Ω a,∞ ) circular flow v(x) = 1 |x| 2 e θ (x), solving (1.1)-(1.2) with stream function u(x) = −1/|x| and pressure p(x) = −1/(4|x| 2 ): one has |v| > 0 in Ω a,∞ and v · e θ > 0 on C a , but inf Ωa,∞ |v| = 0 and ∂v 2 Notice that the condition lim ε→0 Cε |v · e r | = 0 is fulfilled in particular if v(x) · e r (x) = o(1/|x|) as |x| > → 0. We do not know what could be the critical behavior of v ·e r at 0, or another type of asymptotic condition at the origin, for the conclusion of Theorem 1.5 to hold. However we can say that, without the condition lim ε→0 Cε |v · e r | = 0, the conclusion of Theorem 1.5 does not hold in general. Let us give a counter-example similar to (1.9) above (which was there defined in Ω a,∞ ). More precisely, consider the in the usual polar coordinates. The function u satisfies ∆u = 0 in Ω 0,b \ {0} with Dirichlet boundary condition u = 0 on C b , and the C ∞ with pressure p = −|v| 2 /2 (and vorticity equal to 0). In the usual polar coordinates, the field v is given by It has only two stagnation points in Ω 0,b \ {0} and they both lie on C b . Hence, the first part of condition (1.10) is fulfilled. But Cε |v · e r | = 4(ε/b − b/ε) → 0 as ε > → 0, and v is not a circular flow.
Lastly, in Theorem 1.5, the flow v is not assumed to be bounded. Actually, there are unbounded circular flows satisfying all assumptions of Theorem 1.5: consider for instance the The same observation holds good in a smooth annulus Ω a,b with 0 < a < b < ∞.
The case of the punctured plane Ω 0,∞ The last geometric configuration considered in the paper is that the punctured plane Theorem 1.7 Let v be a C 2 (Ω 0,∞ ) flow solving (1.1) and such that |v| > 0 in Ω 0,∞ and lim inf |x|→+∞ |v(x)| > 0. Assume moreover that v(x) · e r (x) = o 1 |x| as |x| → +∞ and Cε |v · e r | → 0 as ε > → 0, (1.13) and that the flow has one streamline which is a Jordan curve surrounding the origin. Then v is a circular flow. Furthermore, there is a C 2 ((0, +∞)) function V with constant strict sign The conclusion says that, under some conditions on |v| and under the same conditions as in Theorems 1.3 and 1.5 on the behavior of the radial component of v at infinity and at the origin, the existence of a streamline surrounding the origin implies that all streamlines surround the origin and are actually all concentric circles. Remark 1.8 Let us mention here other rigidity results for the stationary solutions of (1.1) in various geometrical configurations. The analyticity of the streamlines under a condition of the type v 1 > 0 in the unit disk was shown in [16]. The local correspondence between the vorticities of the solutions of (1.1) and the co-adjoint orbits of the vorticities for the non-stationary version of (1.1) in more general annular domains was investigated in [8]. In a previous paper [12] (see also [13]), we considered the case of a two-dimensional strip with bounded section and the case of bounded flows in a half-plane, assuming in both cases that the flows v are tangential on the boundary and that inf |v| > 0: all streamlines are then proved to be lines which are parallel to the boundary of the domain (in other words the flow is a parallel flow). Earlier results by Kalisch [15] were concerned with flows in two-dimensional strips under the additional assumption v · e = 0, where e is the main direction of the strip. Lastly, in [14], we considered the case of the whole plane R 2 and we showed that any C 2 (R 2 ) bounded flow v is still a parallel flow under the condition inf R 2 |v| > 0.

Some Serrin-type free boundary problems with overdetermined boundary conditions
The last main results on the solutions of the Euler equations (1.1) are two Serrin-type results in smooth simply or doubly connected bounded domains whose boundaries are free but on which the flow is assumed to satisfiy an additional condition.

Theorem 1.9
Let Ω be a C 2 non-empty simply connected bounded domain of R 2 . Let v ∈ C 2 (Ω) satisfy the Euler equations (1.1) and assume that v · n = 0 and |v| is constant on ∂Ω, where n denotes the outward unit normal on ∂Ω. Assume moreover that v has a unique stagnation point in Ω. Then, up to a shift, In the proof, we will show that the C 3 (Ω) stream function u defined by (1.4) satisfies a semilinear elliptic equation ∆u+f (u) = 0 in Ω. Furthermore, up to normalization, the function u vanishes on ∂Ω and is positive in Ω. Lastly, since |v| is assumed to be constant along ∂Ω, the normal derivative ∂u ∂n of u along ∂Ω is constant. This problem is therefore an elliptic equation with overdetermined boundary conditions. Since the celebrated paper by Serrin [22], it has been known that these overdetermined boundary conditions on ∂Ω determine the geometry of Ω, namely, Ω is then a ball and the function u is radially symmetric (hence, here, v would then be a circular flow). The proof is based on the method of moving planes developed in [3,6,10,22] and on the maximum principle, and it relies on the Lipschitz continuity of the function f . In our case, the function f is given in terms of the function u itself and it may not be Lipschitz continuous on the whole range [0, max Ω u]. More precisely, it may not be Lipschitz continuous in a neighborhood of the maximal value max Ω u. One therefore has to adapt the proof to this case by removing small neighborhoods of size ε around the maximal point of u (which is the unique stagnation point of v): one shows the symmetry of the domain in all directions up to ε and one concludes by passing to the limit as ε > → 0. In connection with Theorems 1.5 and 1.9, we state the following conjecture. Conjecture 1.10 Let D be an open non-empty disk and let z ∈ D. Let v be a C 2 (D \{z}) and bounded flow solving (1.1) and v · n = 0 on ∂D, where n denotes the outward unit normal on ∂D. Assume that |v| > 0 in D\{z}. Then z is the center of the disk and the flow is circular with respect to z.
Up to shift, one can assume that D = Ω 0,b for some b ∈ (0, +∞), hence n = e r on ∂D. If the point z is a priori assumed to be the center of the disk, namely the origin, then Theorem 1.5 implies that v is a circular flow. Up to rotation, assume now that z = (α, 0) for some α ∈ (0, b) and, without loss of generality, that the stream function u is positive in D\{z} and vanishes on ∂D. The goal would be to reach a contradiction. As far as Theorem 1.9 is concerned, the method of proof described in the paragraph following the statement shows simultaneously the symmetry of the domain and the symmetry of the function u (which obeys an equation of the type ∆u+f (u) = 0), thanks to the overdetermined boundary conditions satisfied by u. Here in Conjecture 1.10, the same technics based on the method of moving method implies for instance on the one hand that the function u is even in x 2 in Ω 0,b \{z}, and on the other hand that u(x 1 , x 2 ) < u(2α − x 1 , x 2 ) for all (x 1 , x 2 ) ∈ Ω 0,b such that x 1 > α. But, regarding the second property, the Hopf lemma might not apply to the function ( at the point z = (α, 0) since the vorticity function f might not be Lipschitz continuous around the limiting value of u at z (see also Remark 2.5 below, and notice that u is not differentiable at z, unless one further assumes that |v(x)| → 0 as x → z). Therefore, the same arguments as the ones in the proof of Theorem 1.9 do not lead to an obvious contradiction if z is not the center of the disk. However, Conjecture 1.10 seems natural and will be the purpose of further investigation.
