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

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

Thierry Gallay Institut Fourier UMR CNRS 5582

Universit´e de Grenoble I, BP 74 38402 Saint-Martin-d’H`eres, France [email protected]

Vladim´ır ˇSver´ak School of Mathematics University of Minnesota 127 Vincent Hall, 206 Church St. SE

Minneapolis, MN 55455, USA [email protected] September 30, 2015

Abstract

Motivated by applications to vortex rings, we study the Cauchy problem for the three- dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant func- tion spaces. If the axisymmetric vorticityω^θis integrable with respect to the two-dimensional measure drdz, where (r, θ, z) denote the cylindrical coordinates inR³, we show the existence of a unique global solution, which converges to zero in L¹ norm ast → ∞. 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.

1 Introduction

Among all three-dimensional incompressible flows,axisymmetric flows without swirlform a par- ticular 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 [17] and Ukhovksii & Yudovich [23]. In the viscous case, the original approach of [17, 23] applies to velocity fields in the Sobolev space H²(R³), see [18], but it is possible to obtain the same conclusions under the weaker assumption that the initial velocity belongs toH^1/2(R³) [1]. 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

∗This paper is dedicated to Prof. Denis Serre on the occasion of his sixtieth birthday.

only. As was already mentioned, all previous works deal with finite energy solutions, and energy is not a scale invariant quantity for the three-dimensional viscous flows. In particular, the long- time 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 [3, 11, 14]. 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 incom- pressible flows is described by the Navier-Stokes equations

∂_tu+ (u· ∇)u = ∆u− ∇p , divu = 0 , (1) where u = u(x, t) ∈ R³ 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 . (2) Here (r, θ, z) are the usual cylindrical coordinates in R³, defined by x = (rcosθ, rsinθ, z) for any x∈R³, ande_r, e_θ, e_z denote the unit vectors in the radial, toroidal, and vertical directions, respectively :

er =



 cosθ sinθ

0



 , e_θ =





−sinθ cosθ

0



 , ez =



 0 0 1



 .

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 = curluassociated with the velocity field (2) is purely toroidal :

ω(r, z, t) = ω_θ(r, z, t)e_θ , where ω_θ =∂_zu_r−∂_ru_z . (3) 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

∂_ru_r+1

ru_r+∂_zu_z = 0, ∂_zu_r−∂_ru_z = ω_θ , (4) in the half space Ω = {(r, z) ∈ R²|r > 0, z ∈ R}, with boundary conditions u_r = ∂_ru_z = 0 at r = 0. System (4) 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

∂_tω_θ+u· ∇ω_θ− u_r

r ω_θ = ∆ω_θ− ω_θ

r² , (5)

where u· ∇=ur∂r+uz∂z and ∆ =∂_r²+ ¹_r∂r+∂_z² denotes the Laplace operator in cylindrical coordinates. Equation (5) is considered in the half-plane Ω with homogeneous Dirichlet condition

at the boundary r = 0. As was already observed in [17, 23], it is useful to consider also the related quantity

η(r, z, t) = ω_θ(r, z, t)

r , (6)

which satisfies the advection-diffusion equation

∂tη+u· ∇η = ∆η+2

r∂rη , (7)

with homogeneous Neumann condition at the boundaryr = 0. Systems (5) and (7) are, of course, perfectly equivalent. In what follows we find it more convenient to work with the axisymmetric vorticity equation (5), at least to prove local existence of solutions, but equation (7) 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) ∈ R²|r > 0, z ∈ R} with the two-dimensional measure drdz, as opposed to the 3D measurerdrdzwhich could appear more natural for axisymmetric problems.

Thus, given anyp ∈[1,∞), we denote by L^p(Ω) the space of measurable functions ω_θ : Ω→ R for which the following norm is finite :

kω_θkL^p(Ω) = Z

Ω|ω_θ(r, z)|^pdrdz 1/p

, 1≤p <∞ .

The spaceL^∞(Ω) is defined similarly. Sometimes, however, it is more convenient to use the 3D measure rdrdz, and the corresponding spaces are then denoted byL^p(R³) to avoid confusion.

For instance, we define

kηkL^p(R3) = Z

Ω|η(r, z)|^prdrdz 1/p

, 1≤p <∞. We are now in position to state our first main result.

