Thierry Gallay
email: thierry.gallay@ujf-grenoble.fr

Vladimír Šverák
email: sverak@math.umn.edu

Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations *

Motivated by applications to vortex rings, we study the Cauchy problem for the threedimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity ω θ is integrable with respect to the two-dimensional measure dr dz, where (r, θ, z) denote the cylindrical coordinates in R 3 , we show the existence of a unique global solution, which converges to zero in L 1 norm as t → ∞. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between both situations. The solutions we construct have infinite energy in general, so that energy dissipation cannot be invoked to control the long-time behavior. We also treat the more general case where the initial vorticity is a finite measure whose atomic part is small enough compared to viscosity. Such data include point masses, which correspond to vortex filaments in the three-dimensional picture.

Introduction

Among all three-dimensional incompressible flows, axisymmetric flows without swirl form a particular class that is relatively simple to study and yet contains interesting examples, such as circular vortex filaments or toroidal vortex rings. For the evolutions defined by both the Euler and the Navier-Stokes equations, global well-posedness in that class was established almost fifty years ago by Ladyzhenskaya [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF] and Ukhovksii & Yudovich [START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF]. In the viscous case, the original approach of [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF][START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF] applies to velocity fields in the Sobolev space H 2 (R 3 ), see [START_REF] Leonardi | On axially symmetric flows in R 3[END_REF], but it is possible to obtain the same conclusions under the weaker assumption that the initial velocity belongs to H 1/2 (R 3 ) [START_REF] Abidi | Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes[END_REF]. In all these works, global existence for arbitrary large data is shown by combining the standard energy estimate, which holds for general solutions of the Navier-Stokes equations, with a priori bounds on the vorticity that are specific to the axisymmetric case.

In this paper, we revisit the Cauchy problem for the axisymmetric Navier-Stokes equations (without swirl) for the following reasons. First, motivated by a future study of vortex filaments, we wish to formulate a global well-posedness result involving scale invariant function spaces only. As was already mentioned, all previous works deal with finite energy solutions, and energy is not a scale invariant quantity for three-dimensional viscous flows. In particular, the longtime behavior of axisymmetric solutions has not been studied in terms of scale invariant norms.

Our second motivation is to emphasize the analogy between the axisymmetric case and the two-dimensional situation where the velocity field is planar and depends on two variables only. Indeed, we shall see that, if appropriate function spaces are used, local existence of solutions can be established in the axisymmetric case using literally the same proof as in the two-dimensional situation, which has been studied by many authors [START_REF] Ben-Artzi | Global solutions of two-dimensional Navier-Stokes and Euler equations[END_REF][START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF][START_REF] Kato | The Navier-Stokes equation for an incompressible fluid in R 2 with a measure as the initial vorticity[END_REF]. However, significant differences appear when one considers a priori estimates and long-time asymptotics.

To formulate our results, we introduce some notation. The time evolution of viscous incompressible flows is described by the Navier-Stokes equations

∂ t u + (u • ∇)u = ∆u -∇p , div u = 0 , (1)
where u = u(x, t) ∈ R 3 denotes the velocity field and p = p(x, t) ∈ R the pressure field. For simplicity, we assume throughout this paper that the kinematic viscosity and the fluid density are both equal to 1. We restrict ourselves to axisymmetric solutions without swirl for which the velocity field has the following particular form :

u(x, t) = u r (r, z, t)e r + u z (r, z, t)e z .

Here (r, θ, z) are the usual cylindrical coordinates in R 3 , defined by x = (r cos θ, r sin θ, z) for any x ∈ R 3 , and e r , e θ , e z denote the unit vectors in the radial, toroidal, and vertical directions, respectively :

e r =   cos θ sin θ 0   , e θ =   -sin θ cos θ 0   , e z =   0 0 1   . (3)
We emphasize that the "swirl" u • e θ is assumed to vanish identically. This means that the velocity field (2) is not only invariant under rotations about the vertical axis, but also under reflections by any plane containing the vertical axis.

A direct calculation shows that the vorticity ω = curl u associated with the velocity field (2) is purely toroidal : ω(r, z, t) = ω θ (r, z, t)e θ ,

where

ω θ = ∂ z u r -∂ r u z . ( 4
)
The flow is entirely determined by the single quantity ω θ , because the velocity field u can be reconstructed by solving the linear elliptic system

∂ r u r + 1 r u r + ∂ z u z = 0 , ∂ z u r -∂ r u z = ω θ , (5)
in the half space Ω = {(r, z) ∈ R 2 | r > 0 , z ∈ R}, with boundary conditions u r = ∂ r u z = 0 at r = 0. System [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF] is the differential formulation of the axisymmetric Biot-Savart law, which will be studied in more detail in Section 2 below. The evolution equation for ω θ reads

∂ t ω θ + u • ∇ω θ - u r r ω θ = ∆ω θ - ω θ r 2 , (6)
where u • ∇ = u r ∂ r + u z ∂ z and ∆ = ∂ 2 r + 1 r ∂ r + ∂ 2 z denotes the Laplace operator in cylindrical coordinates. Equation [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF] is considered in the half-plane Ω with homogeneous Dirichlet condition at the boundary r = 0. As was already observed in [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF][START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF], it is useful to consider also the related quantity η(r, z, t) = ω θ (r, z, t) r , [START_REF] Th | Stability and interaction of vortices in two-dimensional viscous flows[END_REF] which satisfies the advection-diffusion equation

∂ t η + u • ∇η = ∆η + 2 r ∂ r η , (8)
with homogeneous Neumann condition at the boundary r = 0. Systems [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF] and ( 8) are, of course, perfectly equivalent. In what follows we find it more convenient to work with the axisymmetric vorticity equation [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF], at least to prove local existence of solutions, but equation [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF] will be useful to derive a priori estimates and to study the long-time behavior. Throughout this paper, to emphasize the similarity with the two-dimensional case, we equip the half-plane Ω = {(r, z) ∈ R 2 | r > 0 , z ∈ R} with the two-dimensional measure dr dz, as opposed to the 3D measure r dr dz which could appear more natural for axisymmetric problems. Thus, given any p ∈ [1, ∞), we denote by L p (Ω) the space of measurable functions ω θ : Ω → R for which the following norm is finite :

ω θ L p (Ω) = Ω |ω θ (r, z)| p dr dz 1/p , 1 ≤ p < ∞ .
The space L ∞ (Ω) is defined similarly. Sometimes, however, it is more convenient to use the 3D measure r dr dz, and the corresponding spaces are then denoted by L p (R 3 ) to avoid confusion.

For instance, we define

η L p (R 3 ) = Ω |η(r, z)| p r dr dz 1/p , 1 ≤ p < ∞ .
We are now in position to state our first main result.

Theorem 1.1 For any initial data ω 0 ∈ L 1 (Ω), the axisymmetric vorticity equation (6) has a unique global mild solution

ω θ ∈ C 0 ([0, ∞), L 1 (Ω)) ∩ C 0 ((0, ∞), L ∞ (Ω)) . (9)
The solution satisfies ω θ (t) L 1 (Ω) ≤ ω 0 L 1 (Ω) for all t > 0, and

lim t→0 t 1-1 p ω θ (t) L p (Ω) = 0 , for 1 < p ≤ ∞ , (10)
lim t→∞ t 1-1 p ω θ (t) L p (Ω) = 0 , for 1 ≤ p ≤ ∞ . (11)
If, in addition, the axisymmetric vorticity is non-negative and has finite impulse :

I = Ω r 2 ω 0 (r, z) dr dz < ∞ , (12)
then lim t→∞ t 2 ω θ (r √ t, z √ t, t) = I 16 √ π r e -r 2 +z 2 4 , (r, z) ∈ Ω , (13)
where convergence holds in

L p (Ω) for 1 ≤ p ≤ ∞. In particular ω θ (t) L p (Ω) = O(t -2+ 1 p ) as t → ∞ in that case. Remark 1.2 A mild solution of (6) on R + = [0, ∞)
is a solution of the associated integral equation, namely Eq. (49) below. As will be verified in Section 4, both sides of (49) are welldefined if ω θ satisfies (9) and [START_REF] Th | Long-time asymptotics of the Navier-Stokes and vorticity equations on R 3[END_REF]. In the uniqueness claim, we only assume that ω θ satisfies (9) and is a mild solution of (6) for strictly positive times. The proof then shows that (10) automatically holds.

The statement of Theorem 1.1 has several aspects, and it is worth discussing them separately. The local well-posedness claim is certainly not surprising, because the class of initial data we consider is covered by at least two existence results in the literature. Indeed, if ω θ ∈ L 1 (Ω), it is easy to verify that the vorticity ω = ω θ e θ belongs to the Morrey space M 3/2 (R 3 ) defined by the norm ω M 3/2 = sup

x∈R 3 sup R>0 1 R B(x,R) |ω(x)| dx ,
where B(x, R) ⊂ R 3 denotes the ball of radius R > 0 centered at x ∈ R 3 . In addition ω can be approximated in M 3/2 (R 3 ) by smooth and compactly supported functions. As was proved by Giga & Miyakawa [START_REF] Giga | Navier-Stokes flow in R 3 with measures as initial vorticity and Morrey spaces[END_REF], the Navier-Stokes equations in R 3 thus have a unique local solution with initial vorticity ω, which is even global in time if the norm ω M 3/2 is sufficiently small. On the other hand, under the same assumption on the vorticity, one can show that the velocity field u given by the Biot-Savart law in R 3 belongs to the space BMO -1 (R 3 ) defined by the norm

u BMO -1 = sup x∈R 3 sup R>0 1 R 3 B(x,R) R 2 0 |e t∆ u| 2 dt dx 1/2
, where e t∆ denotes the heat semigroup in R 3 . In fact u can be approximated by smooth and compactly supported functions in BMO -1 (R 3 ), so that u ∈ VMO -1 (R 3 ). Thus we can also invoke the celebrated result by Koch & Tataru [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF] to obtain the existence of a unique local solution to the Navier-Stokes equation in R 3 , which is again global in time if the norm u BMO -1 is sufficiently small. In contrast to the general results in [START_REF] Giga | Navier-Stokes flow in R 3 with measures as initial vorticity and Morrey spaces[END_REF][START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF], the approach we follow to solve the Cauchy problem for Eq. ( 6) uses specific features of the axisymmetric case. As we shall see in Section 4, it is elementary and completely parallel to the two-dimensional situation which was studied e.g. in [START_REF] Ben-Artzi | Global solutions of two-dimensional Navier-Stokes and Euler equations[END_REF][START_REF] Th | Stability and interaction of vortices in two-dimensional viscous flows[END_REF].