A related weaker conjecture (with stronger assumptions) can also be formulated: if D is an open non-empty disk, if z ∈ D, if v ∈ C 2 (D) solves (1.1), if v · n = 0 on ∂D and if z is the only stagnation point of v in D, then z is the center of the disk and v is circular with respect to z. For the same reasons as in the previous paragraph (since the vorticity function f might not be Lipschitz continuous around u(z)), the proof of that second conjecture not clear either.
The last main result of the paper is concerned with the case of doubly connected bounded domains.
Let v ∈ C 2 (Ω) satisfy the Euler equations (1.1). Assume that v · n = 0 on ∂Ω = ∂ω 1 ∪ ∂ω 2 , where n denotes the outward unit normal on ∂Ω, and that |v| is constant on ∂ω 1 and on ∂ω 2 . Assume moreover that |v| > 0 in Ω. Then ω 1 and ω 2 are two concentric disks and, up to shift, for some 0 < a < b < ∞ and v is a circular flow satisfying the conclusion of Theorem 1.1 in Ω = Ω a,b .
In this case, by using similar arguments as in the proof of Theorem 1.1 in smooth annuli Ω a,b with 0 < a < b < ∞, it follows that the stream function u of the flow v satisfies a semilinear elliptic equation ∆u + f (u) = 0 in Ω, with u = c 1 on ∂ω 1 and u = c 2 on ∂ω 2 , for some real numbers c 1 = c 2 . Furthermore, min(c 1 , c 2 ) < u < max(c 1 , c 2 ) in Ω and ∂u ∂n is constant along ∂ω 1 and along ∂ω 2 . Since v has no stagnation point in Ω, the function f is then shown to be Lipschitz continuous in the whole interval [min(c 1 , c 2 ), max(c 1 , c 2 )], and known results of Reichel [18] and Sirakov [24] then imply that, up to shift, Ω = Ω a,b for some 0 < a < b < ∞, and u is radially symmetric.
Further symmetry results have been obtained for nonlinear elliptic equations of the type ∆u + f (u) = 0 or more general ones in exterior domains with overdetermined boundary conditions (see e.g. [1,19,24]), or in the whole space (see e.g. [11,17,23]), in both cases with further assumptions on the solution u at infinity and on the function f . Such conditions are in general not satisfied by the stream function u and the vorticity function f of a flow v that would be defined in the complement of a simply connected bounded domain or in the whole or punctured plane. Lastly, we refer to [5,9,20,21] for further references on overdetermined boundary value elliptic problems in domains with more complex topology or in unbounded epigraphs.

Outline of the paper
In Section 2, we prove Theorems 1.1 and 1.2 dealing with the case of bounded smooth annuli Ω a,b . Section 3 is devoted to the proof of Theorems 1.3 and 1.4 in the exterior domains Ω a,∞ . Sections 4 and 5 are concerned with the proof of Theorems 1.5 and 1.7 in the punctured disks Ω 0,b and in the punctured plane Ω 0,∞ . Lastly, the proof of the Serrin-type Theorems 1.9 and 1.11 is carried out in Section 6. The strategies of the proofs of Theorems 1.1, 1.2, 1.3, 1.4, 1.5 and 1.7 share some common features: we show some properties of the streamlines of the flow and we prove some symmetry results for the equation satisfied by the stream function u, after checking that this equation is well defined. However, the cases of the exterior domains Ω a,∞ and the punctured disks Ω 0,b and plane Ω 0,∞ involve some additional technicalities and require specific additional assumptions. They also require some further Liouville type results for the semilinear elliptic equations ∆u + f (u) = 0 in these domains. For the sake of clarity of the paper, that is why we preferred to first deal with the case of smooth annuli Ω a,b (with 0 < a < b < ∞) and to carry out the whole proof of Theorems 1.1 and 1.2 separately in Section 2. This section is devoted to the proof of Theorem 1.2 (we recall that Theorem 1.1 is a particular case of Theorem 1.2). Throughout this section, we consider two positive real numbers a < b and a C 2 (Ω a,b ) solution v of (1.1)-(1.2) satisfying (1.3), namely Before going into further details, let us first explain the general strategy of the proof of Theorem 1.2. As already mentioned in the introduction, since div v = 0 in the two-dimensional annulus Ω a,b and since v · e r = 0 on C a , there is a C 3 (Ω a,b ) stream function u : Ω a,b → R satisfying (1.4), that is, By definition, the stream function u is constant along the streamlines of the flow, parametrized by the solutions ξ x of (1.5). In order to show that the flow v is circular, one will show that the stream function u is radially symmetric, that is, there is a . Furthermore, since |v| is continuous and does not vanish in Ω a,b nor in the whole circle C a nor in the whole circle C b , the function V then has a constant strict sign in [a, b].
To show that u is radially symmetric, we will prove that, up to changing v into −v, u satisfies a semilinear elliptic equation of the type for some real numbers c 1 < c 2 and some C 1 ([c 1 , c 2 ]) function f . Lastly, we use a result of Sirakov [24] to complete the proof.
The first step of the proof consists in proving that u strictly ranges between its, different, values on C a and C b . Lemma 2.1 There are two real numbers c 1 = c 2 such that u = c 1 on C a , u = c 2 on C b , and Proof. First of all, since u satisfies (1.2), the C 3 (R) functions t → u(a cos t, a sin t) and t → u(b cos t, b sin t) are constant. There are then two real numbers c 1 and c 2 such that Since ∇u is (at least) Lipschitz-continuous in Ω a,b , each σ x is defined in a neighborhood of 0 and the quantities is the trajectory of the gradient flow in Ω a,b containing x. Notice that the functions σ x and u • σ x are of class ) meets the streamline of the flow v containing x (parametrized by the solution ξ x of (1.5)), orthogonally at x, since v = ∇ ⊥ u.