Theorem 1.1 For any initial data ω₀ ∈ L¹(Ω), the axisymmetric vorticity equation (5) has a unique global mild solution

ω_θ∈C⁰([0,∞), L¹(Ω))∩C⁰((0,∞), L^∞(Ω)) . (8) The solution satisfies kω_θ(t)kL¹(Ω)≤ kω0kL¹(Ω) for allt >0, and

limt→0t^1−1pkω_θ(t)kL^p(Ω) = 0 , for 1< p≤ ∞ , (9)

tlim→∞t^1−1pkω_θ(t)kL^p(Ω) = 0 , for 1≤p≤ ∞ . (10) If, in addition, the axisymmetric vorticity is non-negative and has finite impulse :

I = Z

Ω

r²ω₀(r, z) drdz < ∞ , (11)

then

tlim→∞t²ω_θ(r√ t, z√

t, t) = I

16√πr e^−r2+z

2

4 , (r, z)∈Ω, (12)

where convergence holds in L^p(Ω) for 1 ≤ p ≤ ∞. In particular kω_θ(t)kL^p(Ω) = O(t^−2+1p) as t→ ∞ in that case.

Remark 1.2 A mild solution of (5) on R₊ = [0,∞) is a solution of the associated integral equation, namely Eq. (48) below. As will be verified in Section 4, both sides of (48) are well- defined if ω_θ satisfies (8) and (9). In the uniqueness claim, we only assume that ω_θ satisfies (8) and is a mild solution of (5) for strictly positive times. The proof then shows that (9) 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, ifω_θ ∈L¹(Ω), it is easy to verify that the vorticity ω =ω_θe_θ belongs to the Morrey space M^3/2(R³) defined by the norm

kωkM^3/2 = sup

x∈R3

sup

R>0

1 R

Z

B(x,R)|ω(x)|dx ,

whereB(x, R)⊂R³ denotes the ball or radius R >0 centered at x∈R³. In additionω can be approximated in M^3/2(R³) by smooth and compactly supported functions. As was proved by Giga & Miyakawa [12], the Navier-Stokes equations inR³thus have a unique local solution with initial vorticityω, which is even global in time if the normkωkM^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 inR³ belongs to the space BMO⁻¹(R³) defined by the norm

kukBMO⁻¹ = sup

x∈R3

sup

R>0

1 R³

Z

B(x,R)

Z _R²

0 |e^t∆u|²dtdx

!1/2

,

where e^t∆ denotes the heat semigroup in R³. In fact u can be approximated by smooth and compactly supported functions in BMO⁻¹(R³), so that u ∈ VMO⁻¹(R³). Thus we can also invoke the celebrated result by Koch & Tataru [16] to obtain the existence of a unique local solution to the Navier-Stokes equation inR³, which is again global in time if the normkukBMO⁻¹

is sufficiently small. In contrast to the general results in [12, 16], the approach we follow to solve the Cauchy problem for Eq. (5) 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 [3, 7].

The assertion of global well-posedness in Theorem 1.1 is also quite natural in view of the historical results by Ladyzhenskaya [17] and Ukhovkii & Yudovich [23]. As in [17, 23] we use the structure of equation (7) to derive a priori estimates on the quantityηinL^p(R³), 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 inL²(R³). Instead we prove that any solution of (5) with initial dataω₀ ∈L¹(Ω) satisfieskω_θ(t)kL¹(Ω) ≤ kω₀kL¹(Ω)

for all t >0 and

sup

t>0

tkω_θ(t)kL^∞(Ω) ≤ C kω₀kL¹(Ω)

, (13) whereC(s) =O(s) as s→0. This new a priori estimate is scale invariant, and implies that all solutions of (5) with initial data in L¹(Ω) are global. Using the axisymmetric Biot-Savart law, we also deduce the following optimal bound on the velocity field :

sup

t>0

t^1/2ku(t)kL^∞(Ω) ≤ C kω₀kL¹(Ω)

. (14) Our last comment on Theorem 1.1 concerns thelong-time behavior, which differs significantly from what happens in the two-dimensional case. In the latter situation, the L¹ 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 [8, 9]. In contrast, the axisymmetric vorticity ω_θ vanishes on the boundary of the half-plane Ω, so that theL¹ norm is strictly decreasing for all nontrivial solutions. As we shall see in Section 6, this implies that kω_θ(t)kL¹(Ω) → 0 whent → ∞, as asserted in (10). 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 (11). 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 (34), whose profile is explicitly determined in (12).