The assertion of global well-posedness in Theorem 1.1 is also quite natural in view of the historical results by Ladyzhenskaya [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF] and Ukhovkii & Yudovich [START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF]. As in [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF][START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF] we use the structure of equation [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF] to derive a priori estimates on the quantity η in L p (R 3 ), for 1 ≤ p ≤ ∞. However, since the solutions we consider do not have finite energy in general, we cannot apply the classical energy estimate to obtain a uniform bound on the velocity field in L 2 (R 3 ). Instead we prove that any solution of [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF] with initial data ω 0 ∈ L 1 (Ω) satisfies ω θ (t) L 1 (Ω) ≤ ω 0 L 1 (Ω) for all t > 0 and sup

t>0 t ω θ (t) L ∞ (Ω) ≤ C ω 0 L 1 (Ω) , (14)
where C(s) = O(s) as s → 0. This new a priori estimate is scale invariant, and implies that all solutions of (6) with initial data in L 1 (Ω) are global. Using the axisymmetric Biot-Savart law, we also deduce the following optimal bound on the velocity field :

sup t>0 t 1/2 u(t) L ∞ (Ω) ≤ C ω 0 L 1 (Ω) . (15)
Our last comment on Theorem 1.1 concerns the long-time behavior, which differs significantly from what happens in the two-dimensional case. In the latter situation, the L 1 norm of the vorticity is non-increasing in time, but does not converge to zero in general (in particular, it is constant for solutions with a definite sign). The long-time behavior is described by self-similar solutions, called Oseen vortices, which have a nonzero total circulation [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF][START_REF] Th | Global Stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF]. In contrast, the axisymmetric vorticity ω θ vanishes on the boundary of the half-plane Ω, so that the L 1 norm is strictly decreasing for all nontrivial solutions. As we shall see in Section 6, this implies that ω θ (t) L 1 (Ω) → 0 when t → ∞, as asserted in [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF]. In other words, the long-time behavior of the axisymmetric vorticity is trivial when measured in scale invariant function spaces. More can be said when the initial data have a definite sign and a finite impulse, given by [START_REF] Giga | Navier-Stokes flow in R 3 with measures as initial vorticity and Morrey spaces[END_REF]. In that case, the axisymmetric vorticity ω θ inherits the same properties for all positive times and converges as t → ∞ to a self-similar solution of the linearized equation (35), whose profile is explicitly determined in [START_REF] Kato | Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions[END_REF].

As in the two-dimensional case, it is possible to extend the local existence claim in Theorem 1.1 to a larger class of initial data, so as to include finite measures as initial vorticities. Let M(Ω) denote the set of all real-valued finite measures on the half-plane Ω, equipped with the total variation norm

µ tv = sup Ω φ dµ φ ∈ C 0 (Ω) , φ L ∞ (Ω) ≤ 1 , for µ ∈ M(Ω) ,
where C 0 (Ω) is the set of all real-valued continuous functions on Ω that vanish at infinity and on the boundary ∂Ω. If µ ∈ M(Ω) is absolutely continuous with respect to Lebesgue's measure, then µ = ω θ dr dz for some ω θ ∈ L 1 (Ω), and µ tv = ω θ L 1 (Ω) . More generally, one can decompose any µ ∈ M(Ω) as µ = µ ac + µ sc + µ pp , where µ ac is absolutely continuous with respect to Lebesgue's measure, µ pp is a countable collection of Dirac masses, and µ sc has no atoms but is supported on a set of zero Lebesgue measure. In the original three-dimensional picture, each Dirac mass in the atomic part µ pp corresponds to a circular vortex filament, whereas vortex sheets are included in the singularly continuous part µ sc . The proof of Theorem 1.1 can be adapted to initial vorticities in M(Ω), and gives the following statement, which is our second main result.

Theorem 1.3 There exist positive constants ǫ and C such that, for any initial data ω 0 ∈ M(Ω) with (ω 0 ) pp tv ≤ ǫ, the axisymmetric vorticity equation (6) has a unique global mild solution

ω θ ∈ C 0 ((0, ∞), L 1 (Ω) ∩ L ∞ (Ω)) satisfying lim sup t→0 ω θ (t) L 1 (Ω) < ∞ , lim sup t→0 t 1/4 ω θ (t) L 4/3 (Ω) ≤ Cǫ , (16)
and such that ω θ (t) ⇀ ω 0 as t → 0. Moreover, the asymptotic estimates for t → ∞ given in Theorem 1.1 hold without change.

Observe that we now have a limitation on the size of the data, which however only affects the atomic part of the initial vorticity. This technical restriction inevitably occurs if local existence is established using a fixed point argument in scale invariant spaces, as we do in Section 4. In the two-dimensional case, early results by Giga, Miyakawa, & Osada [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF] and by Kato [START_REF] Kato | The Navier-Stokes equation for an incompressible fluid in R 2 with a measure as the initial vorticity[END_REF] had a similar limitation, which was then relaxed in [START_REF] Th | Global Stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF][START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF] using completely different techniques. In the axisymmetric situation, existence of a global solution to [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF] with a large Dirac mass as initial vorticity has recently been established by Feng and Šverák [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF], using an approximation argument, but uniqueness is still under investigation. For a general initial vorticity ω 0 ∈ M(Ω), both existence and uniqueness are open.

Even if we restrict ourselves to initial vorticities with a small atomic part, the uniqueness claim in Theorem 1.3 is probably not optimal. Indeed, although the solutions we construct satisfy both estimates in [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF], we believe that uniqueness should hold (as in the two-dimensional case) under the sole assumptions that ω θ (t) is uniformly bounded in L 1 (Ω) for t > 0 and converges weakly to the initial vorticity ω 0 as t → 0. However, technical difficulties arise when adapting the two-dimensional proof to the axisymmetric case, and for the moment we need an additional assumption, such as the second estimate in [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF], to obtain uniqueness. We hope to clarify that question in a future work. Remark 1.4 It is interesting to ask whether our results can be extended to the case where a non-zero axisymmetric forcing is included in the Navier-Stokes equations. For example, one can add a forcing term f = f (r, z, t) to the right-hand side of equation [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF]. If f is "sufficiently regular" and decays "sufficiently fast" as r + |z| + t → ∞, our results remain true, with quite straightforward modifications of the proofs. However, if we wish to obtain results under optimal, scale invariant assumptions such as

∞ 0 Ω |f (r, z, t)| dr dz dt < ∞ ,
non-trivial modifications seem to be needed. We thank the anonymous referee for raising this interesting point, which we plan to address in a future work.

The rest of this paper is organized as follows. In Section 2, which is devoted to the axisymmetric Biot-Savart law, we estimate various norms of the velocity field u in terms of the axisymmetric vorticity ω θ . In Section 3, we show that the semigroup generated by the linearization of (6) about the origin satisfies the same L p -L q estimates as the two-dimensional heat kernel. After these preliminaries, we prove the local existence claims in Theorems 1.1 and 1.3 in Sections 4.1 and 4.2, respectively. Global existence follows from a priori estimates which are established in Section 5. Finally, the long-time behavior is investigated in Section 6. We prove that all solutions of (6) converge to zero in L 1 (Ω), and we also compute the leading term in the long-time asymptotics for vorticities with a definite sign and a finite impulse.

Acknowledgements. This project started during visits of the first named author to the University of Minnesota, whose hospitality is gratefully acknowledged. Our research was supported in part by grants DMS 1362467 and DMS 1159376 from the National Science Foundation (V.S.), and by the grant "Dyficolti" ANR-13-BS01-0003-01 from the French Ministry of Research (Th.G.).

The axisymmetric Biot-Savart law

In this section, we assume that the axisymmetric vorticity ω θ : Ω → R is given, and we study the properties of the velocity field u = (u r , u z ) satisfying the linear elliptic system [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF]. The divergence-free condition ∂ r (ru r ) + ∂ z (ru z ) = 0 implies that there exists a function

ψ : Ω → R such that u r = - 1 r ∂ψ ∂z , u z = 1 r ∂ψ ∂r . (17)
As ∂ z u r -∂ r u z = ω θ , the axisymmetric stream function ψ satisfies the following linear elliptic equation in the half-space Ω :

-∂ 2 r ψ + 1 r ∂ r ψ -∂ 2 z ψ = rω θ . (18)
Boundary conditions for [START_REF] Leonardi | On axially symmetric flows in R 3[END_REF] are determined by observing that, for a smooth axisymmetric vector field u : R 3 → R 3 with div u = 0, the stream function defined by [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF] satisfies the asymptotic expansion

ψ(r, z) = ψ 0 + r 2 ψ 2 (z) + O(r 4 ) , as r → 0 , (19)
see [START_REF] Liu | Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation[END_REF] for an extensive discussion of these regularity issues. Without loss of generality, we can assume that the constant ψ 0 in ( 19) is equal to zero, in which case we conclude that ψ(0, z) = ∂ r ψ(0, z) = 0.

The solution of ( 18) with these boundary conditions is well known, see e.g. [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF]. If we assume that the vorticity ω θ decays sufficiently fast at infinity, we have the explicit representation

ψ(r, z) = 1 2π Ω √ rr F (r -r) 2 + (z -z) 2 rr ω θ (r, z) dr dz , (20)
where the function F : (0, ∞) → R is defined by

F (s) = π 0 cos φ dφ 2(1-cos φ) + s 1/2 = π/2 0 cos(2φ) dφ sin 2 φ + s/4 1/2 , s > 0 . ( 21
)
Useful properties of F are collected in the following lemma, whose proof can be found in [START_REF] Šverák | Selected Topics in Fluid Mechanics[END_REF]Section 19].

Lemma 2.1 The function F : (0, ∞) → R defined by ( 21) is decreasing and satisfies the asymptotic expansions :

i) F (s) = log 8 √ s -2 + O s log 1 s and F ′ (s) = - 1 2s + O log 1 s as s → 0; ii) F (s) = π 2s 3/2 + O 1 s 5/2 and F ′ (s) = - 3π 4s 5/2 + O 1 s 7/2 as s → ∞.
Remark 2.2 It follows in particular from Lemma 2.1 that the maps s → s α F (s) and s → s β F ′ (s) are bounded if 0 < α ≤ 3/2 and 1 ≤ β ≤ 5/2. These observations will be constantly used in the subsequent proofs.

Combining ( 17) and ( 20), we obtain explicit formulas for the axisymmetric Biot-Savart law :

u r (r, z) = Ω G r (r, z, r, z)ω θ (r, z) dr dz , u z (r, z) = Ω G z (r, z, r, z)ω θ (r, z) dr dz , (22)
where

G r (r, z, r, z) = - 1 π z - z r 3/2 r1/2 F ′ (ξ 2 ) , ξ 2 = (r -r) 2 + (z -z) 2 rr , (23)
G z (r, z, r, z) = 1 π r - r r 3/2 r1/2 F ′ (ξ 2 ) + 1 4π r1/2 r 3/2 F (ξ 2 ) -2ξ 2 F ′ (ξ 2 ) . (24)
Our first result gives elementary estimates for the axisymmetric Biot-Savart law in usual Lebesgue spaces. We emphasize the striking similarity with the corresponding bounds for the two-dimensional Biot-Savart law in the plane R 2 , see e.g. [8, Lemma 2.1].

