Ë L Rapha

Danchin
email: danchin@univ-paris12.fr

REMARKS ON THE LIFESPAN OF THE SOLUTIONS TO SOME MODELS OF INCOMPRESSIBLE FLUID MECHANICS

Keywords: 2010 Mathematics Subject Classification, 35Q35,76B03, 1

We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the N -dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl.

Introduction

The evolution of the velocity u = u(t, x) and pressure P = P (t, x) fields of a perfect homogeneous incompressible fluid is governed by the following Euler equations: (0.1)

∂ t u + u • ∇u + ∇P = 0, div u = 0.
There is a huge literature concerning the well-posedness issue for Euler equations. Roughly, they may be solved locally in time in any reasonable Banach space embedded in the set C 0,1 of bounded Lipschitz functions (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][START_REF] Bertozzi | Vorticity and incompressible flow[END_REF][START_REF] Chemin | Fluides parfaits incompressibles[END_REF][START_REF] Hmidi | Existence globale pour le système d'Euler incompressible 2-D dans B 1 ∞,1[END_REF][START_REF] Hmidi | Incompressible viscous flows in borderline Besov spaces[END_REF][START_REF] Pak | Existence of solution for the Euler equations in a critical Besov space B 1 ∞,1 (R n )[END_REF][START_REF] Vishik | Hydrodynamics in Besov spaces[END_REF][START_REF] Zhou | Local well-posedness for the incompressible Euler equations in the critical Besov spaces[END_REF]).

In the two-dimensional case, it is well known that Euler equations are globally well-posed for sufficiently smooth initial data. This noticeable fact relies on the conservation of the vorticity ω := ∂ 1 u 2 -∂ 2 u 1 along the flow of the velocity field, and has been first proved rigorously in the pioneering works by W. Wolibner [START_REF] Wolibner | Un théorème d'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long[END_REF] and V. Yudovich [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF].

This conservation property is no longer true, however, in more physically relevant contexts such as

(1) the three-dimensional setting for (0.1), (2) nonhomogeneous incompressible perfect fluids, (3) inviscid fluids subjected to a buoyancy force which is advected by the velocity fluid (the so-called inviscid Boussinesq system below). As a consequence, the problem of global existence for general (even smooth or small) data is still open for the above three cases.

In a recent work [START_REF] Danchin | The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces[END_REF], it has been shown that for slightly nonhomogeneous two-dimensional incompressible fluids, the lifespan tends to infinity when the nonhomogeneity tends to zero. The present paper is mainly dedicated to the study of the lifespan for the first and third item.

More precisely, in the first section of the paper, we shall consider the inviscid Boussinesq system:

(0.2)      ∂ t θ + u • ∇θ = 0, ∂ t u + u • ∇u + ∇P = θe N , div u = 0.
Here the relative temperature θ = θ(t, x) is a real valued function 1 and e N stands for the unit vertical vector.

As for the standard incompressible Euler equations, any functional space embedded in C 0,1 is a good candidate for the study of the well-posedness issue for (0.2). This stems from the fact that System (0.2) is a coupling between transport equations. Hence preserving the initial regularity requires the velocity field to be at least locally Lipschitz with respect to the space variable. By arguing as in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 7, one may show that, indeed, System (0.2) is locally well-posed in B s p,q whenever B s p,q is embedded in C 0,1 or, in other words, for any (s, p, q

) ∈ R × [1, +∞] 2 satisfying (0.3) s > 1 + N p or s = 1 + N p and q = 1 .
As a by-product of estimates for transport equations, we shall get various continuation criteria which generalize those of [START_REF] Shu | Small-scale structures in Boussinesq convection[END_REF] and of [START_REF] Liu | Local well-posedness and blow-up criterion of the Boussinesq equations in critical Besov spaces[END_REF]. We shall finally establish lower bounds for the lifespan of the solutions to (0.2) which show that in the two-dimensional case and for small initial temperature, the solution tends to be global-in-time.

As pointed out in many works (see e.g. [START_REF] Shu | Small-scale structures in Boussinesq convection[END_REF]), there is a formal similarity between the twodimensional Boussinesq system and general axisymmetric solutions to the three-dimensional Euler system -the so-called axisymmetric solutions with swirl. In the second part of this paper, we adapt the method of the first part so as to establish new lower bounds for the lifespan to those solutions in the case where the swirl is small. In particular, we find out that the solution tends to be global if the swirl goes to zero.

In the Appendix, we briefly recall the definition and a few basic properties of Besov space, and prove a commutator estimate.

Before going further into the description of our results, let us introduce a few notation.

• Throughout the paper, C stands for a harmless "constant" the meaning of which depends on the context. • The vorticity ω associated to a vector field u over R N is the matrix valued function with entries

ω ij := ∂ j u i -∂ i u j .
If N = 2 then the vorticity is identified with the scalar function ω := ∂ 1 u 2 -∂ 2 u 1 and if N = 3, with the vector field ∇ × u. • For all Banach space X and interval I of R, we denote by C(I; X) the set of continuous functions on I with values in X. If X has predual X * then we denote by C w (I; X) the set of bounded measurable functions f : I → X such that for any φ ∈ X * , the function t → f (t), φ X×X * is continuous over I.

The inviscid Boussinesq system

This section is devoted to the well-posedness issue for the inviscid Boussinesq system (0.2). We first establish a local-in-time existence result and continuation criteria in the spirit of those for the incompressible Euler equation. Next, we provide a new lower bound for the lifespan. Roughly, we establish that if θ 0 is of order ε, then the lifespan is at least of order log | log ε|.