We now claim that, for each x ∈ Ω a,b , Assume first by way of contradiction that (2.4) does not hold. There exist then an increasing sequence of positive real numbers (t n ) n∈N and a point y ∈ Ω a,b such that t n → t + x and σ x (t n ) → y as n → +∞. Since y ∈ Ω a,b and the continuous field |∇u| = |v| does not vanish in Ω a,b by (1.3), there are three real numbers r > 0, η > 0 and τ > 0 such that Since σ x (t n ) → y as n → +∞, one has σ x (t n ) ∈ B(y, r/2) for all n large enough, hence σ x is defined in [t n − τ, t n + τ ] with σ x (t) ∈ B(y, r) ⊂ Ω a,b for all t ∈ [t n − τ, t n + τ ] and n large enough. This implies that t + x = +∞. Furthermore, for all n large enough, one has x , t + x ) and since t n → t + x = +∞ as n → +∞, one then gets that u(σ x (t)) → +∞ as t → t + x = +∞, contradicting the boundedness of u (u is continuous in the compact set Ω a,b ). Therefore, (2.4) has been proved. Similarly, (2.5) holds.
Finally, for each In both cases, one has Using again that u • σ x is increasing, one infers that min(c 1 , In particular, at t = 0, one concludes that min(c 1 , c 2 ) < u(x) < max(c 1 , c 2 ). This property holds for every x ∈ Ω a,b and the proof of Lemma 2.1 is thereby complete.
The next lemma shows that all streamlines of the flow which are included in the open annulus Ω a,b are closed and surround the origin. Lemma 2.2 For every x ∈ Ω a,b , the solution ξ x of (1.5) is defined in R and periodic. Furthermore, there are a continuous periodic function ρ x : R → (a, b) and a continuous function In other words, the Jordan curve ξ x (R) surrounds the origin.
Proof. Throughout the proof, we fix any . It then follows from Lemma 2.1 and the uniform continuity of u in Ω a,b that a < inf In the sequel, we call and u(ξ x (ϕ(n))) = u(y) for all n ∈ N. Furthermore, since |∇u(y)| = |v(y)| > 0 by (1.3) (remember that y ∈ Ω a,b ), there are some real numbers r > 0 and τ − < 0 < τ + such that B(y, r) ⊂ Ω a,b and Therefore, ξ x (ϕ(n)) ∈ ξ y ((τ − , τ + )) for all n large enough, that is, ξ x (ϕ(n)) = ξ y (τ n ) with τ n ∈ (τ − , τ + ) (notice in particular that this implies that the streamlines Ξ x and Ξ y coincide). Since for all n large enough and ϕ(2n 3) and (2.7), we claim that the non-empty open connected subset of R 2 surrounded by the curve Ξ x is not included in Ω a,b . Indeed, otherwise, the function u, which is constant on Ξ x , would have a critical point in this domain: that is impossible since |∇u| = |v| > 0 in Ω a,b . As a conclusion, the streamline Ξ x surrounds the origin. That implies that θ x (R) = R, where θ x : R → R is any continuous function given as in (2.6). In (2.6), the function ρ x = |ξ x | is necessarily continuous and periodic, as is ξ x . The proof of Lemma 2.2 is thereby complete. Remark 2.3 Lemma 2.2 was concerned with the description of the streamlines Ξ x when x belongs to the open annulus Ω a,b . If |v| > 0 on C a (resp. on C b ), then the boundary conditions (1.2) imply that, for any x ∈ C a (resp. x ∈ C b ), ξ x is still defined and periodic in R with Ξ x = C a (resp. Ξ x = C b ). If x ∈ ∂Ω a,b and |v(x)| = 0, then ξ x (t) = x for all t ∈ R and Ξ x = {x}. If x ∈ C a (resp. x ∈ C b ) with |v(x)| > 0 and if v has some stagnation points on C a (resp. C b ), then it is easy to see that ξ x is still defined in R, but it is not periodic anymore and Ξ x is a proper arc of C a (resp. C b ) which is open relatively to C a (resp. C b ).
Up to changing v into −v and u into −u, one can assume without loss of generality that the real numbers c 1 = c 2 given in Lemma 2.1 are such that (2.8) The last preliminary lemma is the derivation of an equation of the type (1.6) in Ω a,b .
Let us consider the first case only (the second one can be handled similarly). Then, let σ A be the solution of (2. The function σ A is of class C 1 ([0, t + A )) and, for every t ∈ (0, t + A ), one has σ A (t) ∈ Ω a,b with By setting σ A (t + A ) = A + and remembering that the field ∇u is (at least) continuous in Ω a,b , it follows that the function σ A : . On the other hand, the stream function u is constant along the streamline Ξ x , and the C 1 (Ω a,b ) vorticity ∂v 2 ∂x 1 − ∂v 1 ∂x 2 = ∆u satisfies v·∇(∆u) = 0 in Ω a,b from the Euler equations (1.1), hence ∆u is constant along the streamline Ξ x too. As a consequence, (2.10) yields Lastly, since the functions ∆u and f • u are (at least) continuous in Ω a,b , the equation ∆u + f (u) = 0 is satisfied every in Ω a,b and the proof of Lemma 2.4 is thereby complete.
Proof of Theorem 1.2. It follows from Lemmas 2.1 and 2.4, together with (2.8) that the function u is a classical solution of (2.1) for some real numbers c 1 < c 2 and some (at least) Lipschitz-continuous function f : [c 1 , c 2 ] → R. If then follows from [24, Theorem 5] 4 that u is radially symmetric and increasing with respect to |x| in Ω a,b . Therefore, there is a Lastly, since |v| is continuous in Ω a,b and does not vanish in Ω a,b nor in the whole circle C a nor in the whole circle C b , the function V then has a constant strict (positive) sign in [a, b]. The proof of Theorem 1.2 is thereby complete.