As in the two-dimensional case, it is possible to extend the local existence claim in Theo- rem 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

kµk^tv = sup Z

Ω

φdµ

φ∈C0(Ω), kφkL^∞(Ω)≤1

, forµ∈ M(Ω),

where C0(Ω) is the set of all real-valued continuous functions on Ω that vanish at infinity and on the boundary∂Ω. Ifµ∈ M(Ω) is absolutely continuous with respect to Lebesgue’s measure, then µ = ω_θdrdz for some ω_θ ∈ L¹(Ω), and kµktv = kω_θkL¹(Ω). More generally, one can decompose any µ ∈ M(Ω) as µ = µac +µsc +µpp, where µac 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 partµ_ppcorresponds to a circular vortex filament, whereas vortex sheets are included in the singularly continuous partµ_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 constantsǫandC such that, for any initial dataω₀ ∈ M(Ω) with k(ω₀)_ppktv ≤ǫ, the axisymmetric vorticity equation (5) has a unique global mild solution ω_θ ∈C⁰((0,∞), L¹(Ω)∩L^∞(Ω)) satisfying

lim sup

t→0 kω_θ(t)kL¹(Ω) < ∞ , lim sup

t→0

t^1/4kω_θ(t)kL^4/3(Ω) ≤ Cǫ , (15) and such that ω_θ(t) ⇀ ω₀ 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 [11] and by Kato [14]

had a similar limitation, which was then relaxed in [9, 6] using completely different techniques.

In the axisymmetric situation, existence of a global solution to (5) with a large Dirac mass as initial vorticity has recently been established by Feng and ˇSver´ak [5], using an approximation argument, but uniqueness is still under investigation. For a general initial vorticityω₀∈ 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 (15), we believe that uniqueness should hold (as in the two-dimensional case)

under the sole assumptions that ω_θ(t) is uniformly bounded in L¹(Ω) for t > 0 and converges weakly to the initial vorticity ω0 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 (15), to obtain uniqueness. We hope to clarify that question in a future work.

The rest of this paper is organized as follows. In Section 2, which is devoted to the ax- isymmetric Biot-Savart law, we estimate various norms of the velocity field u in terms of the axisymmetric vorticityω_θ. In Section 3, we show that the semigroup generated by the lineariza- tion of (5) 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 (5) converge to zero inL¹(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 Univer- sity 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 (VS) and by the grant “Dyficolti” ANR-13-BS01-0003-01 from the French Ministry of Research (ThG).

2 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 (4). The divergence-free condition ∂r(rur) +∂z(ruz) = 0 implies that there exists a function ψ: Ω→ R such that

u_r = −1 r

∂ψ

∂z , u_z = 1 r

∂ψ

∂r . (16)

As ∂zur−∂ruz = ω_θ, theaxisymmetric stream function ψ satisfies the following linear elliptic equation in the half-space Ω :

−∂_r²ψ+1

r∂_rψ−∂_z²ψ = rω_θ . (17)

Boundary conditions for (17) are determined by observing that, for a smooth axisymmetric vector field u : R³ → R³ with divu = 0, the stream function defined by (16) satisfies the asymptotic expansion

ψ(r, z) = ψ₀+r²ψ₂(z) +O(r⁴) , asr →0, (18) see [19] for an extensive discussion of these regularity issues. Without loss of generality, we can assume that the constant ψ₀ in (18) is equal to zero, in which case we conclude that ψ(0, z) =∂_rψ(0, z) = 0.

The solution of (17) with these boundary conditions is well known, see e.g. [5]. If we assume that the vorticity ω_θ decays sufficiently fast at infinity, we have the explicit representation

ψ(r, z) = 1 2π

Z

Ω

√rr F¯

(r−r)¯²+ (z−z)¯ ² rr¯

ω_θ(¯r,z) d¯¯ rd¯z , (19) where the function F : (0,∞)→Ris defined by

F(s) = Z _π

0

cosφdφ

2(1−cosφ) +s1/2 = Z _π/2

0

cos(2φ) dφ

sin²φ+s/41/2 , s >0. (20)

Useful properties of F are collected in the following lemma, whose proof can be found in [22, Section 19].

Lemma 2.1 The functionF : (0,∞) →Rdefined by (20)is decreasing and satisfies the asymp- totic expansions :

i) F(s) = log 8

√s

−2 +O slog1

s

and F^′(s) =− 1 2s+O

log1 s

as s→0;

ii)F(s) = π

2s^3/2 +O 1 s^5/2

andF^′(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 7→ s^αF(s) and s 7→

s^βF^′(s) are bounded if 0 < α ≤ 3/2 and 1 ≤ β ≤ 5/2. These observations will be constantly used in the subsequent proofs.