1.1. Well-posedness and continuation criteria. The present subsection is devoted to the proof of the following result.

Theorem 1.1. Let (s, p, q) ∈ R × [1, +∞] 2 satisfy (0.3). Assume that u 0 (with div u 0 ≡ 0) and θ 0 belong to B s p,q and that, in addition, (u 0 , ∇θ 0 ) ∈ L r for some

r ∈]1, ∞[ if p = ∞. Then (0.2) admits a unique local-in-time solution (θ, u, ∇P ) in C w (] -T, T [; B s p,∞ ) if q = ∞ and in C(] -T, T [; B s p,q ) if q < ∞.
Besides, ∇θ and u are in C(] -T, T [; L r ) if (u 0 , ∇θ 0 ) ∈ L r . Furthermore, the solution may be continued beyond 2 T whenever one of the following three conditions is satisfied: 2 For expository purpose, we just consider positive times i)

T 0 ∇u L ∞ dt < ∞; ii) T 0 ω L ∞ + ∇θ L ∞ dt < ∞ and s > 1 + N/p; iii) N = 2, T 0 ∇θ L ∞ dt < ∞ and s > 1 + 2/p.
Before proving this result, a few comments are in order.

1. If it is assumed that ω 0 ∈ L r instead of u 0 ∈ L r then the vorticity of the constructed solution is continuous in time with values in L r . 2. In the two-dimensional case and in the Hölder spaces framework, the above statement has been established in [START_REF] Chae | Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations[END_REF]. The critical Besov case (that is p = 1 + 2/p, p ∈]1, ∞[) has been investigated in [START_REF] Liu | Local well-posedness and blow-up criterion of the Boussinesq equations in critical Besov spaces[END_REF]. 3. In [START_REF] Shu | Small-scale structures in Boussinesq convection[END_REF], a continuation criterion involving the L ∞ norm of the vorticity only has been stated. However, as the first inequality below [START_REF] Chae | Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations[END_REF] therein fails if m ≥ 2, we do not know whether that criterion is correct. 4. The first item has been proved recently in [START_REF] Liu | Local well-posedness and blow-up criterion of the Boussinesq equations in critical Besov spaces[END_REF] in the two dimensional case. 5. Let us finally mention that one may replace ω L ∞ with ω Ḃ0 ∞,∞ ∩L r in the second criterion.

Proof of Theorem 1.1. The proof of the local well-posedness in the Besov spaces framework is a straightforward adaptation to that of the corresponding result for the Euler system in B s p,q , and is thus omitted. The reader may refer to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 7 for more details. So let us go for the proof of the continuation criteria. Let us first assume that 1 < p < ∞. In this case, the Marcinkiewicz theorem for Calderon-Zygmund operators ensures that

(1.1) ∇u L p ≤ C ω L p .
Therefore, decomposing ω into low and high frequencies as follows 3 :

ω = ∆ -1 ω + (Id -∆ -1 )ω,
and taking advantage of the remark that follows Proposition A.1 in the appendix, we gather that

(1.2) ∇u B s-1 p,q ≤ C ω B s-1 p,q . Now, in dimension N the vorticity equation reads ∂ t ω + u • ∇ω + A(∇u, ω) = T ∇(θe N ) -∇(θe N ) with A(∇u, ω) := ω • ∇u + T ∇u • ω.
Hence applying ∆ j to the vorticity equation yields

∂ t ω j + u • ∇ω j = -∆ j A(∇u, ω) + ∆ j T ∇(θe N ) -∇(θe N ) + [u, ∆ j ] • ∇ω
with ω j := ∆ j ω and θ j := ∆ j θ. Therefore, because div u = 0,

(1.3) ω j (t) L p ≤ ω j (0) L p + t 0 ∇θ j L p dτ + t 0 ∆ j A(∇u, ω) L p dτ + t 0 [u, ∆ j ] • ∇ω L p dτ.
Next, let us use (see the appendix) that

(1.4) 2 j(s-1) [u, ∆ j ] • ∇ω L p ℓ q ∇u L ∞ ω B s-1 p,q
whenever s > 0.

3 The notation ∆-1 is defined in the appendix

If s > 1 + N/p then standard tame estimates (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2) imply that

A(∇u, ω) B s-1 p,q ≤ C ω L ∞ ∇u B s-1 p,q + ∇u L ∞ ω B s-1 p,q , ≤ C ∇u L ∞ ∇u B s-1 p,q .
The last inequality remains true in the limit case s = 1 + N/p and q = 1, a consequence of the algebraic structure of A(∇u, ω) (see e.g. Inequality (52) in [START_REF] Danchin | The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces[END_REF]).

Hence, multiplying (1.3) by 2 j(s-1) , taking the ℓ q norm with respect to j and taking advantage of (1.2) yields

(1.5) ω(t) B s-1 p,q ≤ ω 0 B s-1 p,q + C t 0 ∇θ B s-1 p,q dτ + C t 0 ∇u L ∞ ω B s-1 p,q dτ.
Next, in order to bound the B s p,q norm of θ, we use the fact that

∂ t θ j + u • ∇θ j = [u, ∆ j ] • ∇θ, whence (1.6) θ j (t) L p ≤ θ j (0) L p + t 0 [u, ∆ j ] • ∇θ L p dτ.
Given that, according to (a slight modification of) Lemma 2.100 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], we have