Proposition 2.3

The following properties hold for the velocity field u defined from the vorticity ω θ via the axisymmetric Biot-Savart law [START_REF] Šverák | Selected Topics in Fluid Mechanics[END_REF].

i) Assume that 1 < p < 2 < q < ∞ and 1 q = 1 p -1 2 . If ω θ ∈ L p (Ω), then u ∈ L q (Ω) 2 and u L q (Ω) ≤ C ω θ L p (Ω) . (25) ii) If 1 ≤ p < 2 < q ≤ ∞ and ω θ ∈ L p (Ω) ∩ L q (Ω), then u ∈ L ∞ (Ω) 2 and u L ∞ (Ω) ≤ C ω θ σ L p (Ω) ω θ 1-σ L q (Ω) , where σ = p 2 q -2 q -p ∈ (0, 1) . ( 26
)
Proof. Both assertions follow from the basic estimate

|G r (r, z, r, z)| + |G z (r, z, r, z)| ≤ C (r -r) 2 + (z -z) 2 1/2 , (27)
which holds for all (r, z) ∈ Ω and all (r, z) ∈ Ω. At a heuristic level, estimate (27) follows quite naturally from the scaling properties of G r (r, z, r, z) and G z (r, z, r, z) if we observe that these functions behave for (r, z) close to (r, z) like the two-dimensional velocity field generated by a Dirac mass located at (r, z) in R 2 . To prove (27) rigorously, we first bound the radial component G r . We distinguish two cases : a) If r ≤ 2r, we use the fact that ξ 2 F ′ (ξ 2 ) is bounded, and obtain the estimate

|G r (r, z, r, z)| ≤ C |z -z| r 3/2 r1/2 rr (r -r) 2 + (z -z) 2 = C r1/2 r 1/2 |z -z| (r -r) 2 + (z -z) 2 ≤ C (r -r) 2 + (z -z) 2 1/2 , because r/r ≤ 2 and |z -z| ≤ (r -r) 2 + (z -z) 2 1/2 . b) If r > 2r, observing that ξ 3 F ′ (ξ 2 ) is bounded, we deduce |G r (r, z, r, z)| ≤ C |z -z| r 3/2 r1/2 r 3/2 r3/2 (r -r) 2 + (z -z) 2 3/2 = C r |z -z| (r -r) 2 + (z -z) 2 3/2 ≤ C (r -r) 2 + (z -z) 2 1/2 , because r < 2(r -r) ≤ 2 (r -r) 2 + (z -z) 2 1/2 . This proves estimate (27) for G r .
Similar calculations give the same bound for the second component G z too. Indeed, the first term in the right-hand side of (24) can be estimated exactly as above, using the obvious fact that |r -r| ≤ (r -r) 2 + (z -z) 2 1/2 . The second term involves the quantity F (ξ 2 ) -2ξ 2 F ′ (ξ 2 ), which is bounded by Cξ -1 in case a) and by Cξ -3 in case b). This concludes the proof of (27). Now, since the integral kernel G = (G r , G z ) in [START_REF] Šverák | Selected Topics in Fluid Mechanics[END_REF] satisfies the same bound as the twodimensional Biot-Savart kernel in R 2 , properties i) and ii) in Proposition 2.3 can be established exactly as in the 2D case. Estimate (25) thus follows from the Hardy-Littlewood-Sobolev inequality, and the bound (26) can be proved by splitting the integration domain and applying Hölder's inequality, see e.g. [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF]Lemma 2.1].

The proof of Proposition 2.3 shows that the axisymmetric Biot-Savart law [START_REF] Šverák | Selected Topics in Fluid Mechanics[END_REF] has the same properties as the usual Biot-Savart law in the whole plane R 2 . In fact, it is possible to obtain in the axisymmetric situation weighted inequalities (involving powers of the distance r to the vertical axis) which have no analogue in the 2D case. As an example, we state here an interesting extension of estimate (25).

Proposition 2.4 Let α, β ∈ [0, 2] be such that 0 ≤ β -α < 1, and assume that p, q ∈ (1, ∞) satisfy 1 q = 1 p - 1 + α -β 2 .
If r β ω θ ∈ L p (Ω), then r α u ∈ L q (Ω) 2 and we have the bound

r α u L q (Ω) ≤ C r β ω θ L p (Ω) . (28)
Proof. As in the proof of Proposition 2.3, all we need is to establish the pointwise estimate

r α rβ |G r (r, z, r, z)| + |G z (r, z, r, z)| ≤ C (r -r) 2 + (z -z) 2 λ , (29)
for some λ ∈ (0, 1). Indeed, once (29) is known, the bound (28) follows immediately from the Hardy-Littlewood-Sobolev inequality if q -1 = p -1 + λ -1. To prove (29), we proceed as before and distinguish three regions : a) r/2 ≤ r ≤ 2r ; b) r > 2r ; c) r > 2r . In each region, we bound the quantities F ′ (ξ 2 ) and F (ξ 2 ) -2ξ 2 F ′ (ξ 2 ) appropriately using Remark 2.2. These calculations shows that inequality (29) holds with λ = (1 + β -α)/2 if and only if the exponents α, β satisfy 0 ≤ α ≤ β ≤ 2. Since we need λ < 1 we assume in addition that β -α < 1. The details are straightforward and can be left to the reader.

Remarks 2.5 1. Similarly, one can establish a weighted analogue of inequality (26).

2. If u is replaced by u r , inequality (28) holds for all α, β ∈ [-1, 2] such that 0 ≤ β -α < 1.

3. If we take α = 1/q and β = 1/p, so that 1 q = 1 p -1 3 , inequality (28) is equivalent to

u L q (R 3 ) ≤ C ω L p (R 3 ) ,
which is well known for the three-dimensional Biot-Savart law, see e.g. [10, Lemma 2.1].

Finally, we prove other weighted inequalities, which are similar to those considered in [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF].

Proposition 2.6

The following estimates hold :

u L ∞ (Ω) ≤ C rω θ 1/2 L 1 (Ω) ω θ /r 1/2 L ∞ (Ω) , (30) u r r L ∞ (Ω) ≤ C ω θ 1/3 L 1 (Ω) ω θ /r 2/3 L ∞ (Ω) . (31)
Proof. Estimate (30) is stated in [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF] in the slightly weaker form

u L ∞ (Ω) ≤ C ω θ 1/4 L 1 (Ω) r 2 ω θ 1/4 L 1 (Ω) ω θ /r 1/2 L ∞ (Ω) ,
but the proof given there actually yields the stronger bound (30), which can be compared with (26) in the case p = 1, q = ∞. To prove (31), we follow the same approach as in [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF]. Using translation and scaling invariance, it is sufficient to show that the quantity

u r r r=1,z=0 = u r (1, 0) = Ω z πr 1/2 F ′ (r -1) 2 + z 2 r ω θ (r, z) dr dz (32) is bounded by C ω θ 1/3 L 1 (Ω) ω θ /r 2/3 L ∞ (Ω) . We decompose Ω = I 1 ∪ I 2 ,
where

I 1 = (r, z) ∈ Ω 1 2 ≤ r ≤ 2 , -1 ≤ z ≤ 1 , I 2 = Ω \ I 1 .
When integrating over the first region I 1 , we use the fact that |F ′ (s)| ≤ Cs -1 and r ≈ 1. We thus obtain

|u (1) r (1, 0)| ≤ C I 1 |z| (r -1) 2 + z 2 |ω θ (r, z)| dr dz ≤ C I 1 |ω θ (r, z)| (r -1) 2 + z 2 1/2 dr dz .
As in [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF], or in part ii) or Proposition 2.3, we deduce that

|u (1) r (1, 0)| ≤ C ω θ 1/2 L 1 (I 1 ) ω θ 1/2 L ∞ (I 1 ) ≤ C ω θ 1/3 L 1 (I 1 ) ω θ /r 2/3 L ∞ (I 1 ) . (33)
In the complementary region, we use the optimal bound |F ′ (s)| ≤ Cs -5/2 and we observe that, if (r, z) ∈ I 2 , then (r -1) 2 + z 2 ≥ C(r 2 + z 2 ) for some C > 0. We thus have

|u (2) r (1, 0)| ≤ C I 2 |z| r 2 (r -1) 2 + z 2 5/2 |ω θ (r, z)| dr dz ≤ C I 2 |ω θ (r, z)| r 2 + z 2 dr dz .
Fix any R > 0 and denote Ω R = {(r, z) ∈ Ω | ρ ≤ R}, where ρ = (r 2 + z 2 ) 1/2 . Extending the integration domain from

I 2 to Ω = Ω R ∪ (Ω \ Ω R ), we compute |u (2) r (1, 0)| ≤ C Ω R |ω θ (r, z)| ρ 2 dr dz + C Ω\Ω R |ω θ (r, z)| ρ 2 dr dz ≤ C ω θ /r L ∞ (Ω) Ω R 1 ρ dr dz + CR -2 Ω\Ω R |ω θ (r, z)| dr dz ≤ CR ω θ /r L ∞ (Ω) + CR -2 ω θ L 1 (Ω) .
Optimizing over R > 0 gives the bound |u

(2) r (1, 0)| ≤ C ω θ 1/3 L 1 (Ω) ω θ /r 2/3 L ∞ (Ω)
, which together with (33) implies (31).

Remark 2.7 Estimate (31) can also be obtained by observing that

u r r L ∞ (R 3 ) ≤ C ω θ r L 3,1 (R 3 ) ≤ C ω θ r 1/3 L 1 (R 3 ) ω θ r 2/3 L ∞ (R 3 ) , (34)
where L 3,1 (R 3 ) denotes the Lorentz space. The first inequality in (34) is proved in [2, Proposition 4.1], and the second one follows by real interpolation.

The semigroup associated with the linearized equation

This section is devoted to the study of the linearized vorticity equation ( 6) :

∂ t ω θ = ∂ 2 r + ∂ 2 z + 1 r ∂ r - 1 r 2 ω θ , (35)
which is considered in the half-plane Ω = {(r, z) | r > 0, z ∈ R}, with homogeneous Dirichlet condition at the boundary r = 0. Given initial data ω 0 ∈ L 1 (Ω), we denote by ω θ (t) = S(t)ω 0 the solution of (35) at time t > 0.