Combining (16) and (19), we obtain explicit formulas for the axisymmetric Biot-Savart law : u_r(r, z) =

Z

Ω

G_r(r, z,r,¯ z)ω¯ _θ(¯r,z) d¯¯ rd¯z , u_z(r, z) = Z

Ω

G_z(r, z,r,¯ z)ω¯ _θ(¯r,z) d¯¯ rd¯z , (21) where

G_r(r, z,r,¯ z) =¯ −1 π

z−¯z

r^3/2r¯^1/2 F^′(ξ²) , ξ² = (r−r)¯ ²+ (z−¯z)²

r¯r , (22)

G_z(r, z,r,¯ z) =¯ 1 π

r−¯r

r^3/2r¯^1/2F^′(ξ²) + 1 4π

¯ r^1/2 r^3/2

F(ξ²)−2ξ²F^′(ξ²)

. (23)

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 planeR², see e.g. [8, Lemma 2.1].

Proposition 2.3 The following properties hold for the velocity fieldudefined from the vorticity ω_θ via the axisymmetric Biot-Savart law (21).

i) Assume that 1< p <2< q <∞ and ¹_q = ¹_p−¹₂. If ω_θ∈L^p(Ω), then u∈L^q(Ω)² and

kukL^q(Ω) ≤ Ckω_θkL^p(Ω) . (24) ii) If 1≤p <2< q≤ ∞ andω_θ∈L^p(Ω)∩L^q(Ω), then u∈L^∞(Ω)² and

kukL^∞(Ω) ≤ Ckω_θk^σL^p(Ω)kω_θk¹_L^−q_(Ω)^σ , where σ = p 2

q−2

q−p ∈ (0,1) . (25) 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)¯ ²1/2 , (26) which holds for all (r, z)∈Ω and all (¯r,z)¯ ∈Ω. At a heuristic level, estimate (26) follows quite naturally from the scaling properties of Gr(r, z,r,¯ z) and¯ Gz(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¯ R². To prove (26) rigorously, we first bound the radial component Gr. We distinguish two cases :

a) If ¯r ≤2r, we use the fact thatξ²F^′(ξ²) is bounded, and obtain the estimate

|G_r(r, z,r,¯ z)¯ | ≤ C |z−z¯| r^3/2¯r^1/2

r¯r

(r−¯r)²+ (z−z)¯²

= Cr¯^1/2 r^1/2

|z−z¯|

(r−r)¯²+ (z−z)¯ ² ≤ C

(r−¯r)²+ (z−z)¯²1/2 , because ¯r/r ≤2 and |z−z¯| ≤ (r−r)¯²+ (z−z)¯ ²1/2

. b) If ¯r >2r, observing that ξ³F^′(ξ²) is bounded, we deduce

|G_r(r, z,r,¯ z)¯| ≤ C |z−z¯| r^3/2r¯^1/2

r^3/2r¯^3/2

(r−¯r)²+ (z−z)¯²3/2

= C r¯|z−z¯|

(r−¯r)²+ (z−z)¯ ²3/2 ≤ C

(r−r)¯²+ (z−z)¯ ²1/2 , because ¯r <2(¯r−r)≤2 (r−¯r)²+ (z−z)¯ ²1/2

. This proves estimate (26) forG_r.

Similar calculations give the same bound for the second component G_z too. Indeed, the first term in the right-hand side of (23) can be estimated exactly as above, using the obvious fact that|r−r¯| ≤ (r−r)¯²+ (z−z)¯ ²1/2

. The second term involves the quantityF(ξ²)−2ξ²F^′(ξ²), which is bounded byCξ⁻¹ in case a) and byCξ⁻³ in case b). This concludes the proof of (26).

Now, since the integral kernel G = (G_r, G_z) in (21) satisfies the same bound as the two- dimensional Biot-Savart kernel in R², properties i) and ii) in Proposition 2.3 can be established exactly as in the 2D case. Estimate (24) thus follows from the Hardy-Littlewood-Sobolev in- equality, and the bound (25) can be proved by splitting the integration domain and applying

H¨older’s inequality, see e.g. [8, Lemma 2.1].

The proof of Proposition 2.3 shows that the axisymmetric Biot-Savart law (21) has the same properties as the usual Biot-Savart law in the whole plane R². 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 (24).