(1.7) 2 js [u, ∆ j ] • ∇θ L p ℓ q ∇u L ∞ θ B s p,q + ∇θ L ∞ ω B s-1 p,q , we eventually get (1.8) θ(t) B s p,q ≤ θ 0 B s p,q + C t 0 ∇u L ∞ θ B s p,q + ∇θ L ∞ ω B s-1 p,q dτ.
Finally, from the equation for θ, we easily get where P stands for the Leray projector over divergence-free vector-fields. As it is continuous over L p (recall that 1 < p < ∞), we deduce that

(1.9) ∇θ(t) L ∞ ≤ ∇θ 0 L ∞ + t 0 ∇u L ∞ ∇θ L ∞ dτ. So if ∇u is in L 1 ([0, T [; L ∞ ) then ∇θ is in L ∞ ([0, T [×R N ).
(1.11) u(t) L p ≤ u 0 L p + C t 0 ∇u L ∞ u L p dτ.
Now, the standard continuation criterion for hyperbolic PDEs ensures that the solution (θ, u) may be continued beyond T.

Let us now treat the case where s > 1 + N/p and (1.12)

T 0 ω L ∞ + ∇θ L ∞ dt < ∞.
We first bound ω and ∇θ in L ∞ ([0, T [; L p ) by taking advantage of (1.1) and of the vorticity and temperature equations. We get

ω(t) L p ≤ ω 0 L p + t 0 ∇θ L p dτ + C t 0 ω L p ω L ∞ dτ, (1.13) ∇θ(t) L p ≤ ∇θ 0 L p + C t 0 ∇θ L ∞ ω L p dτ. (1.14) Hence, (ω, ∇θ)(t) L p ≤ (ω 0 , ∇θ 0 ) L p + C t 0 (1 + (ω, ∇θ) L ∞ ) (ω, ∇θ) L p dτ.
So Gronwall's lemma provides us with a bound for ω and ∇θ in L ∞ ([0, T [; L p ).

Next, we use the following classical logarithmic interpolation inequality (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]):

(1.15) ∇u L ∞ ω L p ∩L ∞ log e + ω B s-1 p,q
.

Plugging this inequality in (1.5) and (1.8), and summing up, we get

ω(t) B s-1 p,q + θ(t) B s p,q ≤ ω 0 B s-1 p,q + θ 0 B s p,q +C t 0 1 + ∇θ L ∞ + ω L p ∩L ∞ ω B s-1 p,q + θ B s p,q log e + ω B s-1 p,q dτ. So Osgood's lemma implies that ω B s-1 p,r
and θ B s p,r are bounded on [0, T [. Bounding u L p may be done by combining Inequalities (1.11) and (1.15). Hence the solution (θ, u) may be continued beyond T.

Let us finally assume that N = 2 and that

T 0 ∇θ L ∞ dt < ∞.
Then Equation (1.16) gives

ω(t) L ∞ ≤ ω 0 L ∞ + t 0 ∂ 1 θ L ∞ dτ. Hence ω ∈ L ∞ ([0, T [×R 2
) and the previous continuation criterion implies that the solution (θ, u) may be continued beyond T.

Let us end the proof with a few comments concerning the cases p = 1, ∞. If p = ∞ and the solution also satisfies (∇θ, ω) ∈ L ∞ ([0, T [; L r ) for some r ∈]1, ∞[, then arguing as for proving (1.2) yields ∇u B s-1 ∞,q ∩L r ≤ C ω B s-1 ∞,q ∩L r . From the vorticity and temperature equations, we get

ω(t) L r ≤ ω 0 L r + 2 t 0 ∇u L ∞ ω L r dτ + 2 t 0 ∇θ L r dτ, ∇θ(t) L r ≤ ∇θ 0 L r + t 0 ∇u L ∞ ∇θ L r dτ.
So one may conclude that (1.5) and (1.8) hold true if replacing the norm in B s-1 ∞,q by the norm in B s-1 ∞,q ∩ L r . In order to bound u B s ∞,q , one may write that (using Bernstein's inequality to get the second line),

u(t) B s ∞,q ∆ -1 u(t) L ∞ + ω(t) B s-1 ∞,q , ∆ -1 u 0 L ∞ + ∆ -1 (u(t) -u 0 ) L r + ω(t) B s-1
∞,q . Now, according to (1.10), we have

u(t) -u 0 L r ≤ C t 0 u L r ∇u L ∞ dτ.
From this, it is easy to complete the proof.

Finally, if p = 1 then embedding ensures that ∇θ and u are in L ∞ ([0, T [; L r ) for some finite r, so that one may conclude as in the case p = ∞.

1.2. Lower bounds for the lifespan of the solutions to (0.2). Let (θ 0 , u 0 ) satisfy the assumptions of Theorem 1.1. Then is is clear that (θ ε , u ε , ∇Π ε ) satisfies (0.2) on [T -/ε, T + /ε] with initial data θ ε 0 = ε 2 θ 0 and u ε 0 = εu 0 if and only if the triplet (θ, u, ∇Π) defined by

θ ε (t, x) := ε 2 θ(εt, x), u ε (t, x) := εu(εt, x) and Π ε (t, x) := ε 2 Π(εt, x) satisfies (0.2) on [T -, T + ] with data (θ 0 , u 0 ).
From this, we gather that for initial temperature and velocity of size ε 2 and ε, respectively, the lifespan is (at least) of order ε -1 .