Lemma 3.1 For any t > 0, the evolution operator S(t) associated with Eq. ( 35) is given by the explicit formula

(S(t)ω 0 )(r, z) = 1 4πt Ω r1/2 r 1/2 H t rr exp - (r -r) 2 + (z -z) 2 4t ω 0 (r, z) dr dz , ( 36
)
where the function H : (0, +∞) → R is defined by

H(τ ) = 1 √ πτ π/2 -π/2 e -sin 2 φ τ cos(2φ) dφ , τ > 0 . ( 37
)
Proof. If ω θ is a solution of (35), we observe that the vector valued function ω = ω θ e θ satisfies the usual heat equation ∂ t ω = ∆ω in the whole space R 3 . For any t > 0, we thus have the solution formula

ω(x, t) = 1 (4πt) 3/2 R 3 e -|x-x| 2 4t ω(x, 0) dx , x ∈ R 3 . (38)
Denoting x = (r cos θ, r sin θ, z) and x = (r cos θ, r sin θ, z), we can write (38) in the form

ω θ (r, z, t)   -sin θ cos θ 0   = 1 (4πt) 3/2 ∞ 0 R π -π e -|x-x| 2 4t ω 0 (r, z)   -sin θ cos θ 0   r d θ dz dr , (39)
where

|x -x| 2 = (r -r) 2 + (z -z) 2 + 4rr sin 2 θ - θ 2 .
Now, if we integrate over the angle θ in (39) and use the definition (37) of H, we see that (39) is equivalent to ω θ (t) = S(t)ω 0 with S(t) defined by (36).

The function H cannot be expressed in terms of elementary functions, but all we need to know is the behavior of H(τ ) as τ → 0 and τ → ∞.

Lemma 3.2

The function H : (0, ∞) → R defined by (37) is smooth and satisfies the asymptotic expansions :

i) H(τ ) = 1 - 3τ 4 + O(τ 2 ) and H ′ (τ ) = -3/4 + O(τ ) as τ → 0; ii) H(τ ) = π 1/2 4τ 3/2 + O 1 τ 5/2 and H ′ (τ ) = - 3π 1/2 8τ 5/2 + O 1 τ 7/2 as τ → ∞.
Proof. Expansion ii) follows immediately from (37) if, in the expression e -sin 2 φ τ , we replace the exponential function by its Taylor series at the origin. Expansion i) can be deduced from the formula

H(τ ) = 1 √ π 1 √ τ -1 √ τ e -x 2 1 -2τ x 2 √ 1 -τ x 2 dx ,
which is obtained by substituting x = sin φ √ τ in the integral (37).

Remark 3.3

We believe that the function H is decreasing, although we do not have a simple proof. In what follows, we only use the fact that the maps τ → τ α H(τ

) and τ → τ β H ′ (τ ) are bounded if 0 ≤ α ≤ 3/2 and 0 ≤ β ≤ 5/2.
Let (S(t)) t≥0 be the family of linear operators defined by (36) for t > 0 and by S(0) = 1 (the identity operator). By construction, we have the semigroup property : S(t 1 + t 2 ) = S(t 1 )S(t 2 ) for all t 1 , t 2 ≥ 0. Further important properties are collected in the following proposition.

p ∈ [1, ∞). Moreover, if 1 ≤ p ≤ q ≤ ∞, the following estimates hold : i) If ω 0 ∈ L p (Ω), then S(t)ω 0 L q (Ω) ≤ C t 1 p -1 q ω 0 L p (Ω) , t > 0 . ( 40
) ii) If f = (f r , f z ) ∈ L p (Ω) 2 , then S(t) div * f L q (Ω) ≤ C t 1 2 + 1 p -1 q f L p (Ω) , t > 0 , (41)
where div * f = ∂ r f r + ∂ z f z denotes the two-dimensional divergence of f .

Proof. We claim that

1 4πt r1/2 r 1/2 H t rr exp - (r -r) 2 + (z -z) 2 4t ≤ C t exp - (r -r) 2 + (z -z) 2 5t , (42)
for all (r, z) ∈ Ω, all (r, z) ∈ Ω, and all t > 0. Indeed, since H is bounded, estimate (42) is obvious when r ≤ 2r. If r > 2r, we observe that τ 1/2 H(τ ) is bounded, so that

r1/2 r 1/2 H t rr ≤ C r1/2 r 1/2 r 1/2 r1/2 t 1/2 = C r t 1/2 ≤ C |r -r| t 1/2 ,
where in the last inequality we used the fact that r < 2(r -r) = 2|r -r|. As xe -x 2 /4 ≤ Ce -x 2 /5 for all x ≥ 0, we conclude that (42) holds in all cases. This provides for the integral kernel in (36) a pointwise upper bound in terms of the usual heat kernel in the whole plane R 2 , with a diffusion coefficient equal to 5/4 instead of 1. Thus estimate (40) follows immediately from Young's inequality, as in the 2D case.

To prove (41), we assume that ω 0 = div * f = ∂ r f r + ∂ z f z , and we integrate by parts in (36) to obtain the identity

(S(t) div * f )(r, z) = 1 4πt Ω r1/2 r 1/2 exp - (r -r) 2 + (z -z) 2 4t (A r f r + A z f z ) dr dz , where A r = t rr 2 H ′ t rr - 1 2r + r - r 2t H t rr , A z = - z - z 2t H t rr .
Proceeding as above and using Remark 3.3, it is straightforward to verify that

1 4πt r1/2 r 1/2 exp - (r-r) 2 + (z-z) 2 4t |A r | + |A z | ≤ C t 3/2 exp - (r-r) 2 + (z-z) 2 5t , (43)
for all (r, z) ∈ Ω, all (r, z) ∈ Ω, and all t > 0. Thus estimate (41) follows again from Young's inequality, as in the 2D case. Finally, we show that the semigroup (S(t)) t≥0 is strongly continuous in L p (Ω) if 1 ≤ p < ∞. All we need to verify is the continuity at the origin. Given ω 0 ∈ L p (Ω), we denote by ω 0 : R 2 → R the function obtained by extending ω 0 by zero outside Ω. Using the change of variables r = r + √ tρ, z = z + √ tζ in (36), we obtain the identity

S(t)ω 0 -ω 0 (r, z) = 1 4π R 2 e -ρ 2 +ζ 2 4 Ψ(r, z, ρ, ζ, t) dρ dζ ,
for all (r, z) ∈ Ω, where

Ψ(r, z, ρ, ζ, t) = 1 + √ tρ r 1/2 H t r(r + √ tρ) ω 0 (r + √ tρ, z + √ tζ) -ω 0 (r, z) .
By Minkowski's integral inequality, we deduce that

S(t)ω 0 -ω 0 L p (Ω) ≤ 1 4π R 2 e -ρ 2 +ζ 2 4 Ψ(•, •, ρ, ζ, t) L p (Ω) dρ dζ . (44)
Now, the estimates we used in the proof of (42) show that

1 + √ tρ r 1/2 H t r(r + √ tρ) ≤ C(1 + |ρ|) , (45)
whenever r > 0 and r + √ tρ > 0. This immediately implies that

Ψ(•, •, ρ, ζ, t) L p (Ω) ≤ C(1 + |ρ|) ω 0 L p (Ω) ,
for all (ρ, ζ) ∈ R 2 and all t > 0. Since the left-hand side of (45) converges to 1 as t → 0, and since translations act continuously in L p (R 2 ) for p < ∞, it also follows from estimate (45) and Lebesgue's dominated convergence theorem that

Ψ(•, •, ρ, ζ, t) L p (Ω) -→ 0 as t → 0 , for all (ρ, ζ) ∈ R 2 .
Thus another application of Lebesgue's theorem implies that the right-hand side of (44) converges to zero as t → 0, which is the desired result.

As in the previous section, it is possible to derive weighted estimates for the linear semigroup (36). The proof of the following result is very similar to that of Propositions 2.4 and 3.4, and can thus be left to the reader.

Proposition 3.5 Let 1 ≤ p ≤ q ≤ ∞ and -1 ≤ α ≤ β ≤ 2. If r β ω 0 ∈ L p (Ω), then r α S(t)ω 0 L q (Ω) ≤ C t 1 p -1 q + β-α 2 r β ω 0 L p (Ω) , t > 0 . ( 46
)
Moreover, if -1 ≤ α ≤ β ≤ 1 and r β f ∈ L p (Ω) 2 , then r α S(t) div * f L q (Ω) ≤ C t 1 2 + 1 p -1 q + β-α 2 r β f L p (Ω) , t > 0 . ( 47
)

Local existence of solutions

Equipped with the results of the previous sections, we now take up the proof of Theorems 1.1 and 1.3. In view of the divergence-free condition in (5), the evolution equation ( 6) for the axisymmetric vorticity ω θ can be written in the equivalent form

∂ t ω θ + div * (u ω θ ) = ∂ 2 r + ∂ 2 z + 1 r ∂ r - 1 r 2 ω θ , (48)
where div * (u ω θ ) = ∂ r (u r ω θ ) + ∂ z (u z ω θ ). Given initial data ω 0 , the integral equation associated with (48) is

ω θ (t) = S(t)ω 0 - t 0 S(t -s) div * (u(s)ω θ (s)) ds , t > 0 , (49)
where S(t) denotes the linear semigroup defined in (36). In this section, our goal is to prove local existence and uniqueness of solutions to (49) using a standard fixed point argument, in the spirit of Kato [START_REF] Kato | Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions[END_REF]. For the sake of clarity, we first treat the case where ω 0 ∈ L 1 (Ω), and then consider the more complicated situation where ω 0 is a finite measure with sufficiently small atomic part. This will establish the local well-posedness claims in Theorems 1.1 and 1.3, respectively.

Local existence when the initial vorticity is integrable

Proposition 4.1 For any initial data ω 0 ∈ L 1 (Ω), there exists T = T (ω 0 ) > 0 such that the integral equation (49) has a unique solution

ω θ ∈ C 0 ([0, T ], L 1 (Ω)) ∩ C 0 ((0, T ], L ∞ (Ω)) . (50)
Moreover ω θ (t) L 1 (Ω) ≤ ω 0 L 1 (Ω) for all t ∈ [0, T ], and estimate (10) holds. Finally, if ω 0 L 1 (Ω) is small enough, the local existence time T > 0 can be taken arbitrarily large.

Proof. We follow the same approach as in the two-dimensional case, see e.g. [START_REF] Ben-Artzi | Global solutions of two-dimensional Navier-Stokes and Euler equations[END_REF][START_REF] Th | Stability and interaction of vortices in two-dimensional viscous flows[END_REF][START_REF] Kato | The Navier-Stokes equation for an incompressible fluid in R 2 with a measure as the initial vorticity[END_REF]. Given T > 0 we introduce the function space

X T = ω θ ∈ C 0 ((0, T ], L 4/3 (Ω) ω θ X T < ∞ , (51)
equipped with the norm

ω θ X T = sup 0<t≤T t 1/4 ω θ (t) L 4/3 (Ω) .
For any t ≥ 0, we denote ω lin (t) = S(t)ω 0 , where S(t) is the linear semigroup (36). It then follows from Proposition 3.4 that ω lin ∈ X T for any T > 0. For later use, we define

C 1 (ω 0 , T ) = ω lin X T = sup 0<t≤T t 1/4 S(t)ω 0 L 4/3 (Ω) . (52)
In view of (40), there exists a universal constant

C 2 > 0 such that C 1 (ω 0 , T ) ≤ C 2 ω 0 L 1 (Ω) for any T > 0. Moreover, since L 1 (Ω) ∩ L 4/3 (Ω) is dense in L 1 (Ω)
, it also follows from (40) that C 1 (ω 0 , T ) → 0 as T → 0, for any ω 0 ∈ L 1 (Ω).