2), could the assumption (1.3) be slightly relaxed for v still to be necessarily a circular flow? As we mentioned in the introduction, the conclusion does not hold in general if v has stagnation points in Ω a,b . So a natural question is the following one: if |v| > 0 in Ω a,b , then is v a circular flow? It is easy to see from their proofs that Lemmas 2.1 and 2.2 hold good if (1.3) is replaced by |v| > 0 in Ω a,b . Consider then any point y ∈ Ω a,b . With the same notations as in Lemma 2.1, and assuming without loss of generality that c 1 < c 2 , there are some quantities t ± y such that −∞ ≤ t − y < 0 < t + y ≤ +∞ and the solution σ y of (2.3) with y instead of x is of class C 1 ((t − y , t + y )) and ranges in Ω a,b , with |σ y (t)| → a and u(σ y (t)) → c 1 as t → t − y , |σ y (t)| → b and u(σ y (t)) → c 2 as t → t + y . (2.11) is of class C 1 ((c 1 , c 2 )) and, since for every x ∈ Ω a,b the streamline Ξ x intersects σ y ((t − y , t + y )) by Lemma 2.2, the same arguments as in the proof of Lemma 2.4 imply that Furthermore, remembering from Lemma 2.2 that for each x ∈ Ω a,b , the C 1 solution ξ x of (1.5) is periodic and ranges in Ω a,b , we claim that (2.13) Indeed, otherwise, there would exist some sequences (x n ) n∈N in Ω a,b and (t n ) n∈N in R, and a point z such that a < |z| ≤ b together with |x n | → a and ξ xn (t n ) → z as n → +∞. Hence, u(x n ) → c 1 by Lemma 2.1 and the uniform continuity of u, while u(x n ) = u(ξ xn (t n )) → u(z) > c 1 by Lemma 2.1 again, leading to a contradiction. Therefore, (2.13) holds and, similarly, one has min t∈R |ξ x (t)| → b as |x| → b with |x| < b. Since the function ∆u is constant along any streamline of the flow from the Euler equations (1.1) and since ∆u is uniformly continuous in Ω a,b , it then follows from the previous observations and Lemma 2.2 that ∆u is constant on C a and constant on C b . Call d 1 and d 2 the values of ∆u on C a and C b , respectively, and set f (c 1 ) = −d 1 and f (c 2 ) = −d 2 . One then infers from (2.11) and (2.12) and that the equation ∆u + f (u) = 0 holds in the closed annulus Ω a,b (u is then a classical C 2 (Ω a,b ) solution of (2.1)). However, since and since |∇u(σ y (g −1 (τ )))| may converge to 0 as τ → c 1 or c 2 (this happens if |v| = 0 on C a or if |v| = 0 on C b ), it is not sure whether the function f is bounded in (c 1 , c 2 ) or not (it is not sure whether or not there exists a maximal curve σ X ((t − X , t + X )) lying in Ω a,b , for some X ∈ Ω a,b , along which |∇u| is bounded from below by a positive constant). The argument used in the proof of Theorem 1.2 to conclude that the solution u of (2.1) is radially symmetric relies on [24,Theorem 5], which itself uses the Lipschitz-continuity of f over the range of u. Thus, the same argument can not be applied as such in general in the case where v is just assumed to have no stagnation point in Ω a,b , without the assumption (1.3). Other arguments should then be used to prove that v is circular or to disprove this property in general. We leave this question open for a further work.

and 1.4
This section is devoted to the proof of Theorems 1.3 and 1.4. Throughout this section, we fix a positive real number a and we consider a C 2 (Ω a,∞ ) flow v solving (1.1)-(1.2) and such that for some positive real number η > 0. Notice that these conditions are fulfilled in both Theorems 1.3 and 1.4. The C 3 (Ω a,∞ ) stream function u given by (1.4) is well defined since v is divergence free and tangent on C a , and u satisfies Since ∇u · e θ = −v · e r = 0 on C a , the function u is constant on C a and, since u is unique up to an additive constant, one can also assume without loss of generality that We first show in Section 3.1 a preliminary lemma, namely Lemma 3.1 below, holding for both Theorems 1.3 and 1.4. It is concerned with the limit of u along the trajectory of the gradient flow starting from the point A. Then Sections 3.2 and 3.3 are devoted to the proof of Theorems 1.3 and 1.4. In Section 3.4, we do the proof of an independent lemma, Lemma 3.8 below, which is itself used not only in the proof of Theorems 1.3 and 1.4, but also for Theorems 1.5 and 1.7 as well as for the Serrin-type Theorem 1.9.

A preliminary common lemma
Let us consider here the trajectory of ∇u starting from the boundary point A satisfying (3.3)-(3.4). More precisely, let σ be the solution of (2.3) with x = A, that is, Proof. Since ∇u is (at least) of class C 1 (Ω a,∞ ) and ∇u(A) · e r (A) > 0 by (3.4), there is t * ∈ (0, +∞) such that σ is defined and of class C 1 at least in [0, t * ), and σ(s) ∈ Ω a,∞ for all 0 < s < t * . Define T = sup t > 0 : σ is defined and of class C 1 in [0, t) and σ((0, t)) ⊂ Ω a,∞ .
Assume now by way of contradiction that |σ(t)| does not converge to +∞ as t < → T . Then there are a sequence (t n ) n∈N in (0, T ) and a point y ∈ Ω a,∞ such that t n → T and σ(t n ) → y as n → +∞. As in the proof of Lemma 2.1, there are three positive real numbers r > 0, ρ > 0 and τ > 0 such that B(y, r) ⊂ Ω a,∞ , |v| ≥ ρ in B(y, r), and σ z is defined (at least) in [−τ, τ ] and ranges in B(y, r) for all z ∈ B(y, r/2). Owing to the definition of T , one gets that T = +∞ and u(σ(t n + τ )) ≥ u(σ(t n − τ )) + 2ρ 2 τ for all n large enough, hence u(σ(t)) → +∞ as t → +∞ since u • σ is increasing on [0, T ) = [0, +∞). This leads to a contradiction since u(σ(t n )) → u(y) ∈ R as n → +∞. Therefore, |σ(t)| → +∞ as t Consequently, u(σ(t)) → +∞ as t < → T , and the proof of Lemma 3.1 is thereby complete.  Let us describe in this paragraph the main scheme of the proof of Theorem 1.3. We first show that the stream function u is positive in Ω a,∞ and converges to +∞ at infinity (see Lemma 3.3 below). This implies that all streamlines of the flow surround the origin and we further show that the far streamlines converge to families of concentric circles at infinity (Lemma 3.4). Therefore, u satisfies a semilinear elliptic equation of the type ∆u + f (u) = 0 in Ω a,∞ with Dirichlet boundary conditions on C a , for some function f of class C 1 ([0, +∞)) (Lemma 3.6).
If some streamlines were true circles centered at the origin, then [24,Theorem 5] would imply that the stream function u is radially symmetric in the bounded region between C a and these streamlines. To circumvent the fact that the streamlines are not known to be true circles a priori, we use Lemmas 3.7 and 3.8 to compare the stream function u with its reflection with respect to some lines approximating any line containing the origin. We then proceed by passing to the limit as the approximation parameter goes to 0. With Lemma 3.8, it then easily follows that u is radially symmetric and that all streamlines are truly circular, thus completing the proof of Theorem 1.3. The first lemma is concerned with the positivity of u and with its limit at infinity. Proof. For every r ≥ a, the C 3 (R) function w r : θ → w r (θ) = u(r cos θ, r sin θ) is 2π-periodic and w r (θ) = r∇u(r cos θ, r sin θ) · e θ (r cos θ, r sin θ) = −r v(r cos θ, r sin θ) · e r (r cos θ, r sin θ) Furthermore, it follows from Lemma 3.1 that, for every r ≥ a, there is s r ∈ [0, T ) such that |σ(s r )| = r. Therefore, s r → T as r → +∞ (since σ is at least continuous in [0, T )) and u(σ(s r )) → +∞ by Lemma 3.1. Together with (3.9), there holds min Cr u → +∞ as r → +∞. In other words, u(x) → +∞ as |x| → +∞.
Let now R > a be any large real number such that min C R u > 0. Since u = 0 on C a and u has no critical point in Ω a,∞ , one gets that u > 0 in Ω a,R . Since R can be as large as wanted, one concludes that u > 0 in Ω a,∞ .