The above result is, obviously, independent of the dimension. At the same time, in the case θ 0 ≡ 0 (corresponding to the incompressible Euler equation) global existence holds true in dimension 2. In the case θ 0 ≡ 0, the question of global existence has remained unsolved, even in the two-dimensional case. We here want to study whether, nevertheless, dimension 2 is somehow "better". To answer this question, we shall take advantage of the fact that the vorticity equation in dimension 2 has no stretching term: it reduces to (1.16)

∂ t ω + u • ∇ω = ∂ 1 θ.
Hence, taking advantage of the special a priori estimates for the transport equation in Besov spaces with null regularity index (as discovered by M. Vishik in [START_REF] Vishik | Hydrodynamics in Besov spaces[END_REF] and by T. Hmidi and S. Keraani in [START_REF] Hmidi | Incompressible viscous flows in borderline Besov spaces[END_REF]), one may write

(1.17) ω(t) B 0 ∞,1 ≤ ω 0 B 0 ∞,1 + t 0 ∂ 1 θ B 0 ∞,1 dτ 1 + C t 0 ∇u L ∞ dτ .
This will be the key to our result below.

Theorem 1.2. Assume that N = 2. Let (θ 0 , u 0 ) be in B s p,q with (s, p, q) satisfying (0.3). If p ∈ {1, +∞}, suppose in addition that (∇θ 0 , ω 0 ) ∈ L r for some 1 < r < ∞. There exists a constant C depending only on r and such that (setting p = r if p ∈ (1, +∞)), the lifespan T * of (0.2) satisfies

T * ≥ 1 C ω 0 B 0 ∞,1 ∩L r log 1 + 1 2 log 1 + C ω 0 2 B 0 ∞,1 ∩L r ∇θ 0 B 0 ∞,1 ∩L r • Proof.
Let us first notice that, according to the continuation criteria derived in Theorem 1.1, it suffices to show that if the solution is defined on [0, T [×R n with T ≤ T * and T * as above, then ω and ∇θ are bounded in L ∞ (0, T ; B 0 ∞,1 ∩ L r ). Now, estimates for the transport equation in Besov spaces (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 3) yield

(1.18) ∇θ(t) B 0 ∞,1 ≤ ∇θ 0 B 0 ∞,1 e C t 0 ∇u B 0 ∞,1 dτ .
Of course, standard L r estimates for the transport equation imply that

(1.19) ω(t) L r ≤ ω 0 L r + t 0 ∂ 1 θ L r and ∇θ(t) L r ≤ ∇θ 0 L r e t 0 ∇u L ∞ dτ .
Let us finally notice that putting together embedding, Inequality (1.1) and the remark that follows Proposition A.1, we have

∇u L ∞ ∇u B 0 ∞,1 ω B 0 ∞,1 ∩L r . Therefore, denoting Ω(t) := ω(t) B 0 ∞,1 ∩L r and Θ(t) := ∇θ(t) B 0 ∞,1 ∩L r
and taking advantage of (1.17), (1.18), (1.19), we conclude that

Θ(t) ≤ Θ 0 e C t 0 Ω dτ , Ω(t) ≤ Ω 0 + t 0 Θ dτ 1 + C t 0 Ω dτ .
Now, plugging the inequality for Θ(t) in the inequality for Ω(t), we get

(1.20) Ω(t) ≤ Ω 0 + tΘ 0 e C t 0 Ω dτ 1 + C t 0 Ω dτ .
Let us assume for a while that Therefore, for Condition (1.21) to be satisfied, it suffices that

Θ 0 T exp e 2CT Ω 0 -1 ≤ Ω 0 that is to say (1.23) X exp(e X -1) ≤ Y with X := 2CT Ω 0 and Y = 2CΩ 2 0 Θ 0 • Let us notice that X ≤ e X -1 ≤ exp(e X -1) -1 for any X ∈ R + .
Hence Inequality (1.23) is satisfied provided that exp 2(e X -1) ≤ 1 + Y.

So we easily gather from a bootstrap argument that the lifespan T * satisfies

T * ≥ 1 2CΩ 0 log 1 + 1 2 log 1 + 2CΩ 2 0 Θ 0 ,
which is exactly the desired inequality.

Remark 1.3. In the case where the solution is C 1,r for some r ∈ (0, 1) (an assumption which is not satisfied in the critical regularity case), one may first write estimates for ω L ∞ and ω C r , and next use the classical logarithmic inequality for bounding ∇u L ∞ in terms of ω L ∞ and ω C r . This does not improve the lower bound for the lifespan, though.

The axisymmetric incompressible Euler equations

We now consider the incompressible Euler equations (0.1). As recalled in the introduction, Euler equations are globally well-posed in dimension 2. In dimension d ≥ 3, the global wellposedness issue has remained unsolved unless some property of symmetry is satisfied : it is known that axisymmetric or helicoidal without swirl data generate global solutions (see e.g. [START_REF] Danchin | Axisymmetric incompressible flows with bounded vorticity[END_REF] and the references therein for more details).

In the general case, an easy scaling argument similar to that of the Boussinesq system yields that for data of size ε, the lifespan is at least of order ε -1 .

Here we want to focus on the axisymmetric solutions to Euler equations with swirl, that is on solutions u to (0.1) such that, in cylindrical coordinates, (2.1) u(r, z) = u r (r, z)e r + u θ (r, z)e θ + u z (r, z)e z .