Given ω θ ∈ X T and p ∈ [1, 2), we define a map Fω θ : (0, T ] → L p (Ω) in the following way :

(Fω θ )(t) = t 0 S(t -s) div * (u(s)ω θ (s)) ds , 0 < t ≤ T , ( 53
)
where it is understood that u(s) is the velocity field obtained from ω θ (s) via the axisymmetric Biot-Savart law [START_REF] Šverák | Selected Topics in Fluid Mechanics[END_REF]. Using estimate (41), Hölder's inequality, and the bound (25), we obtain for t ∈ (0, T ] :

t 1-1 p (Fω θ )(t) L p (Ω) ≤ t 1-1 p t 0 C (t -s) 3 2 -1 p u(s)ω θ (s) L 1 (Ω) ds ≤ t 1-1 p t 0 C (t -s) 3 2 -1 p u(s) L 4 (Ω) ω θ (s) L 4/3 (Ω) ds ≤ t 1-1 p t 0 C (t -s) 3 2 -1 p ω θ (s) 2 L 4/3 (Ω) ds (54) ≤ t 1-1 p t 0 C (t -s) 3 2 -1 p ω θ 2 X T s 1 2 ds ≤ C ω θ 2 X T .
It is also straightforward to verify that the quantity (Fω θ )(t) depends continuously on the time parameter t ∈ (0, T ] in the topology of L p (Ω). Choosing p = 4/3, we deduce that Fω θ ∈ X T and Fω θ X T ≤ C 3 ω θ 2 X T for some C 3 > 0. More generally, we have the Lipschitz estimate

Fω θ -F ωθ X T ≤ C 3 ω θ X T + ωθ X T ω θ -ωθ X T , (55)
for all ω θ , ωθ ∈ X T . Now we consider the map G : X T → X T defined by Gω θ = ω lin -Fω θ . We fix R > 0 such that 2C 3 R < 1, and denote by B R the closed ball of radius R centered at the origin in X T . If C 1 (ω 0 , T ) ≤ R/2, the estimates above show that G maps B R into B R and is a strict contraction there, so that (by the Banach fixed point theorem) G has a unique fixed point ω θ in B R . By construction, ω θ is a solution to the integral equation ( 49) in X T . The condition C 1 (ω 0 , T ) ≤ R/2 can be fulfilled in two different ways. If the initial data are small enough so that C 2 ω 0 L 1 (Ω) ≤ R/2, the existence time T > 0 can be chosen arbitrarily, and the fixed point argument therefore establishes global existence for small data in L 1 (Ω). On the other hand, for larger initial data ω 0 , we can always choose T > 0 small enough so that C 1 (ω 0 , T ) ≤ R/2, hence we also have local existence for arbitrary data. Remarks 4.2 1. For large data, the local existence time T > 0 given by the fixed point argument depends on the initial data, and it is not possible to bound T from below using the norm ω 0 L 1 (Ω) only. However, if ω 0 ∈ L 1 (Ω) ∩ L p (Ω) for some p > 1, then (by Proposition 3.4) an upper bound on ω 0 L p (Ω) provides a lower bound on the local existence time T .

2.

For later use, we note that the fixed point argument also proves that the solution ω θ depends continuously on the initial data. More precisely, if K ⊂ L 1 (Ω) is any compact set, or any sufficiently small neighborhood of a given point, we can take the same local existence time T > 0 for all initial data ω 0 ∈ K, and there exists C > 0 such that

sup t∈[0,T ] ω θ (t) -ωθ (t) L 1 (Ω) + ω θ -ωθ X T ≤ C ω 0 -ω0 L 1 (Ω) , (56)
for all ω 0 , ω0 ∈ K, where ω θ , ωθ ∈ X T denote the solutions corresponding to the initial data ω 0 , ω0 , respectively.

To conclude the proof of Proposition 4.1, we establish a few additional properties of the local solution ω θ ∈ X T . We first note that

lim t→0 t 1/4 ω θ (t) L 4/3 (Ω) = lim T →0 ω θ X T = 0 , (57)
because, when T > 0 is small, the fixed point argument holds in the ball B R with R = 2C 1 (ω 0 , T ). This proves [START_REF] Th | Long-time asymptotics of the Navier-Stokes and vorticity equations on R 3[END_REF] for p = 4/3. Next, using (54) with p = 1, we see that that the map t → (Fω θ )(t) is continuous in the topology of L 1 (Ω) and satisfies (Fω

θ )(t) L 1 (Ω) ≤ C ω θ 2
X T for all t ∈ (0, T ]. In view of (57), this implies that the map ω θ -ω lin ≡ -Fω θ belongs to C 0 ([0, T ], L 1 (Ω) and vanishes at t = 0. In particular, using Proposition 3.4, we conclude that ω θ ∈ C 0 ([0, T ], L 1 (Ω)). That ω θ (t) L 1 (Ω) is a non-increasing function of time is a well known fact, which will be discussed in Lemma 5.1 below. Finally, to prove that ω θ ∈ C 0 ((0, T ], L p (Ω)) for any p ∈ (1, ∞] and that (10) holds, we use a standard bootstrap argument which we explain in some detail because it will be used again in Section 5. For p ∈ (1, ∞] we define

M p (T ) = sup 0<t≤T t 1-1 p S(t)ω 0 L p (Ω) , and
N p (T ) = sup 0<t≤T t 1-1 p ω θ (t) L p (Ω) . ( 58
)
Then M p (T ) → 0 as T → 0, and we already know that N p (T ) → 0 as T → 0 for p = 4/3, hence for all p ∈ (1, 4/3] by interpolation. To prove the same result for p > 4/3, we split the integral in (49) in two parts and estimate the nonlinear term as follows :

u ω θ L r (Ω) ≤ C u L s (Ω) ω θ L q 2 (Ω) ≤ C ω θ L q 1 (Ω) ω θ L q 2 (Ω) , s = 2q 1 2 -q 1 ,
where 4 3 ≤ q 1 < 2, 4 3 ≤ q 2 ≤ ∞, and 1 r = 1 q 1 + 1 q 2 -1 2 . We thus obtain

ω θ (t) L p (Ω) ≤ S(t)ω 0 L p (Ω) + C t/2 0 ω θ (s) 2 L q (Ω) (t -s) 2 q -1 p ds + C t t/2 ω θ (s) L q 1 (Ω) ω θ (s) L q 2 (Ω) (t -s) 1 q 1 + 1 q 2 -1 p ds , (59)
where the exponents p ∈ [1, ∞], q, q 1 ∈ [4/3, 2) and q 2 ∈ [4/3, ∞] are assumed to satisfy

1 2 ≤ 2 q - 1 p , and 1 2 ≤ 1 q 1 + 1 q 2 - 1 p < 1 . ( 60
)
Multiplying both sides of (59) by t 1-1 p and taking the supremum over t ∈ (0, T ] we obtain the useful bound

N p (T ) ≤ M p (T ) + C p,q N q (T ) 2 + C p,q 1 ,q 2 N q 1 (T )N q 2 (T ) , (61)
where C p,q and C p,q 1 ,q 2 are positive constants. If we choose q = q 1 = q 2 = 4/3, we deduce from (61) that N p (T ) → 0 as T → 0 for any p < 2. Then, taking q = 4/3 and q 1 = q 2 sufficiently close to 2, we obtain the same result for any p < ∞. Finally, choosing q = 4/3, q 1 = 3/2, and q 2 = 4, we conclude from (61) that N ∞ (T ) → 0 as T → 0, which proves [START_REF] Th | Long-time asymptotics of the Navier-Stokes and vorticity equations on R 3[END_REF].

Our final task is to discuss the uniqueness of the solution to (49). We first observe that, given ω 0 ∈ L 1 (Ω), the solution ω θ ∈ X T given by the fixed point argument is, by construction, the only solution of (49) in X T satisfying (57). In fact, using a nice argument due to Brezis [START_REF] Brezis | Global solutions of twodimensional Navier-Stokes and Euler equations[END_REF], it is possible to prove uniqueness in a much larger class. Indeed, given any T > 0, assume that

ω θ ∈ C 0 ([0, T ], L 1 (Ω)) ∩ C 0 ((0, T ], L ∞ (Ω)
) is a mild solution of (48) on (0, T ] in the following sense :

ω θ (t) = S(t -t 0 )ω θ (t 0 ) - t t 0 S(t -s) div * (u(s)ω θ (s)) ds , 0 < t 0 ≤ t ≤ T . ( 62
)
The set K = {ω θ (t) | t ∈ [0, T ]} is compact in L 1 (Ω), hence the fixed point argument allows us to construct a local solution in X T for all initial data ω0 ∈ K, with a common existence time T > 0 (without loss of generality, we assume henceforth that T ≤ T /2). That solution is denoted by ωθ (t) = Σ(t)ω 0 for t ∈ [0, T ]. By the observation above, for all t 0 ∈ (0, T ] we have the relation

ω θ (t) = Σ(t -t 0 )ω θ (t 0 ) , t ∈ [t 0 , t 0 + T ] , (63)
because the left-hand side is a solution of (62) on the time interval [t 0 , t 0 + T ] and we obviously have (t -t 0 ) 1/4 ω θ (t) L 4/3 (Ω) → 0 as t → t 0 , which is the analogue of condition (57). Now, for any fixed t ∈ (0, T ], it follows from (56) that

Σ(t -t 0 )ω θ (t 0 ) -Σ(t -t 0 )ω θ (0) L 1 (Ω) ≤ C ω θ (t 0 ) -ω θ (0) L 1 (Ω) -→ 0 ,
as t 0 → 0, and it is also clear that Σ(t -t 0 )ω θ (0) -Σ(t)ω θ (0) L 1 (Ω) → 0 as t 0 → 0. Thus taking the limit t 0 → 0 in (63) we obtain the relation ω θ (t) = Σ(t)ω θ (0) for t ∈ [0, T ], which means that the solution ω θ we started with coincides on the time interval [0, T ] with the solution constructed from the initial data ω θ (0) by the fixed point argument.

But this is impossible, because the intersection above is empty if ξ ∈ ∂Ω and equal to the singleton { ξ} if ξ ∈ Ω, and we assumed that the measure µ is non-atomic. Hence property (67), which can be interpreted as a weak form of absolute continuity with respect to Lebesgue's measure, must hold. Now, for any given t > 0, there exists ξ(t) ∈ Ω such that

|(S(t)µ)( ξ(t))| = S(t)µ L ∞ (Ω) ,
because the map ξ → (S(t)µ)(ξ) is continuous and vanishes at infinity as well as on the boundary ∂Ω. Using definition (64) and the pointwise estimate (42), we thus obtain

t S(t)µ L ∞ (Ω) ≤ C B( ξ(t),δ) e -|ξ-ξ(t)| 2 5t d|µ|(ξ) + C Ω\B( ξ(t),δ) e -|ξ-ξ(t)| 2 5t d|µ|(ξ) ,
where ǫ and δ are as in (67). The first integral is bounded by C|µ|(B( ξ(t), δ)) ≤ Cǫ, and the second one by Ce -δ 2 /(5t) |µ|(Ω). It follows that

L ∞ (µ) = lim sup t→0 t S(t)µ L ∞ (Ω) ≤ Cǫ ,
and since ǫ > 0 was arbitrary we conclude that L ∞ (µ) = 0, which is the desired result.