Before stating the next lemma on the property of all streamlines and the almost radial symmetry of the far streamlines, we recall that the streamlines of the flow can be parametrized by the solutions ξ x of (1.5).
Proof. Consider any x ∈ Ω a,∞ . From Lemma 3.3, there is r > |x| such that min Cr u > u(x) > 0. Since u = 0 on C a and u equal to the constant u(x) along Ξ x , it follows from the continuity of u that a < inf y∈Ξx |y| ≤ sup y∈Ξx |y| < r. Therefore, as in Lemma 2.2, the solution ξ x of (1.5) is defined in R and periodic, and the streamline Ξ x = ξ x (R) surrounds the origin.
Let us now introduce a few notations which will be used in this section, as well as in the proof of Theorems 1.4, 1.5, 1.7 and 1.9 in the following sections. For e ∈ S 1 = C 1 and λ ∈ R, we denote T e,λ = x ∈ R 2 : x · e = λ , H e,λ = x ∈ R 2 : x · e > λ , (3.14) and, for x ∈ R 2 , R e,λ (x) = x e,λ = x − 2(x · e − λ)e. (3.15) In other words, R e,λ is the orthogonal reflection with respect to the line T e,λ . For x ∈ Ω a,∞ , let Ω x denote the bounded connected component of R 2 \ Ξ x . Notice that Ω x is well defined and contains the origin, by Lemma 3.4. Notice also that u is equal to the positive constant u(x) along Ξ x , while u vanishes along C a and has no critical point in Ω a,∞ . Hence, where Ω x ∩ Ω a,∞ is the bounded domain located between Ξ x and C a . As a consequence, ∇u(z) points outwards Ω x at each point z ∈ Ξ x . The following lemma says that, for any ε > 0, the set Ω x ∩ H e,λ will be an admissible set for the method of moving planes for any e ∈ S 1 and λ > ε > 0, provided |x| is large enough.
Proof. Fix ε > 0, and assume by way of contradiction that the conclusion of the lemma does not hold. Then there are some sequences (x n ) n∈N in Ω a,∞ , (e n ) n∈N in S 1 , (λ n ) n∈N in (ε, +∞) and (y n ) n∈N such that lim n→+∞ |x n | = +∞, and y n ∈ H en,λn ∩ Ω xn and z n := R en,λn (y n ) ∈ Ω xn for all n ∈ N.
Since lim n→+∞ |y n − z n | = 0 and lim n→+∞ |y n | = lim n→+∞ |z n | = lim n→+∞ |x n | = +∞, one also infers that θ n − θ n → 0 as n → +∞. We also recall from (3.12) that Ξ xn = ( xn (θ) cos θ, xn (θ) sin θ) : θ ∈ R for all n large enough. It then follows from Lemma 3.4 and from the assumptions on y n and z n that |y n | ≤ xn (ϕ n + θ n ) and |z n | ≥ xn (ϕ n + θ n ) for all n large enough. Denote, for n large enough, y n = ( xn (ϕ n + θ n ) cos(ϕ n + θ n ), xn (ϕ n + θ n ) sin(ϕ n + θ n )) ∈ Ξ xn , z n = ( xn (ϕ n + θ n ) cos(ϕ n + θ n ), xn (ϕ n + θ n ) sin(ϕ n + θ n )) ∈ Ξ xn , and observe that y n ∈ (0, y n ] and z n ∈ (0, z n ]. We now claim that θ n = θ n for all n large enough. Indeed, otherwise, up to extraction of a subsequence, y n = z n and the four points 0, y n , y n = z n and z n would be aligned in that order. But since y n − z n = 2(y n · e n − λ n )e n with y n · e n − λ n > 0, the vectors y n and z n would Figure 2: The points y n , y n , z n , z n , and ζ n (with here e n = (1, 0) and ϕ n = 0) be parallel to e n . Hence, y n = (y n · e n )e n with y n · e n > λ n > ε > 0 and z n = (z n · e n )e n with z n · e n = 2λ n − y n · e n < λ n < y n · e n . This contradicts the fact that 0, y n and z n lie on the half-line R + e n in that order. Thus, θ n = θ n for all n large enough, thus for all n without loss of generality. Notice that the same arguments also imply that θ n = 0 and θ n = 0 for all n large enough (since otherwise in either case one would have θ n = θ n = 0 up to extraction of a subsequence), thus for all n without loss of generality. In particular, either 0 < θ n < π/2 or −π/2 < θ n < 0.
Assume first that, up to extraction of a subsequence, 0 < θ n < π/2 for all n. One then infers from the definition of z n = R enλn (y n ) and the previous paragraph that 0 < θ n < θ n < π 2 .
The second case, for which, up to extraction of a subsequence, −π/2 < θ n < 0 for all n (and then −π/2 < θ n < θ n < 0) can be handled similarly and leads to a contradiction as well. The proof of Lemma 3.7 is thereby complete.
Let us finally state the following important Lemma 3.8, that will be used in the proof of Theorems 1.3, 1.4, 1.5, 1.7 and 1.9.
Lemma 3.8 Let Ξ and Ξ be two C 1 Jordan curves surrounding the origin, and let Ω and Ω be the bounded connected components of R 2 \ Ξ and R 2 \ Ξ , respectively. Assume that Ω ⊂ Ω and let ω = Ω \ Ω be the non-empty and doubly connected domain located between Ξ and Ξ , with boundary ∂ω = Ξ ∪ Ξ .
Proof of Theorem 1.3. We shall show that the stream function u is radially symmetric in Ω a,∞ . Notice that we already know that u = 0 on C a . Let then x = y ∈ Ω a,∞ be such that |x| = |y| (> a).

Proof of Theorem 1.4
This section is devoted to the proof of Theorem 1.4. Instead of (3.1)-(3.2), we assume the stronger condition |v| = |∇u| ≥ η > 0 in Ω a,∞  We first show that all streamlines of the flow surround the origin and that the stream function u satisfies an equation of the type (1.6) in Ω a,∞ , with u > 0 in Ω a,∞ . Then, we prove that if (3.26) does not hold, namely if we assume by way of contradiction that the vorticity ∆u = ∂v 2 ∂x 1 − ∂v 1 ∂x 2 of the flow is nonpositive in Ω a,∞ , then a Kelvin transform of the variables applied to the stream function u leads to a semilinear heterogeneous equation ∆w + |x| −4 f (w) = 0 in the punctured disk Ω 0,1/a with a nonnegative function f . Lemma 3.8, using the good monotonicity of |x| −4 f (w) with respect to |x|, implies that w is radially symmetric, hence u is radially symmetric and the assumption ∆u ≤ 0 contradicts (3.4) and (3.24), leading to the desired conclusion.