Recall that the corresponding vorticity reads ω(r, z) = ω r (r, z)e r + ω θ (r, z)e θ + ω z (r, z)e z with

ω r (r, z) = -∂ z u θ e r , ω θ (r, z) = ∂ z u r -∂ r u z , ω z (r, z) = 1 r ∂ r (ru θ ).
With this notation, axisymmetric solutions satisfy (see e.g. [START_REF] Bertozzi | Vorticity and incompressible flow[END_REF])

(2.2)              D t u r + ∂ r Π = r -1 (u θ ) 2 , D t u θ = -r -1 u r u θ , D t u z + ∂ z Π = 0, ∂ r (ru r ) + ∂ z (ru z ) = 0, with D t := ∂ t + u r ∂ r + u z ∂ z .
As pointed out in [START_REF] Shu | Small-scale structures in Boussinesq convection[END_REF], there is a striking similarity between the two-dimensional Boussinesq system (0.2) satisfied by (θ, ω) in the previous section, and the equations satisfied by (u θ , ω θ ) here. Indeed,

D t (ru θ ) = 0, D t ω θ -1 r u r ω θ -1 r ∂ z (u θ ) 2 = 0 whence, denoting Γ := (ru θ ) 2 and ζ := r -1 ω θ , we have (2.3) D t Γ = 0 and D t ζ = 1 r 4 ∂ z Γ.
Therefore, up to the singular coefficient 1/r 4 , the functions Γ = Γ(r, z) and ζ = ζ(r, z) play the same role as the temperature and the vorticity, respectively, in the 2D Boussinesq system. Keeping in mind that data such that u θ 0 ≡ 0 generate global solutions, it is natural to study whether having r -1 ω θ 0 = O(1) and ru θ 0 = O(ε) gives rise to a family of solutions with lifespan going to infinity when ε goes to 0.

For technical reasons however, due to the singularity near the axis, we shall consider the axisymmetric Euler equations in a smooth bounded axisymmetric domain Ω of R 3 such that, for some given 0 < r 0 < R 0 ,

(2.4) Ω ⊂ (x, y, z) ∈ R 3 : r 0 < x 2 + y 2 < R 0 •
Let us first give a local well-posedness result for the Euler equation in a domain:

Theorem 2.1. Let (s, p, q) satisfy Condition (0.3). Let u 0 be in B s p,q (Ω) with div u 0 = 0 and u 0 tangent to the boundary of Ω. Then System (0.1) with slip boundary conditions has a unique local solution u in C w (] -T, T [; B s p,q (Ω)) (or in C(] -T, T [; B s p,q (Ω)) if q < ∞). If in addition u 0 satisfies (2.1) then u satisfies (2.2).

Proof. This statement has been essentially proved by A. Dutrifoy in [START_REF] Dutrifoy | Precise regularity results for the Euler equations[END_REF] except in the critical case s = 1 + 3/p and r = 1. However, the critical case may be handled by the same method 4 as it relies on a priori estimates for transport equations which are also true in this case.

The last part of the statement is a classical consequence of the uniqueness and of the symmetry of the data u 0 .

One can now state the main result of this part. Theorem 2.2. Let u 0 be an axisymmetric divergence-free vector-field in B s p,q (Ω) with (s, p, q) satisfying (0.3) and Ω a bounded domain satisfying (2.4). Suppose in addition that u 0 | ∂Ω is tangent to the boundary of Ω. Then the lifespan T * to the solution of (0.1) satisfies

T * ≥ 1 C ω θ 0 B 0 ∞,1 log 1 + 1 2 log 1 + C ω θ 0 B 0 ∞,1 (u θ 0 ) 2 B 1 ∞,1
for some constant C depending only on Ω. 4 Proving a continuation criterion involving the vorticity was the main purpose of Dutrifoy's paper, and this requires that s > 1 + 3/p. This is probably the reason why the statement in the critical case is not given therein.

Proof. It suffices to bound the norm of u in B 1 ∞,1 as it controls high norms (see [START_REF] Dutrifoy | Precise regularity results for the Euler equations[END_REF] and notice that B 1 ∞,1 embeds in C 0,1 ). Let u := u r e r +u z e z . Denote by ψ the solution given by Proposition A.4 to the elliptic equation

-∆ ψ = ω θ e θ in Ω, ∂ n ψ = 0 on ∂Ω, Ω ψ dx = 0.
Notice that div u = 0 = div (∇ ∧ ψ) and that ∇ ∧ u = ω θ e θ = ∇ ∧ (∇ ∧ ψ).

As, in addition, both u and ∇∧ ψ have null circulation on the components of ∂Ω (a consequence of the symmetry properties of those two functions and of the domain), they coincide. Hence, Proposition A.4 ensures that

(2.5) ∇ u B 0 ∞,1 ≤ C ω θ e θ B 0 ∞,1
. This inequality will enable us to adapt to the axisymmetric Euler equations the proof of lower bounds for the lifespan of solutions.

We proceed as follows. According to the work by A. Dutrifoy (see in particular Prop. 6 and Cor. 5 in [START_REF] Dutrifoy | Precise regularity results for the Euler equations[END_REF]) for the transport equation in a smooth bounded domain, estimates in Besov spaces B s p,q (Ω) are the same as in the whole space case. From this, one may deduce by following the method of [START_REF] Hmidi | Incompressible viscous flows in borderline Besov spaces[END_REF] that in the particular case s = 0, the estimates improve (as in (1.17)). So we get, bearing (2.3) in mind:

ζ(t) B 0 ∞,1 ≤ ζ 0 B 0 ∞,1 + t 0 r -4 ∂ z Γ B 0 ∞,1 dτ 1 + C t 0 ∇ u L ∞ dτ .
General Dutrifoy's estimates for the transport equation also imply that