Proposition 4.4 There exist positive constants ǫ and C such that, for any initial data ω 0 ∈ M(Ω) with (ω 0 ) pp tv ≤ ǫ, one can choose T = T (ω 0 ) > 0 such that the integral equation (62) has a unique solution ω θ ∈ C 0 ((0, T ], L 1 (Ω) ∩ L ∞ (Ω)) satisfying ( 16) and such that ω θ (t) ⇀ ω 0 as t → 0. Moreover, if ω 0 tv is small enough, the local existence time T > 0 can be taken arbitrarily large.

Proof. We briefly indicate how the proof of Proposition 4.1 has to be modified to handle the case where µ = ω 0 ∈ M(Ω). We use exactly the same function space X T defined in (51), and observe that the fixed point argument works in the ball B R ⊂ X T provided C 1 (µ, T ) ≤ R/2, where C 1 (µ, T ) is defined as in (52) and R > 0 satisfies 2C 3 R < 1 with C 3 as in (55). From (65) we know that C 1 (µ, T ) ≤ C 2 µ tv for any T > 0, hence we again obtain global existence and uniqueness in B R if the initial vorticity is small enough so that C 2 µ tv ≤ R/2. For larger data, we can use (66) which gives lim

T →0 C 1 (µ, T ) = L 4/3 (µ) ≤ C 4 µ pp tv ,
for some positive constant C 4 . Thus, if the atomic part of the initial vorticity is small enough so that C 4 µ pp tv < R/2, we can take T > 0 such that C 1 (µ, T ) ≤ R/2, and the fixed point argument proves local existence and uniqueness in B R . In contrast, if 4C 3 C 4 µ pp tv ≥ 1, it is impossible to choose R > 0 and T > 0 so that the fixed point argument works in the ball B R ⊂ X T , and the method above completely fails. Assuming that the fixed point argument works in the ball B R ⊂ X T , we can establish some additional properties of the solution ω θ ∈ B R as in the case of integrable initial data. For instance, it is straightforward to verify that ω θ -ω lin ∈ C 0 ((0, T ), L 1 (Ω) ∩ L ∞ (Ω)), where ω lin (t) = S(t)µ. However property (57) fails if µ pp = 0, and we cannot argue as in Section 4.1 to show that ω θ (t) -ω lin (t) L 1 (Ω) converges to zero as t → 0. To prove that, we first define

δ = lim sup t→0 t 1/4 ω θ (t) -ω lin (t) L 4/3 (Ω) = lim sup T →0 ω θ -ω lin X T .
Since ω θ -ω lin = (Fω lin -Fω θ ) -Fω lin and ω lin X T + ω θ X T ≤ 2R, we can use (55) to obtain the estimate δ ≤ 2C 3 Rδ + ℓ 4/3 (µ), where

ℓ p (µ) = lim sup t→0 t 1-1 p Fω lin (t) L p (Ω) , 1 ≤ p ≤ ∞ .
As in the two-dimensional case [7, Section 2.3.4], a direct calculation, which exploits some cancellations in the nonlinear term u lin (t) • ∇ω lin (t) for small times, reveals that ℓ p (µ) = 0 for any p ∈ [1, ∞]. This in turn implies that δ = 0, since 2C 3 R < 1. Finally, using again the relation ω θ -ω lin = (Fω lin -Fω θ ) -Fω lin we conclude that lim sup

t→0 ω θ (t) -ω lin (t) L 1 (Ω) ≤ CRδ + ℓ 1 (µ) = 0 . (68)
It follows in particular from (68) that ω θ (t) ⇀ µ as t → 0, because we can use the explicit formula (64) to verify that ω lin (t) ⇀ µ as t → 0. By construction ω θ is a solution of (49), hence of (62), and both inequalities in ( 16) hold.

Finally, if ω θ ∈ C 0 ((0, T ], L 1 (Ω) ∩ L ∞ (Ω)
) is a mild solution on (0, T ) satisfying ( 16) and such that ω θ (t) ⇀ ω 0 as t → 0, we can take the limit t 0 → 0 in (62) and conclude that ω θ satisfies (49), so that ω θ coincides with the solution constructed by the fixed point argument.

Remark 4.5 In Proposition 4.4 we only claim uniqueness of solutions under assumption [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF], which means (after restricting the existence time) that ω θ belongs to the ball B R ⊂ X T where the fixed point argument works. As in the two-dimensional case, one may conjecture that uniqueness holds among all solutions

ω θ ∈ C 0 ((0, T ), L 1 (Ω) ∩ L ∞ (Ω)) such that ω θ (t) L 1 (Ω)
is uniformly bounded and ω θ (t) ⇀ µ as t → 0. We hope to come back to that interesting question in a future work.

Remark 4.6 The solutions constructed in Propositions 4.1 and 4.4 are in fact smooth for positive times, and satisfy the axisymmetric vorticity equation (48) in the classical sense. This can be proved using standard smoothing properties of the Navier-Stokes equations that are not specific to the axisymmetric case, see e.g. [START_REF] Koch | Liouville theorems for the Navier-Stokes equations and applications[END_REF] and Proposition 5.5 below.

A priori estimates and global existence

We continue the proof of Theorems 1.1 and 1.3 by showing that the local solutions constructed in Sections 4.1 and 4.2 can be extended to global solutions for positive times. Since we are not interested in the behavior for small times, we can assume without loss of generality that the initial vorticity is integrable. Let thus

ω θ ∈ C 0 ([0, T ], L 1 (Ω)) ∩ C 0 ((0, T ], L ∞ (Ω)
) be a solution of the integral equation (49), hence also of the differential equation ( 48), with initial data ω 0 ∈ L 1 (Ω). Our goal here is to derive a priori estimates on various norms of ω θ .

Lemma 5.1 The solution of (49

) satisfies ω θ (t) L 1 (Ω) ≤ ω 0 L 1 (Ω) for all t ∈ [0, T ]. Moreover, if ω 0 ≡ 0, the map t → ω θ (t) L 1 (Ω) is strictly decreasing.
Proof. We first assume that ω 0 ≥ 0 and ω 0 ≡ 0. By the strong maximum principle, the solution ω θ (t) of ( 48) is strictly positive for t ∈ (0, T ]. Integrating by parts and using the fact that ω θ (t) satisfies the homogeneous Dirichlet boundary condition on ∂Ω, we easily find

d dt Ω ω θ (r, z, t) dr dz = -2 R ∂ r ω θ (0, z, t) dz < 0 , (69)
For t > 0, the second inequality is strict if either ω + θ (0) or ω - θ (0) is nonzero, and if both quantities are nonzero the first inequality is also strict (by the strong maximum principle). If ω 0 ≡ 0, this proves that the L 1 norm of the solution ω θ (t) is strictly decreasing at initial time, and a similar argument shows that it is strictly decreasing over the whole interval [0, T ].

Higher L p norms of the vorticity ω θ are more difficult to control, because the velocity field in (48) does not satisfy div * u = 0. As in [START_REF] Ladyzhenskaya | Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF][START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF], we thus consider the related quantity η = ω θ /r, which satisfies Eq. ( 8) with initial data η 0 = ω 0 /r ∈ L 1 (R 3 ). Using the existence result in Proposition 4.1 and the weighted estimates on the linear semigroup given in Proposition 3.5, it is easy to verify that η ∈ C 0 ([0, T ], L 1 (R 3 )) ∩ C 0 ((0, T ], L ∞ (R 3 )). Moreover, by Lemma 5.1, the map t → η(t) L 1 (R 3 ) ≡ ω θ (t) L 1 (Ω) is decreasing for nonzero solutions. Since the advection field u -(2/r)e r in (8) satisfies div u -2e r r = -2 div e r r = -4πδ r=0 ≤ 0 , a classical method due to Nash [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF] gives the following a priori estimate :

Lemma 5.2 [START_REF] Feng | On the Cauchy problem for axi-symmetric vortex rings[END_REF]Lemma 3.8] For any initial data η 0 ∈ L 1 (R 3 ), the solution of (8) satisfies, for 1 ≤ p ≤ ∞,

η(t) L p (R 3 ) ≤ C t 3 2 (1-1 p ) η 0 L 1 (R 3 ) , 0 < t ≤ T . (71)
Equivalently, the axisymmetric vorticity ω θ satisfies, for p ∈ [1, ∞], the weighted estimate

r 1 p -1 ω θ (t) L p (Ω) ≤ C t 3 2 (1-1 p ) ω 0 L 1 (Ω) , 0 < t ≤ T . (72)
Using (72), we now establish our main a priori estimate on the solutions of (6).

Proposition 5.3 Any solution ω θ ∈ C 0 ([0, T ], L 1 (Ω)) ∩ C 0 ((0, T ], L ∞ (Ω)) of (49) with initial data ω 0 ∈ L 1 (Ω) satisfies, for all p ∈ [1, ∞], ω θ (t) L p (Ω) ≤ C p ( ω 0 L 1 (Ω) ) t 1-1 p , 0 < t ≤ T , (73)
where C p (s) = O(s) as s → 0.

Proof. We can assume without loss of generality that M := ω 0 L 1 (Ω) > 0. We know from Lemma 5.1 that ω θ (t) L 1 (Ω) ≤ M for t ∈ [0, T ], hence (73) holds for p = 1. To prove (73) for p = 2, we compute

d dt Ω ω 2 θ dr dz = -2 Ω |∇ω θ | 2 dr dz + Ω u r r - 1 r 2 ω 2 θ dr dz . (74)
The celebrated Nash inequality [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF] asserts that

Ω ω 2 θ dr dz ≤ C Ω |ω θ | dr dz Ω |∇ω θ | 2 dr dz 1/2 ≤ CM Ω |∇ω θ | 2 dr dz 1/2
.