As a first step of this scheme, let us consider the solutions ξ x of (1.5) with x ∈ Ω a,∞ and let us recall that Ξ x denotes the streamline of the flow containing x. Our goal is to show that E = [0, T ). To do so, we prove that E is not empty (it contains 0), open relatively to [0, T ) and that the largest interval containing 0 and contained in E is actually equal to [0, T ).
Note first that, since v · e r = 0 and v · e θ = 0 on C a , the streamline Ξ σ(0) = Ξ A is equal to the circle C a and it surrounds the origin. In other words, 0 ∈ E.
Let us now show that E is open relatively to [0, T ). Let s 0 ∈ E and denote x = σ(s 0 ) ∈ Ω a,∞ . By definition, the function ξ x is periodic, with some period T x > 0. Remember also that u is constant along each streamline of the flow. Therefore, as in Lemma 2.2, since v is (at least) continuous and |v(x)| = |∇u(x)| > 0, there are some real numbers r > 0 and τ ∈ (0, T x ) such that, for every y ∈ B(x, r) ∩ Ω a,∞ , there are some real numbers t ± y such that On the other hand, since ξ x (T x ) = ξ x (0) = x, the Cauchy-Lipschitz theorem provides the existence of a real number r ∈ (0, r] such that, for every z ∈ B(x, r ) ∩ Ω a,∞ , the function ξ z is defined (and of class C 1 ) at least on the interval [0, T x ] and ξ z (T x ) ∈ B(x, r) ∩ Ω a,∞ . Furthermore, by continuity of σ, there is ε > 0 such that s 0 + ε < T and As a consequence, for every s ∈ [max(0, s 0 − ε), s 0 + ε], the points z := σ(s) ∈ B(x, r ) ∩ Ω a,∞ and y := ξ z (T x ) ∈ B(x, r) ∩ Ω a,∞ satisfy u(z) = u(y), hence z ∈ B(x, r ) ∩ u −1 ({u(y)}) and z = ξ y (t) for some t ∈ (t − y , t + y ) (⊂ (−τ, τ )). Thus, ξ y (−T x ) = z = ξ y (t) and since |t| < τ < T x , the function ξ y is defined in R and (T x + t)-periodic. So is ξ z since z ∈ Ξ y . In other words, for every s ∈ [max(0, s 0 − ε), s 0 + ε], the function ξ σ(s) = ξ z is defined in R and periodic. One then concludes as in the last paragraph of the proof of Lemma 2.2 that Ξ σ(s) surrounds the origin. Finally, the set E is open relatively to [0, T ). Denote The previous paragraphs imply that 0 < T * ≤ T . The proof of Lemma 3.9 will be complete once we show that T * = T . Assume by way of contradiction that T * < T (in particular, T * is then a positive real number). Consider any increasing sequence (s n ) n∈N in (0, T * ) and converging to T * . Owing to the definition of T * , each function ξ σ(sn) is periodic and each streamline Ξ σ(sn) surrounds the origin. Furthermore, since each s n is positive and u • σ is increasing in [0, T ) and u is constant on C a ( A = σ(0)), each streamline Ξ σ(sn) is included in the open set Ω a,∞ . Consider now any n ∈ N and any point x ∈ Ξ σ(sn) . Since ∇u is (at least) Lipschitz-continuous and x ∈ Ω a,∞ , the solution σ x of (2.3) is defined and of class C 1 in at least a neighborhood of 0. Let Therefore, 0 < u(x) < u(σ(T * )). Furthermore,σ x (0) = ∇u(σ x (0)) = ∇u(x) is orthogonal to Ξ σ(sn) at x by definition of u. Since u • σ x is increasing on (t − x , t + x ), since u(σ x (0)) = u(x) > 0 with u = 0 on C a , and since Ξ σ(sn) surrounds the origin, it then follows as in the proof of Lemma 2.1 that |σ x (t)| → a as t → t − x and u(σ x (t)) > 0 for all t ∈ (t − x , t + x ). Then, for any t ∈ (t − x , 0), there holds By passing to the limit as t → t − x , one gets that |x| ≤ a + u(σ(T * ))/η. This property holds for any n ∈ N and any x ∈ Ξ σ(sn) , hence Lastly, consider the streamline Ξ σ(T * ) parametrized by t → ξ σ(T * ) (t). If there is a real number t such that |ξ σ(T * ) (t)| > M , then |ξ σ(sn) (t)| > M for all n large enough, by the Cauchy-Lipschitz theorem. Therefore, Ξ σ(T * ) ⊂ B M and, as in the proof of Lemma 2.2, it follows that ξ σ(T * ) is defined in R and periodic, and it surrounds the origin. In other words, T * ∈ E.
Since E is open relatively to [0, T ) from the previous paragraph, one is led to a contradiction with the definition of T * if T * < T . Eventually, T * = T and the proof of Lemma 3.9 is thereby complete.
The next lemma gives the same conclusion as the previous lemma, for any streamline. It also implies that each level set of the stream function u has only one connected component. Furthermore, for every x ∈ Ω a,∞ , the solution ξ x of (1.5) is defined in R and periodic, and Ξ x surrounds the origin. Lastly, Proof. Fix any R > a, and let C ∈ [0, +∞) be such that |u| ≤ C in Ω a,R . Since u(σ(t)) → +∞ as t < → T by Lemma 3.1, there is τ ∈ (0, T ) such that u(σ(s)) > C for all s ∈ (τ, T ), hence u(ξ σ(s) (t)) = u(σ(s)) > C and |ξ σ(s) (t)| > R for all s ∈ (τ, T ) and t ∈ R. Thus, min R |ξ σ(s) | > R for all s ∈ (τ, T ) (notice that the minimum is well defined by Lemma 3.9). This shows that min R |ξ σ(s) | → +∞ as s < → T . Consider now any point x ∈ Ω a,∞ , and let s ∈ (0, T ) be such that min R |ξ σ(s) | > |x|. Therefore, the point x belongs to the bounded open set surrounded by the Jordan curve Ξ σ(s) . It follows as in the proof of Lemma 2.2 that ξ x is defined in R and periodic, and that it surrounds the origin. Lemma 3.1 then implies that the streamline Ξ x crosses σ([0, T )): there are t ∈ R and s ∈ [0, T ) such that ξ x (t) = σ(s ) (notice that such a s is unique since u • σ is increasing in [0, T ) and u is constant along Ξ x ). One then gets that Ξ x = Ξ σ(s ) and x ∈ ∪ and both sets are equal since the other inclusion is obvious by definition. It only remains to show that min R |ξ x | → +∞ as |x| → +∞. Fix again any R ≥ a. From the first paragraph of the proof, there is s ∈ [0, T ) such that min R |ξ σ(s) | > R. Define R = max R |ξ σ(s) | (one has R > R). For any x with |x| > R , the streamlines Ξ x and Ξ σ(s) do not intersect, and both of them surround the origin. Therefore, min R |ξ x | > min R |ξ σ(s) |, hence min R |ξ x | > R for every |x| > R . The proof of Lemma 3.10 is thereby complete.