Γ(t) B 1 ∞,1 ≤ Γ 0 B 1 ∞,1 exp C t 0 ∇ u B 0 ∞,1 dτ .
Now, the important observation is that 1/r 4 is in C 0,1 (Ω) (for r ≥ r 0 in Ω). Hence

r -4 ∂ z Γ B 0 ∞,1 ≤ C ∂ z Γ B 0 ∞,1 , whence ζ(t) B 0 ∞,1 ≤ ζ 0 B 0 ∞,1 + C t 0 ∂ z Γ B 0 ∞,1 dτ 1 + C t 0 ∇ u L ∞ dτ .
Finally, according to (2.5) and classical embedding properties, we have

∇ u L ∞ ∇ u B 0 ∞,1 ω θ e θ B 0 ∞,1 .
As ω θ e θ = ζ re θ and, under our assumption on Ω, re θ is in C 0,1 , one may thus conclude that

∇ u L ∞ ζ B 0 ∞,1 .
From this point, one may proceed exactly as for the Boussinesq system; we deduce the following lower bound for the lifespan of the solution:

T * ≥ 1 C ζ 0 B 0 ∞,1 log 1 + 1 2 log 1 + C ζ 0 B 0 ∞,1 Γ 2 0 B 1 ∞,1

•

Of course, owing to the shape of Ω, up to an irrelevant constant, one may replace ζ 0 with ω θ 0 and ru θ 0 with u θ 0 , respectively.

Remark 2.3. We believe Theorem 2.2 to be true in the case where Ω satisfying (2.4) is unbounded. However, we refrained from giving the statement as we did not find in the literature the counterpart of Theorem 2.1 and of Proposition A.4. Let us emphasize however that unbounded domains have been considered in [START_REF] Bardos | Finite-time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates[END_REF] (Hölder spaces), and [START_REF] Jellouli | Equation d'Euler sur un domaine extérieur. Existence locale et apparition de singularités[END_REF][START_REF] Kikuchi | The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in R n[END_REF] (weighted Sobolev spaces). By following Dutrifoy's approach, we do not see any obstruction to get similar results in the Besov space framework. This is only a matter of having suitable extension operators available for the domain considered.

We also believe that Proposition A.4 may be extended to unbounded domains provided we prescribe some condition at infinity: the following inequality

(2.6) ∇ u B 0 ∞,1 ∩L r ≤ C ω θ e θ B 0
∞,1 ∩L r . for any r ∈]1, +∞[ seems to be reasonable. However, as proving such inequalities is not the point of this paper, we restricted ourselves to bounded domains. Definition A.3. Let Ω be a domain of R N , and (s, p, q) ∈ R × [1, +∞] 2 . We denote by B s p,q (Ω) the set of distributions u over Ω which are the restriction (in the sense of distributions) to some u in B s p,q (R N ). The space B s p,q (Ω) is endowed with the norm u B s p,q (Ω) := inf u B s p,q (R N )

where the infimum is taken over the set of u in B s p,q (R N ) such that u coincides with the restriction of u to Ω.

The following result will be needed in the proof of Inequality (2.5).

Proposition A.4. Let Ω be a smooth bounded domain of R N and ω be in B 0 ∞,1 (Ω). If in addition the mean value of ω on Ω is zero then the Neumann equation

-∆ψ = ω in Ω, ∂ n ψ = 0 on ∂Ω, Ω ψ dx = 0, has a unique solution ψ in B 2 ∞,1 (Ω) and we have ∇ 2 ψ B 0 ∞,1 (Ω) ≤ C ω B 0 ∞,1 (
Ω) . Proof. In [START_REF] Triebel | On Besov-Hardy-Sobolev spaces in domains and regular elliptic boundary value problems. The case 0 < p < ∞[END_REF], Th. 4.4, it has been proved that, for any s

> -1, if ω ∈ C s (Ω) := B s ∞,∞ (
Ω) (with 0 mean value) then the above system has a unique solution ψ in C s+2 (Ω) satisfying

∇ 2 ψ C s (Ω) ≤ C ω C s (Ω) .
So denoting T : ω → ∇ 2 ψ, the result follows by interpolation : it is only a matter of using the fact that

B 0 ∞,1 (Ω) = (C -1/2 (Ω), C 1/2 (Ω)) 1/2,1 .
Let us now turn to the proof of Inequality (1.4). Let u := u -∆ -1 u. We decompose the commutator as follows: f g = T f g + T g f + R(f, g).

Let us now go to the proof of Inequality (1.4). In all that follows, (c j ) j≥-1 stands for a sequence such that (c j ) ℓ q = 1.

From [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Lemma 2.99, we get, for i ∈ {1, 6}, R i j L p c j 2 -j(s-1) ∇u L ∞ ω B s-1 p,q . As regards R 2 j , we write that

R 2 j = j ′ ≥j-1 S j ′ -1 ∂ k ∆ j ω ∆ j ′ u k .
Note that F(∆ j ω) is supported in an annulus of size 2 j . Hence Bernstein's Inequality ensures that R 2 j L p 2 j j ′ ≥j-1

ω L ∞ ∆ j ′ u L p , whence R 2 j L p 2 -j(s-1) ω L ∞ j ′ ≥j-1
2 (j-j ′ )s 2 j ′ s ∆ j ′ u L p , so that we get if s > 0, R 2 j L p c j 2 -j(s-1) ω L ∞ u B s p,q . As for R 3 j , standard continuity results for the paraproduct operator (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2) imply that R 3 j L p c j 2 -j(s-1) ω L ∞ u B s p,q . For R 4 j , one may write that

R 4 j = ∂ k |j ′ -j|≤2 ∆ j ′ u k ∆ j (∆ j ′ -1 +∆ j ′ +∆ j ′ +1 )ω.
Hence, in view of Bernstein inequality,

R 4 j L p 2 j |j ′ -j|≤2 ω L ∞ ∆ j ′ u L p .
So we get R 4 j L p c j 2 -j(s-1) ω L ∞ u B s p,q . Next, standard continuity results for the remainder operator yield if s > 0,

∂ k R( u k , ω) B s-1 p,q
ω L ∞ u B s p,q . Hence R 5 j L p c j 2 -j(s-1) ω L ∞ u B s p,q . Finally, let us notice that the operator ω → (Id -∆ -1 )u satisfies the hypothesis of the last item of Proposition A.1 with m = -1, hence u B s p,q ω B s-1 p,q . So putting all the above inequalities together completes the proof of (1.4).