On the other hand, using estimate (72) with p = ∞ and Proposition 2.6, we obtain

u r (t) r L ∞ (Ω) ≤ C ω θ (t) 1/3 L 1 (Ω) ω θ (t) r 2/3 L ∞ (Ω) ≤ CM t .
Thus, if we define

f (t) = Ω ω θ (r, z, t) 2 dr dz , 0 ≤ t ≤ T ,
we deduce from (74) that f : [0, T ] → R satisfies the differential inequality

f ′ (t) ≤ - K 1 M 2 f (t) 2 + K 2 M t f (t) , 0 < t ≤ T , (75)
where K 1 , K 2 are positive constants. If we set f (t) = t α g(t) with α = K 2 M , we see that (75) reduces to the simpler differential inequality g ′ (t) ≤ -K 1 M -2 t α g(t) 2 , which can be integrated of the time interval [t 0 , t] ⊂ (0, T ] to give the bound

1 g(t) ≥ 1 g(t 0 ) + K 1 M 2 1 α + 1 t α+1 -t α+1 0 ≥ K 1 M 2 1 α + 1 t α+1 -t α+1 0 ----→ t 0 →0 K 1 M 2 t α+1 α + 1 .
We conclude that

ω θ (t) 2 L 2 (Ω) = f (t) = t α g(t) ≤ α + 1 K 1 M 2 t = K 2 M + 1 K 1 M 2 t , 0 < t ≤ T ,
which proves (73) for p = 2 (hence for 1 ≤ p ≤ 2 by interpolation). To reach the same conclusion for higher values of p, we proceed exactly as in the proof of Proposition 4. With Proposition 5.3 at hand, it is straightforward to show that the solutions of the vorticity equation ( 6) are smooth for positive times, and that estimates similar to (73) hold for the derivatives too. For later use, we state the following result.

∇ω θ (t) L p (Ω) ≤ C p ( ω 0 L 1 (Ω) ) t 3 2 -1 p , t > 0 , (76)
where C p (s) = O(s) as s → 0.

Proof. It is possible to prove estimate (76) by working directly on the integral representation (49), but we find it easier to deduce it from Proposition 5.3 using general smoothing properties of the Navier-Stokes equations. We know from ( 26) and (73) that the velocity field associated with the solution ω θ of (48) satisfies, for any t 0 > 0,

u(t 0 ) L ∞ (Ω) ≤ C ω θ (t 0 ) 1/2 L 1 (Ω) ω θ (t 0 ) 1/2 L ∞ (Ω) ≤ C(M ) √ t 0 , (77)
where M = ω 0 L 1 (Ω) and C(s) = O(s) as s → 0. On the other hand, there exist positive constants a and A such that the solution u(t) of the Navier-Stokes equations ( 1) in R 3 with data

u(t 0 ) ∈ L ∞ (R 3 ) at time t 0 satisfies (t -t 0 ) ∇ 2 u(t) L ∞ (R 3 ) ≤ A u(t 0 ) L ∞ (R 3 ) , t 0 < t < t 0 + a u(t 0 ) -2 L ∞ (R 3 ) , (78)
see [START_REF] Koch | Liouville theorems for the Navier-Stokes equations and applications[END_REF]Proposition 4.1]. Here ∇ 2 u denotes the collection of all second-order derivatives of u.

Since

u(t 0 ) L ∞ (Ω) ≡ u(t 0 ) L ∞ (R 3 )
, we can combine estimates (77), (78) by fixing t > 0 and choosing, for instance,

t 0 = t 2 2C(M ) 2 + a C(M ) 2 + a , so that t 0 < t < t 0 + at 0 C(M ) -2 ≤ t 0 + a u(t 0 ) -2 L ∞ (R 3 )
. We thus obtain

∇ 2 u(t) L ∞ (R 3 ) ≤ A t -t 0 C(M ) √ t 0 = C(M ) t 3/2 , ( 79
)
where C(s) = O(s) as s → 0. Using the pointwise estimate

|∇ω θ | ≤ |∂ r ω θ | + |∂ z ω θ | ≤ C |∇ 2 u| + 1 r |∂ r u z | ,
and the fact that ∂ r u z vanishes at r = 0, we see that (79) implies (76) with p = ∞. The case p < ∞ easily follows by interpolation, in view of (73).

Long-time behavior

In this final section, we study the long-time behavior of the solutions of the axisymmetric vorticity equation ( 6) constructed in Sections 4 and 5. In particular we prove estimate [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF], and we obtain the asymptotic formula (13) in the particular case where the initial vorticity has a definite sign and a finite impulse in the sense of ( 12). This will conclude the proof of Theorems 1.1 and 1.3.

Convergence to zero in scale invariant norms

Let ω θ ∈ C 0 ([0, ∞), L 1 (Ω)) ∩ C 0 ((0, ∞), L ∞ (Ω)) be a global solution of the vorticity equation (48), with initial data ω 0 ∈ L 1 (Ω). We know from Lemma 5.1 that the L 1 norm ω θ (t) L 1 (Ω) is a decreasing function of time, and our goal here is to prove that this quantity actually converges to zero as t → ∞. We first show that ω θ (r, z, t) is essentially confined, for large times, in a ball of radius O( √ t) in Ω.

Proposition 6.1 Let ω 0 ∈ L 1 (Ω) and M = ω 0 L 1 . For any ǫ > 0, there exist positive constants K 3 (ǫ, ω 0 ) and K 4 (ǫ, M ) such that the solution of (48) with initial data ω 0 satisfies, for all t ≥ 0,

Ω(t) |ω θ (r, z, t)| dr dz ≤ ǫ , where Ω(t) = (r, z) ∈ Ω r 2 + z 2 ≥ K 3 + K 4 √ t . ( 80
)
Proof. By the maximum principle, it is sufficient to establish (80) for positive solutions of (48) (in the general case, the result follows by decomposing ω θ as in the proof of Lemma 5.1). We thus assume that ω 0 ≥ 0 and that M = Ω ω 0 dr dz > 0. The only property of the velocity field that will be used to obtain (80) is the a priori estimate (77). We first prove confinement in the radial direction. For R ≥ 0 and t ≥ 0, we define

f (R, t) = ∞ R R ω θ (r, z, t) dz dr . Then f (R, t) is a non-increasing function of R such that f (0, t) = ω θ (t) L 1 (Ω) and f (R, t) → 0 as R → ∞. Moreover, using (48)
, it is easy to verify that f satisfies the evolution equation

∂ t f (R, t) = ∂ 2 R f (R, t) + 1 R ∂ R f (R, t) + R u r (R, z, t)ω θ (R, z, t) dz , R > 0 . (81)
In view of (77), we have the estimate

R u r (R, z, t)ω θ (R, z, t) dz ≤ C(M ) √ t R ω θ (R, z, t) dz = - C(M ) √ t ∂ R f (R, t) , (82)
for some positive constant C(M ). Since ∂ R f ≤ 0, we deduce from (81), (82) that

∂ t f (R, t) ≤ ∂ 2 R f (R, t) - C(M ) √ t ∂ R f (R, t) , R > 0 . (83)
Solving the differential inequality (83), with homogeneous Neumann boundary condition at R = 0, is a straightforward task. For instance, if we extend f (•, t) to the whole real line by setting f (R, t) = f (0, t) for R ≤ 0, the extension satisfies inequality (83) for all R ∈ R. We deduce that

f (R, t) ≤ g(R -2C(M ) √ t, t) , R ≥ 0 , t ≥ 0 ,
where g is the solution of the heat equation

∂ t g = ∂ 2 R g on R with initial data g(R, 0) = f (R, 0). Given any ǫ > 0, we choose R 0 > 0 large enough so that f (R 0 , 0) ≤ ǫ. If R > R 0 we estimate g(R, t) = 1 √ 4πt R 0 -∞ e -(R-r) 2 4t f (r, 0) dr + 1 √ 4πt ∞ R 0 e -(R-r) 2 4t f (r, 0) dr ≤ e -(R-R 0 ) 2 4t √ 4πt R 0 -∞ e -(R 0 -r) 2 4t M dr + ǫ √ 4πt ∞ R 0 e -(R-r) 2 4t dr ≤ M e -(R-R 0 ) 2 4t + ǫ . The right-hand side is smaller than 2ǫ if R ≥ R 0 + 2 √ t log(M/ǫ) 1/2 . Summarizing, we have shown that f (R, t) ≤ 2ǫ provided R ≥ R 0 + K √ t, with K = 2C(M ) + 2 log(M/ǫ) 1/2
. The argument is similar for the confinement in the vertical direction. For Z ∈ R and t ≥ 0, we define

h + (Z, t) = ∞ Z ∞ 0 ω θ (r, z, t) dr dz , h -(Z, t) = Z -∞ ∞ 0 ω θ (r, z, t) dr dz . Then Z → h + (Z, t) is non-increasing, Z → h -(Z, t
) is non-decreasing, and we have the differential inequalities

∂ t h ± (Z, t) ≤ ∂ 2 Z h ± (Z, t) ∓ C(M ) √ t ∂ Z h ± (Z, t) ,
which allow us to compare h ± (Z, t) with suitably translated solutions of the one-dimensional heat equation. Proceeding as above, we find that, for any ǫ > 0, there exist positive constants

Z 0 and K such that h + (Z, t) ≤ 2ǫ if Z ≥ Z 0 + K √ t, and h -(Z, t) ≤ 2ǫ if Z ≤ -Z 0 -K √ t. It follows that ∞ 0 |z|≥Z 0 +K √ t ω θ (r, z, t) dz dz ≤ 4ǫ , t ≥ 0 .
Combining this result with the previous estimate on f (R, t), we obtain (80).

Proposition 6.2 For any initial data ω 0 ∈ L 1 (Ω), the solution of the axisymmetric vorticity equation (48) satisfies ω θ (t) L 1 (Ω) → 0 as t → ∞.

Proof. We know from Lemma 5.1 that ω θ (t) L 1 (Ω) converges to some limit ℓ as t → ∞. To prove that ℓ = 0, we use a standard rescaling argument. Let (λ n ) n∈N be an increasing sequence of positive real numbers such that λ n → ∞ as n → ∞. We define for all n ∈ N :

w n (r, z, t) = λ 2 n ω θ (λ n r, λ n z, λ 2 n (1 + t)) , (r, z) ∈ Ω , t ≥ 0 .
Denoting w n (t) = w n (•, •, t), we claim that the sequence (w n (0)) n∈N is relatively compact in L 1 (Ω). Indeed, by Proposition 6.1, for all ǫ > 0 there exists a compact set Ω 0 ⊂ Ω such that

sup n∈N Ω\Ω 0 |w n (r, z, 0)| dr dz ≤ ǫ . (84)
In addition, Proposition 5.5 asserts that

sup n∈N ∇w n (0) L ∞ (Ω) = sup n∈N λ 3 n ∇ω θ (λ 2 n ) L ∞ (Ω) < ∞ . (85)
Combining ( 84), (85) and using the Riesz criterion [21, Theorem XIII.66], we obtain the desired compactness. Thus, after extracting a subsequence, we can assume that w n (0) converges in L 1 (Ω) to some limit w, which satisfies w

L 1 (Ω) = ℓ because w n (0) L 1 (Ω) = ω θ (λ 2 n ) L 1 (Ω) → ℓ as n → ∞.
Now, we fix t > 0, and repeating the procedure above we extract yet another subsequence so that w n (t) converges in L 1 (Ω) to some limit w(t). By construction, for each n ∈ N, the function w n (r, z, t) solves the vorticity equation (48) with initial data w n (r, z, 0). As in the proof of Proposition 4.1, we thus denote w n (t) = Σ(t)w n (0). Taking the limit n → ∞ and using the fact that the solutions of (48) depend continuously on the initial data in L 1 (Ω), see Remark 4.2.2, we deduce that w(t) = Σ(t)w. But since w(t) L 1 (Ω) = w L 1 (Ω) = ℓ, we get a contradiction with the strict decay of the L 1 norm established in Lemma 5.1, unless ℓ = 0. Corollary 6.3 Under the assumptions of Proposition 6.2, we have

lim t→∞ t 1-1 p ω θ (t) L p (Ω) = 0 , 1 ≤ p ≤ ∞ . (86)
Proof. Given any t 0 > 0, we can apply Proposition 5.3 to the solution of (48) restricted to the time interval [t 0 , ∞). For any p ∈ [1, ∞], we thus find ω θ (t) L p (Ω) ≤ C p ( ω(t 0 ) L 1 (Ω) )

(t -t 0 ) Taking now the limit t 0 → ∞ and using Proposition 6.2 together with the fact that C p (s) = O(s) as s → 0, we obtain (86).