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^βω_θ∈L^p(Ω), then r^αu ∈L^q(Ω)² and we have the bound

kr^αukL^q(Ω) ≤ Ckr^βω_θkL^p(Ω) . (27) Proof. As in the proof of Proposition 2.3, all we need is to establish the pointwise estimate

r^α

¯ r^β

|G_r(r, z,r,¯ z)¯ |+|G_z(r, z,r,¯ z)¯ |

≤ C

(r−¯r)²+ (z−z)¯²λ , (28) for some λ∈ (0,1). Indeed, once (28) is known, the bound (27) follows immediately from the Hardy-Littlewood-Sobolev inequality ifq⁻¹ =p⁻¹+λ−1. To prove (28), we proceed as before and distinguish three regions : a) ¯r/2≤r≤2¯r; b)r >2¯r; c) ¯r >2r. In each region, we bound

the quantitiesF^′(ξ²) andF(ξ²)−2ξ²F^′(ξ²) appropriately using Remark 2.2. These calculations shows that inequality (28) holds withλ= (1 +β−α)/2 if and only if the exponentsα, 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 (25).

2. If u is replaced by ur, inequality (27) holds for allα, β ∈[−1,2] such that0≤β−α <1.

3. If we take α= 1/q and β= 1/p, so that ¹_q = ¹_p −¹₃, inequality (27) is equivalent to kukL^q(R3) ≤ CkωkL^p(R3) ,

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 [5].

Proposition 2.6 The following estimates hold :

kukL^∞(Ω) ≤ Ckrω_θk^1/2_L¹_(Ω)kω_θ/rk^1/2_L∞(Ω) , (29)

u_r r

L^∞(Ω) ≤ Ckω_θk^1/3_L¹_(Ω)kω_θ/rk^2/3_L∞(Ω) . (30) Proof. Estimate (29) is stated in [5] in the slightly weaker form

kukL^∞(Ω) ≤ Ckω_θk^1/4_L¹_(Ω)kr²ω_θk^1/4_L¹_(Ω)kω_θ/rk^1/2_L∞(Ω) ,

but the proof given there actually yields the stronger bound (29), which can be compared with (25) in the case p = 1, q = ∞. To prove (30), we follow the same approach as in [5]. Using translation and scaling invariance, it is sufficient to show that the quantity

u_r r

r=1,z=0 = u_r(1,0) = Z

Ω

z πr^1/2 F^′

(r−1)²+z² r

ω_θ(r, z) drdz (31) is bounded byCkω_θk^1/3_L¹_(Ω)kω_θ/rk^2/3_L∞(Ω). We decompose Ω =I₁∪I₂, where

I₁ = n

(r, z)∈Ω

1

2 ≤r≤2, −1≤z≤1o

, I₂ = Ω\I₁ .

When integrating over the first region I₁, we use the fact that |F^′(s)| ≤ Cs⁻¹ and r ≈ 1. We thus obtain

|u⁽¹⁾_r (1,0)| ≤ C Z

I1

|z|

(r−1)²+z²|ω_θ(r, z)|drdz ≤ C Z

I1

|ω_θ(r, z)|

(r−1)²+z²1/2drdz . As in [5], or in part ii) or Proposition 2.3, we deduce that

|u⁽¹⁾_r (1,0)| ≤ Ckω_θk^1/2_L¹_(I1)kω_θk^1/2_L∞(I1) ≤ Ckω_θk^1/3_L¹_(I1)kω_θ/rk^2/3_L∞(I1) . (32) In the complementary region, we use the optimal bound|F^′(s)| ≤Cs^−5/2 and we observe that, if (r, z)∈I₂, then (r−1)²+z² ≥C(r²+z²) for some C >0. We thus have

|u⁽²⁾_r (1,0)| ≤ C Z

I2

|z|r²

(r−1)²+z²5/2 |ω_θ(r, z)|drdz ≤ C Z

I2

|ω_θ(r, z)|

r²+z² drdz .

Fix any R > 0 and denote Ω_R ={(r, z) ∈ Ω|ρ ≤ R}, where ρ = (r²+z²)^1/2. Extending the integration domain fromI2 to Ω = ΩR∪(Ω\ΩR), we compute

|u⁽²⁾_r (1,0)| ≤ C Z

ΩR

|ω_θ(r, z)|

ρ² drdz+C Z

Ω\ΩR

|ω_θ(r, z)| ρ² drdz

≤ Ckω_θ/rkL^∞(Ω)

Z

ΩR

1

ρdrdz+CR⁻² Z

Ω\ΩR

|ω_θ(r, z)|drdz

≤ CRkω_θ/rkL^∞(Ω)+CR⁻²kω_θkL¹(Ω) .