Furthermore, as in the proof of Lemmas 2.1 and 3.9, the solution σ x of (2.3) is defined in a maximal interval (t − x . For any t ∈ (t − x , 0), there holds u(σ x (t)) > 0 (by applying the result of previous paragraph at the point σ x (t) ∈ Ω a,∞ ). Therefore, for any t ∈ (t − x , 0), one has by passing to the limit as t > → t − x . Since this last inequality holds for any x ∈ Ω a,∞ , one concludes that u(x) → +∞ as |x| → +∞, and the proof of Lemma 3.11 is complete.
The last preliminary lemma provides the existence of a function f such that the elliptic equation (1.6) holds in Ω a,∞ .

Proof of Lemma 3.8
It is based on the method of moving planes developed in [3,6,10,22]. The idea is to compare the function ϕ to its reflection ϕ e,λ in ω e,λ by moving the lines T e,λ and decreasing λ from the value λ to the value ε. We recall that ω e,λ = (H e,λ ∩ ω) \ R e,λ (Ω ).
For any such λ ∈ [λ, λ * ], one then has Φ e,λ ≥ ≡ 0 on the boundary of each connected component of ω e,λ \K and one then infers from the choice of δ and from the strong maximum principle that Φ e,λ > 0 in ω e,λ \ K, and finally Φ e,λ > 0 in ω e,λ . This last property contradicts the definition of λ * . As a conclusion, λ * = ε. Therefore, for every λ ∈ (ε, λ), one has Φ e,λ > 0 in ω e,λ , namely ϕ < ϕ e,λ in ω e,λ and ϕ ≤ ϕ e,λ in ω e,λ by continuity of ϕ. As in the previous paragraph, it also follows by continuity that ϕ ≤ ϕ e,ε in ω e,ε . The proof of Lemma 3.8 is thereby complete. 4 The case of punctured disks Ω 0,b : proof of Theorem 1.5 This section is devoted to the proof of Theorem 1.5. Throughout this section, we fix a positive real number b and we consider a C 2 (Ω 0,b \{0}) flow v solving (1.1)-(1.2) and such that Since v is divergence free, together with the second condition in (4.1), it follows that there is hence v(B) · e θ (B) = 0 since v · e r = 0 on C b . Up to changing v into −v and u into −u, one can assume without loss of generality that v(B) · e θ (B) < 0, that is, Since ∇u · e θ = −v · e r = 0 on C b , the function u is constant on C b and, since u is unique up to an additive constant, one can also assume without loss of generality that Following the general scheme of the proof of Theorem 1.3, we will show that the function u is positive in Ω 0,b , that it has a limit at 0, that all streamlines of the flow in Ω 0,b surround the origin, and that u satisfies a semilinear elliptic equation of the type (1.6) in Ω 0,b \{0}. Finally, we will apply Lemma 3.8 in suitable domains to prove the radial symmetry of u.
The next lemma shows that u has the limit L at the origin and that u is positive in Ω 0,b . Proof. For every r ∈ (0, b], the C 3 (R) function w r : θ → w r (θ) = u(r cos θ, r sin θ) is 2πperiodic and, as in (3.8), one has w r (θ) = −r v(r cos θ, r sin θ) · e r (r cos θ, r sin θ) for all θ ∈ R. Hence, (4.1) implies that Furthermore, it follows from Lemma 4.1 that, for every r ∈ (0, b], there is s r ∈ [0, T ) such that |σ(s r )| = r. Therefore, s r → T as r > → 0 and u(σ(s r )) → L by Lemma 4.1. Together with (4.7), one gets that u(x) → L as |x| > → 0. Let now r ∈ (0, b) be any small real number such that min Cr u > 0. Since u = 0 on C b and u has no critical point in Ω 0,b , one gets that u > 0 in Ω r,b . Since r > 0 can be as small as wanted, one concludes that u > 0 in Ω 0,b .
Similarly, if L ∈ (0, +∞), then for any ε > 0, there is r ε ∈ (0, b) such that max Cr u < L + ε for all r ∈ (0, r ε ]. Then u < L + ε in Ω r,b for any such r, hence u < L + ε in Ω 0,b . Finally, since ε > 0 is arbitrary, one has u ≤ L in Ω 0,b and since u has no critical point in Ω 0,b , one concludes that u < L in Ω 0,b . The proof of Lemma 4.3 is thereby complete. Lemma 4.4 For each x ∈ Ω 0,b , the solution ξ x of (1.5) is defined in R and periodic, and the streamline Ξ x = ξ x (R) surrounds the origin. Furthermore, Proof. Consider any x ∈ Ω 0,b . Since u = 0 on C b and u equal to the constant u(x) ∈ (0, L) along Ξ x , it follows from the continuity of u and the previous lemma that 0 < inf y∈Ξx |y| ≤ sup y∈Ξx |y| < b. Therefore, as in Lemma 2.2, the solution ξ x of (1.5) is defined in R and periodic, and the streamline Ξ x = ξ x (R) surrounds the origin.
With the above lemmas in hand, we shall apply Lemma 3.8 to complete the proof of Theorem 1.5.
Since y · e > 0 and since Ω ⊂ B ε and R e,ε (Ω ) ⊂ B 3ε , it follows that y ∈ ω e,ε for all ε > 0 small enough. As a consequence, u(y) ≤ u(y e,ε ) = u(y − 2(y · e − ε)e) for all ε > 0 small enough and the passage to the limit as ε This section is devoted to the proof of Theorem 1.7. Let v be a C 2 (Ω 0,∞ ) flow solving (1.1) and such that |v| > 0 in Ω 0,∞ and lim inf |x|→+∞ |v(x)| > 0. One assumes that (1.13) holds and that there is X ∈ Ω 0,∞ such that the streamline Ξ X is a Jordan curve surrounding the origin.
Let Ω X be the bounded connected component of R 2 \ Ξ X . Thanks to the second part of assumption (1.13), there is a C 3 (Ω 0,∞ ) function u such that ∇ ⊥ u = v in Ω 0,∞ . Up to normalization, one can assume that u = 0 on Ξ X . Furthermore, since |∇u(X)| = |v(X)| > 0 and ∇u(X) is orthogonal to Ξ X at X, one can assume without loss of generality, up to changing v into −v and u into −u, that ∇u(X) points in the direction of Ω X at X.
Let then σ be the solution of (2.3) with x = X. Since ∇u is at least of class C 1 (Ω 0,∞ ), the function σ is defined in a neighborhood of 0 and there are −∞ ≤ T − < 0 < T + ≤ +∞ such that (T − , T + ) is the maximal interval in which σ is of class C 1 and σ((T − , T + )) ⊂ Ω 0,∞ . Furthermore, because of the normalization of the previous paragraph and since u • σ is increasing in (T − , T + ) and u is constant along Ξ X , one has σ(t) ∈ Ω X for all t ∈ (0, T + ), and σ(t) ∈ R 2 \ Ω X for all t ∈ (T − , 0).