  Therefore, summing up Inequalities (1.5) and (1.8) and using Gronwall's lemma, we easily deduce that ω B s-1 p,r and θ B s p,r are bounded on [0, T [. To complete the proof of the boundedness of the solution in L ∞ ([0, T [; B s p,r ), we still have to bound u in L ∞ ([0, T [; L p ). For that, we use the fact that (1.10) u(t) = u(0) -t 0 P(u • ∇u) dτ

(1. 21 )

 21 T Θ 0 e C T 0 Ω dτ ≤ Ω 0 .Then (1.20) and Gronwall's lemma imply that(1.22) Ω(t) ≤ 2Ω 0 e 2CtΩ 0 for all t ∈ [0, T ].

  the summation convention over repeated indices,R 1 j := [T u k , ∆ j ]∂ k ω, R 2 j := T ∂ k ∆ j ω u k , R 3 j := -∆ j T ∂ k ω u k , R 4 j := ∂ k R( u k , ∆ j ω), R 5 j := -∂ k ∆ j R( u k , ω), R 6 j := [∆ -1 u k , ∆ j ]∂ k ω.Above, T and R stand for the paraproduct and remainder operators, respectively, which are defined as follows (after J.-M. Bony in[START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]):T f g := j S j-1 f ∆ j g and R(f, g) := j |j ′ -j|≤1 ∆ j f ∆ j ′ g with S j := j ′ <j ∆ j .Decomposition (A.1) is obtained after noticing that (A.2)

Acknowledgment. The author is indebted to O. Glass and F. Sueur for pointing out references [14,15,18].

Appendix A

In this appendix, we recall the definition and a few properties of nonhomogeneous Besov spaces B s p,q , then prove a commutator estimate. Let us first introduce a dyadic partition of unity with respect to the Fourier variable (the socalled Littlewood-Paley decomposition): we fix a smooth radial function χ supported in (say) the ball B(0, 4/3), equals to 1 in a neighborhood of B(0, 3/4) and such that r → χ(r e r ) is nonincreasing over R + , and set ϕ(ξ) = χ(ξ/2) -χ(ξ).

The dyadic blocks (∆ j ) j∈Z are defined by

It may be easily checked that the identity u = j ∆ j u holds true in the sense of tempered distributions.

One can now define the Besov space B s p,q as the set of tempered distributions u so that u B s p,q is finite, where

Roughly speaking, the elements of B s p,q have "s derivatives in L p ". For instance, the Besov space B s 2,2 coincides with the nonhomogeneous Sobolev space H s (for any s ∈ R), and B s ∞,∞ coincides with the Hölder space C s , if s ∈ R + \ N.

In this paper, we use freely the following properties for Besov spaces (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2):

• The Besov space B s p,q is (continuously) embedded in the set C 0,1 of Lipschitz bounded functions if and only if Condition (0.3) is satisfied.

• The gradient operator maps B s p,q in B s-1 p,q . • More generally, if F : R N → R is a smooth homogeneous function of degree m away from a neighborhood of the origin then for all (p, q) ∈ [1, ∞] 2 and s ∈ R, Operator F (D) maps B s p,q in B s-m p,q . Remark A.2. From the last property, given that the Biot-Savart operator B : ω → ∇u is an homogeneous smooth multiplier of degree 0, we deduce that (Id -∆ -1 )B is a self-map on B s p,q for any s ∈ R and 1 ≤ p, q ≤ ∞. This implies Inequality (1.2).

The definition of Besov spaces may be extended by restriction to general domains Ω of R N :


Hannah Claire

Eleanor Madden

Penelope A Phillips-Howard

Suzy C Hargreaves

Jennifer Downing

Mark A Bellis

Roberto Vivancos

Chris Morley

Qutub Syed

Access to HIV community services by vulnerable populations: Evidence from an enhanced HIV/AIDS surveillance system

Keywords: HIV, community services, deprivation, third sector, health services HIV, community services, health services, deprivation, third sector, voluntary organisations

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Access to HIV community services by vulnerable populations:

Evidence from an enhanced HIV/AIDS surveillance system HIV disproportionately affects vulnerable populations such as black and minority ethnic (BME) groups, men who have sex with men (MSM) and migrants, in many countries including those in the United Kingdom. Community organisations in the UK are charitable non-governmental organisations with a proportion of the workforce who volunteer, and provide invaluable additional support for people living with HIV. Information on their contribution to HIV care in vulnerable groups is relatively sparse. Data generated from an enhanced HIV surveillance system in North West England, UK, was utilised for this study. We aimed to determine the characteristics of individuals who chose to access community services in addition to clinical services (1,375 out of 4,195 records of people living with HIV in clinical services). Demographic information, risk factors including residency status, uniquely gathered in this region, and deprivation scores were examined.