Optimizing over R >0 gives the bound |u⁽²⁾_r (1,0)| ≤ Ckω_θk^1/3_L¹_(Ω)kω_θ/rk^2/3_L∞(Ω), which together

with (32) implies (30).

Remark 2.7 Estimate (30) can also be obtained by observing that

u_r r

L^∞(R3) ≤ C

ω_θ r

L^3,1(R3) ≤ C

ω_θ r

1/3 L¹(R3)

ω_θ r

2/3

L^∞(R3) , (33) where L^3,1(R³) denotes the Lorentz space. The first inequality in (33) is proved in [2, Proposi- tion 4.1], and the second one follows by real interpolation.

3 The semigroup associated with the linearized equation

This section is devoted to the study of the linearized vorticity equation (5) :

∂_tω_θ =

∂_r²+∂²_z+1

r∂_r− 1 r²

ω_θ , (34)

which is considered in the half-plane Ω = {(r, z)|r > 0, z ∈ R}, with homogeneous Dirichlet condition at the boundaryr = 0. Given initial dataω₀ ∈L¹(Ω), we denote by ω_θ(t) =S(t)ω₀ the solution of (34) at time t >0.

Lemma 3.1 For anyt >0, the evolution operator S(t) associated with Eq. (34)is given by the explicit formula

(S(t)ω0)(r, z) = 1 4πt

Z

Ω

¯ r^1/2 r^1/2 H t

rr¯

exp

−(r−¯r)²+ (z−z)¯² 4t

ω0(¯r,z) d¯¯ rd¯z , (35) where the function H: (0,+∞)→R is defined by

H(τ) = 1

√πτ Z _π/2

−π/2

e^{−sin2τ φ}cos(2φ) dφ , τ >0 . (36) Proof. Ifω_θ is a solution of (34), we observe that the vector valued functionω=ω_θe_θ satisfies the usual heat equation ∂tω = ∆ω in the whole space R³. For any t > 0, we thus have the solution formula

ω(x, t) = 1 (4πt)^3/2

Z

R3

e^−|x−¯x|

2

4t ω(¯x,0) d¯x , x∈R³ . (37) Denoting x= (rcosθ, rsinθ, z) and ¯x= (¯rcos ¯θ,¯rsin ¯θ,z), we can write (37) in the form¯

ω_θ(r, z, t)





−sinθ cosθ

0



 = 1 (4πt)^3/2

Z _∞

0

Z

R

Z _π

−π

e^−|x−¯x|

2

4t ω₀(¯r,z)¯





−sin ¯θ cos ¯θ

0



r¯d¯θd¯zd¯r , (38)

where

|x−x¯|² = (r−r)¯²+ (z−z)¯ ²+ 4rr¯sin²θ−θ¯ 2 .

Now, if we integrate over the angle ¯θ in (38) and use the definition (36) of H, we see that (38) is equivalent toω_θ(t) =S(t)ω₀ with S(t) defined by (35).

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 functionH: (0,∞) →Rdefined by (36)is smooth and satisfies the asymptotic expansions :

i) H(τ) = 1−3τ

4 +O(τ²) 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 (36) if, in the expressione^−sin2τφ, we replace the exponential function by its Taylor series at the origin. Expansion i) can be deduced from the formula

H(τ) = 1

√π Z √¹τ

−^√1_τ

e^−x2 1−2τ x²

√1−τ x² dx , which is obtained by substitutingx= ^sin√^φ

τ in the integral (36).

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 7→ τ^αH(τ) and τ 7→ τ^βH^′(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 (35) fort >0 and byS(0) =1(the identity operator). By construction, we have the semigroup property : S(t₁ +t₂) = S(t₁)S(t₂) for all t₁, t₂ ≥0. Further important properties are collected in the following proposition.

Proposition 3.4 The family (S(t))_t≥0 defined by (35) is a strongly continuous semigroup of bounded linear operators in L^p(Ω) for any p ∈ [1,∞). Moreover, if 1 ≤ p ≤ q ≤ ∞, the following estimates hold :

i) If ω₀ ∈L^p(Ω), then

kS(t)ω₀kL^q(Ω) ≤ C

t^1p−1q kω₀kL^p(Ω) , t >0 . (39) ii) If f = (f_r, f_z)∈L^p(Ω)², then

kS(t) div_∗fkL^q(Ω) ≤ C

t^12+1p−1q kfkL^p(Ω) , t >0 , (40) where div_∗f =∂_rf_r+∂_zf_z denotes the two-dimensional divergence of f.