Asympotic behavior of positive solutions with finite impulse

We now consider the particular situation where the initial vorticity ω 0 ∈ L 1 (Ω) is non-negative and has a finite impulse. We denote

As is well-known, the impulse I is conserved for solutions of (48). The last integral actually vanishes. Indeed, using the explicit formulas ( 22) and ( 23), we obtain Ω ru r ω θ dr dz = -1 π Ω Ω z -z r 1/2 r1/2 F ′ (ξ 2 )ω θ (r, z)ω θ (r, z) dr dz dr dz = 0 , because the integrand is odd with respect to the permutation (r, z) ↔ (r, z). This gives the desired result.

Under the assumption that I < ∞, one can obtain precise information on the long-time behavior of the axisymmetric vorticity. We begin with the linear case :

where convergence holds in L p (Ω) for 1 ≤ p ≤ ∞.

Proof. In view of the explicit formula (36), we have for all (r, z) ∈ Ω : Proof. We estimate the integral term in (49) in the following way. First, using (47) with α = 0 and β = 1, we write Thus it follows from (49) that ω θ (t) L 1 (Ω) ≤ S(t)ω 0 L 1 (Ω) + CM Since S(t)ω 0 L 1 (Ω) = O(t -1 ) as t → ∞ by Lemma 6.5, the integral inequality (90) implies (by a straightforward bootstrap argument) that ω θ (t) L 1 (Ω) = O(t -1 ) as t → ∞. In a similar way, one can show that ω θ (t) L p (Ω) = O(t -2+1/p ) as t → ∞ for all p ∈ [1, ∞].

t
To prove (89), we fix t 0 ≥ 1 and consider the integral equation (62) for t ≥ t 0 . If we bound the nonlinear term exactly as in (90), using the additional information that ω θ (t) L 1 (Ω) = O(t -1 ), we obtain the estimate t ω θ (t) -S(t -t 0 )ω θ (t 0 ) L 1 (Ω) ≤ C(M, I) √ t 0 ,

for some positive constant C(M, I). We now rescale the solution ω θ (r, z, t) as in (89) and take the limit as t → ∞. Using (91) and Lemma 6.5, we obtain lim sup

t→∞ t 2 ω θ (• √ t, • √ t, t) -Φ L 1 (Ω) ≤ C(M, I) √ t 0 , (92)
where Φ : Ω → R denotes the function defined by the expression in the right-hand side of (89).

If we take the limit t 0 → ∞ in (92), we see that (89) holds if convergence is understood in L 1 (Ω).

A similar argument shows that convergence also holds L p (Ω) for all p ∈ [1, ∞].

Proposition 3 . 4

 34 The family (S(t)) t≥0 defined by (36) is a strongly continuous semigroup of bounded linear operators in L p (Ω) for any

1 .Remark 5 . 4

 154 Using the notations (58), we know from Proposition 3.4 that M p (T ) ≤ CM for any p ∈ [1, ∞], and from the argument above that N p (T ) ≤ C(M ) for p ∈[START_REF] Abidi | Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes[END_REF][START_REF] Abidi | On the global well-posedness for the axisymmetric Euler equations[END_REF]. The relation (61) then shows that N p (T ) ≤ C(M ) for all p > 2, and a second iteration gives the desired result for p = ∞ too. Proposition 5.3 shows in particular that the L p norms of the vorticity ω θ (t) cannot blow up in finite time. In view of Remark 4.2, this implies that all solutions constructed in Sections 4.1 and 4.2 are global for positive times, and that the conclusions of Lemma 5.1 and Proposition 5.3 hold for all t > 0.

Proposition 5 . 5

 55 Under the assumptions of Proposition 5.3, we have for all p ∈ [1, ∞] :

M = Ω ω 0

 0 (r, z) dr dz , I = Ω r 2 ω 0 (r, z) dr dz .

Lemma 6 . 4 r 2 ω θ dr dz - Ω r 2 2 Ω

 6422 For any non-negative solution of (48) in L 1 (Ω) with finite impulse, we have Ω r 2 ω θ (r, z, t) dr dz = Ω r 2 ω 0 (r, z) dr dz , t ≥ 0 . Proof. Using (48) we find by a direct calculation d dt Ω r 2 ω θ dr dz = div * (uω θ ) dr dz = ru r ω θ dr dz .

Lemma 6 . 5 √ π r e -r 2 +z 2 4 ,

 654 If ω 0 ∈ L 1 (Ω) is non-negative and I < ∞, one has lim t→∞ t 2 (S(t)ω 0 )(r √ t, z √ t) = I 16 (r, z) ∈ Ω ,

t/ 2 0SL 1 2 L 1 ω θ /r 1 / 2 L ∞ ≤ C ω θ 3 / 4 L 1 r 2 ω θ 3 / 4 L 1 ω θ /r 1 / 2 L 1 / 2 I 3 4 L 1 ,/ 2 S 2 C 1 / 2 I 1

 212123434121234122121 (t -s) div * (u(s)ω θ (s)) ds ) L ∞ (Ω) rω θ (s) L 1 (Ω) ds .From (30) we have u L ∞ rω θ L 1 ≤ C rω θ 3/∞ , hence using Lemmas 5.1 and 5.2 we obtainu(s) L ∞ (Ω) rω θ (s) L 1 (Ω) ≤ C Mwhere M, I are defined in (87). Similarly, we findt t(t -s) div * (u(s)ω θ (s)) ds L 1 (Ω) (t -s) 1/2 u(s) L ∞ (Ω) ω θ (s) L 1 (Ω) ds , where u(s) L ∞ (Ω) ω θ (s) L 1 (Ω) ≤ C M

  (t) L p (Ω) ≤ C p ( ω(t 0 ) L 1 (Ω) ) .

			p 1-1	,	t > t 0 ,
	hence	lim sup t→∞	t 1-1 p ω θ

  (r, z) dr dz , where K(τ ) = τ 3/2 H(τ ) is uniformly bounded and converges to √ π/4 as τ → +∞. The pointwise result (88) thus follows from Lebesgue's dominated convergence theorem. Convergence in L p norms is easy to prove if ω 0 has compact support in Ω, and can be established in the general case by an approximation argument.It follows in particular from (88) that S(t)ω 0 L p (Ω) = O(t Our last result is the extension of Lemma 6.5 to the nonlinear case. Proposition 6.6 If ω 0 ∈ L 1 (Ω) is non-negative and I < ∞, the solution of (48) with initial data ω 0 satisfies where convergence holds in L p (Ω) for all p ∈ [1, ∞]. Thus ω θ (t) L p (Ω) = O(t

	2 (S(t)ω 0 )(r	√	t, z	√	t) =	r 4π Ω	K	√ rr t	exp -	1 4	r-r √	t	2	+ z-z √ t r2 ω 0 -2+ 1 2
			lim t→∞	t 2 ω θ (r	√	t, z	√	t, t) =	I 16 √ π	r e -r 2 +z 2 4	,	(r, z) ∈ Ω ,	(89)
															-2+ 1

p ) as t → ∞.

p ) as t → ∞.

  1/2 I 3/4

	+ CM 1/2 I 1/4	t t/2	t/2 (t -s) 1/2 s 3/4 ds . 0 ω θ (s) (t -s) s 3/4 ds 3/4 L 1 5/4 ω θ (s) L 1	(90)

The case where the initial vorticity is a finite measure

We next consider the more general case where the initial vorticity ω 0 in (49) is a finite measure on Ω, which is no longer absolutely continuous with respect to the Lebesgue measure dr dz. For convenience we denote µ = ω 0 , and we recall the canonical decomposition µ = µ ac + µ sc + µ pp where µ ac is absolutely continuous with respect to Lebesgue's measure, µ pp is a (countable) collection of point masses, and µ sc is the "singularly continuous" part which has no atoms yet is supported on a set of zero Lebesgue measure. We have µ ac ⊥ µ sc ⊥ µ pp , which means that the three measures are mutually singular. In particular, the total variation norm of µ satisfies

The linear semigroup S(t) acts on the measure µ by the formula

which generalizes (36), and we have the following estimates :

Proposition 4.3 Let µ be a finite measure on Ω. Then

and

Proof. Estimate (65) can be established as in Proposition 3.4, using the pointwise upper bound (42). To prove (66) we proceed as in the two-dimensional case [START_REF] Th | Stability and interaction of vortices in two-dimensional viscous flows[END_REF][START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF], with minor modifications. We know from (65) that L p (µ) ≤ C µ tv , hence using the canonical decomposition we find

Therefore, we only need to show that L p (µ ac ) = L p (µ sc ) = 0. In fact, it is sufficient to prove that for p = ∞, because the result then follows for 1 < p < ∞ by interpolation.

From now on, we thus assume that µ is a non-atomic finite measure on Ω, and we denote by |µ| the positive measure which represents the total variation of µ. Given any point ξ = (r, z) ∈ R 2 and any radius δ > 0, we define

where |ξ -ξ| is the Euclidean distance between ξ and ξ. We claim that, for any ǫ > 0, there exists δ > 0 such that sup

Indeed, if that property fails, there exist ǫ > 0, a sequence (ξ n ) of points of Ω, and a sequence (δ n ) of positive real numbers such that δ n → 0 as n → ∞ and |µ|(B(ξ n , δ n )) > ǫ for all n ∈ N. It is clear that the sequence (ξ n ) is bounded, because |µ| is a finite measure. Thus, after extracting a subsequence, we can assume that ξ n converges as n → ∞ to some point ξ = (r, z) ∈ Ω. For any δ > 0 we thus have |µ|(B( ξ, δ)) > ǫ, since B( ξ, δ) ⊃ B(ξ n , δ n ) when n is sufficiently large. We conclude that |µ| δ>0 B( ξ, δ) ≥ ǫ > 0 .