6 Proof of the Serrin-type Theorems 1.9 and 1.11 We start in Section 6.1 with the proof of Theorem 1.11 dealing with the case of doubly connected bounded domains, since the proof follows easily from the arguments used in the proof of Theorems 1.1 and 1.2 and on some known results of Reichel [18] and Sirakov [24] on elliptic overdetermined boundary value problems. Section 6.2 is then devoted to the proof of Theorem 1.9 6.1 Proof of Theorem 1.11 Let ω 1 , ω 2 , Ω = ω 2 \ω 1 and v be as in Theorem 1.11. Since v is divergence free and v · n = 0 on ∂ω 1 , the C 3 (Ω) stream function u given by (1.4) is well defined and is unique up to additive constant. Furthermore, since v · n = 0 on ∂Ω = ∂ω 1 ∪ ∂ω 2 , there are two real numbers c 1 and c 2 such that u = c 1 on ∂ω 1 and u = c 2 on ∂ω 2 .
As in the proof of Lemma 2.1, one can show that, for each x ∈ Ω, the solution σ x of (2.3) is defined in an interval (t − x , t + x ) such that t − x < 0 < t + x , σ x ((t − x , t + x )) ⊂ Ω and dist(σ x (t), ∂Ω) → 0 as t → t ± x . Since u • σ x is increasing in (t − x , t + x ), it follows that c 1 = c 2 and min(c 1 , c 2 ) < u < max(c 1 , c 2 ) in Ω. Up to changing v into −v and u into −u, one can assume without loss of generality that c 1 < c 2 , hence c 1 < u < c 2 in Ω.
Let then X be any point in ω 1 . As in the proof of Lemma 2.2, each streamline Ξ x , with x ∈ Ω, surrounds the point X. Notice that, here, since v · n = 0 on ∂ω 1 ∪ ∂ω 2 and v has no stagnation point in Ω, both Jordan curves ∂ω 1 and ∂ω 2 are streamlines of the flow. Moreover, for an arbitrarily fixed point A ∈ ∂ω 1 , the same arguments as in the proof of Lemma 2.4 imply that the trajectory σ A of the gradient flow is defined in an interval [0, t + A ] with t + A ∈ (0, +∞), σ A ((0, t + A )) ⊂ Ω and σ A (t + A ) ∈ ∂ω 2 . The function g := u • σ A is then a C 1 diffeomorphism from [0, t + A ] onto [c 1 , c 2 ] and the C 1 ([c 1 , c 2 ]) function f defined by (2.9) is such that ∆u + f (u) = 0 along the curve σ A ([0, t + A ]) and finally in the whole set Ω since each streamline of the flow intersects the curve σ A ([0, t + A ]). Since | ∂u ∂n | = |∇u| = |v| along ∂Ω and since |v| is constant along ∂ω 1 and along ∂ω 2 , it follows that ∂u ∂n is constant too along ∂ω 1 and along ∂ω 2 . One concludes from [18,24] (see also [2,25]) that, up to shift, Ω = Ω a,b for some 0 < a < b < ∞ and u is radially symmetric and increasing with respect to |x| in Ω = Ω a,b . The assumptions and the conclusion of Theorem 1.1 are then satisfied and the proof of Theorem 1.11 is thereby complete.

Proof of Theorem 1.9
Let Ω be a C 2 non-empty simply connected bounded domain of R 2 . Let v ∈ C 2 (Ω) satisfy the Euler equations (1.1). We assume that v · n = 0 and |v| is constant on ∂Ω, where n denotes the outward unit normal on ∂Ω, and that v has a unique stagnation point in Ω. Since Ω is simply connected and v is divergence free, there is a C 3 (Ω) stream function u satisfying (1.4). Furthermore, u is constant along ∂Ω since v · n = 0 on ∂Ω. Up to normalization, one can assume without loss of generality that u = 0 on ∂Ω. (6.1) By assumption, the function u has a unique critical point in Ω, and |∇u| = |v| is constant along ∂Ω. Then | ∂u ∂n | = |∇u| = |v| > 0 on ∂Ω. Up to changing v into −v and u into −u, one can assume without loss of generality that ∂u ∂n = γ < 0 on ∂Ω (6.2) for some negative constant γ. Hence, u has a unique maximum point in Ω (which is actually in Ω) and this point is the unique critical point of u in Ω. Up to shift, one can assume without loss of generality that this critical point is the origin 0. One also infers from the uniqueness of the critical point of u that 0 < u < u(0) for all x ∈ Ω\{0}. (6.3) Our goal is to show that Ω is then a ball centered at the origin and that u is radially symmetric and decreasing with respect to |x| in Ω. To do so, we first follow some steps of the proof of Theorem 1.5.
So, let B be any point on ∂Ω. Since ∇u(B) · n(B) < 0 by (6.2) and since 0 is the unique critical point of u, it follows as in the proof of Lemma 4.1 that the solution σ ofσ(t) = ∇u(σ(t)) with σ(0) = B is defined in an interval [0, T ) with T ∈ (0, +∞], and σ((0, T )) ⊂ Ω \ {0} together with |σ(t)| → 0 as t < → T . Furthermore, since ∇u is at least Lipschitz continuous in Ω and |∇u(0)| = 0, one necessarily has T = +∞. Using (6.3) and the fact that Ω is simply connected, it follows as in the proof of Lemma 4.4 that, for each x ∈ Ω\{0}, the streamline Ξ x of the flow containing x surrounds the origin, and that max R |ξ x | → 0 as |x| > → 0. Notice that, since v · n = 0 and |v| > 0 on ∂Ω, the Jordan curve ∂Ω is also a streamline of the flow, surrounding the origin.
Remembering that u satisfies (6.1)-(6.3), it would then follow from [22] that Ω = B R for some R > 0 and u is radially symmetric and decreasing with respect to |x| in Ω, if the function f were known to be Lipschitz continuous in [0, L]. However, by using the same ideas as in Remark 2.5, it is not clear that the function f is bounded in a neighborhood of L and thus the function f may not be Lipschitz continuous in the whole interval [0, L]. We will however still be able to show the desired symmetry of Ω and of u by taking off from Ω small neighborhoods of 0 and applying Serrin's strategy and the method of moving planes in punctured domains. The images by u of the closure of these punctured domains are intervals of the type [0, L ], with L ∈ (0, L), and thus f is Lipschitz continuous in [0, L ].
More precisely, let first ρ > 0 be such that B ρ ⊂ Ω and let e be any unit vector. Let η be any real number in (0, ρ), and denote λ e = max x∈∂Ω x · e > ρ > η.
We will prove that only case a occurs.