Proof. We claim that 1

4πt

¯ r^1/2 r^1/2H t

r¯r exp

−(r−r)¯²+ (z−z)¯ ² 4t

≤ C t exp

−(r−r)¯²+ (z−z)¯ ² 5t

, (41)

for all (r, z) ∈ Ω, all (¯r,z)¯ ∈ Ω, and all t > 0. Indeed, since H is bounded, estimate (41) is obvious when ¯r≤2r. If ¯r >2r, we observe that τ^1/2H(τ) is bounded, so that

¯ r^1/2 r^1/2H t

rr¯

≤ C ¯r^1/2 r^1/2

r^1/2r¯^1/2

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^−x2/4 ≤Ce^−x2/5 for all x ≥ 0, we conclude that (41) holds in all cases. This provides for the integral kernel in (35) a pointwise upper bound in terms of the usual heat kernel in the whole plane R², with a diffusion coefficient equal to 5/4 instead of 1. Thus estimate (39) follows immediately from Young’s inequality, as in the 2D case.

To prove (40), we assume that ω0 = div_∗f =∂rfr+∂zfz, and we integrate by parts in (35) to obtain the identity

(S(t) div_∗f)(r, z) = 1 4πt

Z

Ω

¯ r^1/2 r^1/2 exp

−(r−r)¯²+ (z−z)¯ ² 4t

(A_rf_r+A_zf_z) d¯rd¯z , where

A_r = t

r¯r²H^′ t r¯r

−1

2¯r +r−r¯ 2t

H t

r¯r

, A_z = −z−z¯ 2t H t

rr¯

. Proceeding as above and using Remark 3.3, it is straightforward to verify that

1 4πt

¯ r^1/2 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

, (42) for all (r, z) ∈ Ω, all (¯r,z)¯ ∈Ω, and all t > 0. Thus estimate (40) follows again from Young’s inequality, as in the 2D case.

Finally, we show that the semigroup (S(t))_t≥0 is strongly continuous inL^p(Ω) if 1≤p <∞. All we need to verify is the continuity at the origin. Given ω₀ ∈ L^p(Ω), we denote by ω₀ : R²→Rthe function obtained by extendingω₀ by zero outside Ω. Using the change of variables

¯

r=r+√

tρ, ¯z=z+√

tζ in (35), we obtain the identity

S(t)ω₀−ω₀

(r, z) = 1 4π

Z

R2

e^−ρ2+ζ

2

4 Ψ(r, z, ρ, ζ, t) dρdζ , for all (r, z) ∈Ω, where

Ψ(r, z, ρ, ζ, t) = 1 +

√tρ r

1/2

H t

r(r+√ tρ)

ω₀(r+√

tρ, z+√

tζ)−ω₀(r, z) . By Minkowski’s integral inequality, we deduce that

kS(t)ω₀−ω₀kL^p(Ω) ≤ 1 4π

Z

R2

e^−ρ2+ζ

2

4 kΨ(·,·, ρ, ζ, t)kL^p(Ω)dρdζ . (43) Now, the estimates we used in the proof of (41) show that

1 +

√tρ r

1/2

H t

r(r+√ tρ)

≤ C(1 +|ρ|) , (44)

whenever r >0 andr+√

tρ >0. This immediately implies that kΨ(·,·, ρ, ζ, t)kL^p(Ω) ≤ C(1 +|ρ|)kω₀kL^p(Ω) ,

for all (ρ, ζ) ∈ R² and all t > 0. Since the left-hand side of (44) converges to 1 as t→ 0, and since translations act continuously in L^p(R²) forp <∞, it also follows from estimate (44) and Lebesgue’s dominated convergence theorem that

kΨ(·,·, ρ, ζ, t)kL^p(Ω) −→ 0 ast→0 ,

for all (ρ, ζ)∈R². Thus another application of Lebesgue’s theorem implies that the right-hand side of (43) converges to zero ast→0, which is the desired result.

As in the previous section, it is possible to derive weighted estimates for the linear semigroup (35). 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^βω₀∈L^p(Ω), then kr^αS(t)ω0kL^q(Ω) ≤ C

t^{1p−1q+β−α2} kr^βω0kL^p(Ω) , t >0 . (45) Moreover, if −1≤α≤β ≤1 and r^βf ∈L^p(Ω)², then

kr^αS(t) div_∗fkL^q(Ω) ≤ C

t^{12+1p−1q+β−α2} kr^βfkL^p(Ω) , t >0 . (46)

4 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 (4), the evolution equation (5) for the axisymmetric vorticityω_θ can be written in the equivalent form

∂_tω_θ+ div_∗(u ω_θ) =

∂_r²+∂_z²+1

r∂_r− 1 r²

ω_θ , (47)

where div_∗(u ω_θ) =∂r(urω_θ) +∂z(uzω_θ). Given initial data ω0, the integral equation associated with (47) is

ω_θ(t) = S(t)ω₀− Z t

0

S(t−s) div_∗(u(s)ω_θ(s)) ds , t >0 , (48) whereS(t) denotes the linear semigroup defined in (35). In this section, our goal is to prove local existence and uniqueness of solutions to (48) using a standard fixed point argument, in the spirit of Kato [13]. For the sake of clarity, we first treat the case whereω₀ ∈L¹(Ω), and then consider the more complicated situation whereω₀ 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.

4.1 Local existence when the initial vorticity is integrable

Proposition 4.1 For any initial data ω₀ ∈ L¹(Ω), there exists T = T(ω₀) > 0 such that the integral equation (48) has a unique solution

ω_θ ∈C⁰([0, T], L¹(Ω))∩C⁰((0, T], L^∞(Ω)) . (49) Moreoverkω_θ(t)kL¹(Ω) ≤ kω0kL¹(Ω)for allt∈[0, T], and estimate (9)holds. Finally, ifkω0kL¹(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. [3, 7, 14]. Given T >0 we introduce the function space

XT = n

ω_θ ∈C⁰((0, T], L^4/3(Ω)

kω_θk^XT <∞o

, (50)

equipped with the norm

kω_θkXT = sup

0<t≤T

t^1/4kω_θ(t)kL^4/3(Ω) .

For any t ≥ 0, we denote ω_lin(t) = S(t)ω₀, where S(t) is the linear semigroup (35). It then follows from Proposition 3.4 thatω_lin∈XT for any T >0. For later use, we define

C₁(ω₀, T) = kω_linkXT = sup

0<t≤T

t^1/4kS(t)ω₀kL^4/3(Ω) . (51) In view of (39), there exists a universal constantC2>0 such thatC1(ω0, T)≤C2kω0kL¹(Ω) for any T >0. Moreover, since L¹(Ω)∩L^4/3(Ω) is dense in L¹(Ω), it also follows from (39) that C₁(ω₀, T)→0 as T →0, for anyω₀∈L¹(Ω).

Given ω_θ∈X_T and p∈[1,2), we define a mapFω_θ : (0, T]→L^p(Ω) in the following way : (Fω_θ)(t) =

Z _t

0

S(t−s) div_∗(u(s)ω_θ(s)) ds , 0< t≤T , (52) where it is understood thatu(s) is the velocity field obtained from ω_θ(s) via the axisymmetric Biot-Savart law (21). Using estimate (40), H¨older’s inequality, and the bound (24), we obtain fort∈(0, T] :

t^1−1pk(Fω_θ)(t)kL^p(Ω) ≤ t^1−1p Z _t

0

C

(t−s)^32−1p ku(s)ω_θ(s)kL¹(Ω)ds

≤ t^1−1p Z _t

0

C

(t−s)^32−1pku(s)kL⁴(Ω)kω_θ(s)kL^4/3(Ω)ds

≤ t^1−1p Z _t

0

C

(t−s)^32−1pkω_θ(s)k²_L^4/3_(Ω)ds (53)

≤ t^1−1p Z _t

0

C (t−s)^32−1p

kω_θk²XT

s¹² ds ≤ Ckω_θk²X_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 kFω_θkXT ≤C₃kω_θk²X_T for someC₃ >0. More generally, we have the Lipschitz estimate

kFω_θ− Fω˜_θkXT ≤ C₃ kω_θkXT +kω˜_θkXT

kω_θ−ω˜_θkXT , (54) for all ω_θ,ω˜_θ ∈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 2C3R < 1, and denote by BR the closed ball of radius R centered at the origin in X_T. If C₁(ω₀, 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 BR. By construction, ω_θ is a solution to the integral equation (48) in XT. The condition C₁(ω₀, T) ≤R/2 can be fulfilled in two different ways. If the initial data are small enough so thatC₂kω₀kL¹(Ω)≤R/2, the existence timeT >0 can be chosen arbitrarily, and the fixed point argument therefore establishes global existence for small data in L¹(Ω). On the other hand, for larger initial dataω₀, we can always chooseT >0 small enough so thatC₁(ω₀, T)≤R/2, hence we also have local existence for arbitrary data.