Multivariate logistic regression modelling was conducted to predict the relative effect of patient characteristics on attendance at community services. Attendance at community services was highest in those living in the most, compared with least, deprived areas (p<0.001), and was most evident in MSM and heterosexuals.

Compared to white UK nationals attendance was significantly higher in non-UK nationals of uncertain residency status (Adjusted odds ratio [AOR]=21.91, 95% CI 10. 48-45.83; p<0.001), refugees (AOR=5.75, 95% CI 3.3-10.03; p<0.001), migrant workers (AOR=5.48, 95% CI 2.22-13.51; p<0.001), and temporary visitors (AOR=3.44, 95% CI 1.68-7.05; p<0.001). Community services, initially established predominantly to support MSM, have responded to the changing demography of HIV and reach the most vulnerable members of society. Consequent to their support of migrant populations, community services are vital for the management of HIV in black and minority groups. Paradoxically, this coincides with increasing funding pressures on these services. While the demand on HIV services increases in parallel with the rising HIV prevalence, local government and NHS funding to community organisations has decreased or, at best, stabilised (Weatherburn et al., 2007). NHS and local government commissioners justify funding decisions on the basis of changes in need, restructuring to reduce duplication and substandard or unacceptable service provision (Weatherburn et al., 2007). A recent study found that of 13 different services or opportunities for support, HIV community services were ranked 2 nd for a raft of support needs by 1,929 PLWHIV (Weatherburn et al 2009). It is therefore important to explore the role and impact of community services on the health and wellbeing of persons living with HIV. Few data exist on the systematic measurement on the value of such services.

The North West region of England, UK, comprises of five counties, with a population of 6.9 million people (Office for National Statistics, 2008). The largest city in the region, Manchester, has the highest prevalence of HIV in the UK outside London and the South East (HPA 2008). An enhanced HIV surveillance system in the region collects data from community as well as statutory clinical care services.

The capture of such data provides a unique opportunity to examine the demographic characteristics of people living with HIV who access HIV treatment and care through these different services whilst still retaining confidentiality and anonymity. In this paper we explore characteristics of the population who chose to access community services as well as clinical services, to identify the role and

Methods

North West HIV/AIDS Surveillance.

The enhanced HIV surveillance data collected by the North West HIV/AIDS Monitoring Unit (Centre for Public Health, Liverpool John Moores University) measures the level of health services use, demographics including ethnicity and place of residence, and treatment variables such as route of infection and type of treatment obtained [START_REF] Hargreaves | HIV & AIDS in the North West of England Mid Year[END_REF]. Pseudo-anonymised identifiers are collected and a 'Soundex' (a one-way encryption of the surname that maintains anonymity) is recorded to prevent double counting of individuals attending more than one treatment centre or community service. Reporting is universal across the country from genito-urinary medicine (GUM) clinics, infectious disease departments, haematology units and a number of other specialist units and clinics (Health Protection Agency [HPA], 2006). Data are collected from these clinical services every six months, and include a summary of care from each clinic. A subset of the variables from the hospital dataset is submitted to the national HIV surveillance system at the HPA (2006). Data collection and storage conform to all relevant laws and guidance relating to security and confidentiality (Department of Health, 2003, The Data Protection Act, 1998, Department of Health, 1997). These govern the maintenance and development of the surveillance system and incorporate data extraction for monitoring and research purposes. For this study, data were extracted from the routinely collected data of all HIV positive persons accessing treatment

Community services.

Community organisations in North West England have provided data to the surveillance system, without obligation, since monitoring began in 1996. Of the known community organisations in the region, eight out of the nine provided data in 2005. The missing organisation is a small service offering support to people in one county in the region. Key demographic variables are routinely gathered but, unlike clinical services, no clinical parameters or information on outcomes were documented. We restricted analysis to data records of HIV positive individuals with a valid Soundex, date of birth, and sex. Individuals only seen by community services were excluded from analysis (n=562) where reliable soundex and date of birth data were missing or could not be corroborated with the clinical records.

Relationship with deprivation.

Postal district of residence for all individuals reported by clinical services and community organisations were allocated an Index of Multiple Deprivation (IMD) score based on the lower super output area (LSOA -a small level geographical area consistent with a mean population of 1,500), supplied by the North West Public Health Observatory, England. The higher the IMD score, the more deprived an area. The case reports for people living with HIV were divided into three equal deprivation categories based on IMD score [START_REF] Hargreaves | HIV & AIDS in the North West of England Mid Year[END_REF]. Valid LSOAs and IMD scores were obtained for 82% of case records. IMD scores were allocated to 75% of data records of persons attending only clinical services and to 95% of those also accessing community services. The higher proportion in the overlapping group reflects the greater likelihood of at least one service providing location of residence data. Community and clinical datasets were matched using identifying characteristics (soundex, date of birth and sex).

Statistical analysis.

All those accessing clinical services were categorised as having accessed additional services from community organisations (coded 1) or not (coded 0). The characteristics of those who did and did not attend community organisations were determined, and differences in the proportions assessed using Chi square tests.

Multivariate backward logistic regression analysis was conducted to predict the odds of attendance at community services in addition to clinical services, based on characteristics including individuals' ascribed level of deprivation (based on postcode) and demographic indicators (age, sex, ethnicity/residency status and route of infection). Subgroups with the largest population were selected as the referent cell, apart from deprivation where 'least deprived' was selected. Missing demographic fields were coded as 'unknown' in these analyses. Data were analysed using SPSS version 15 with significance defined at the 5% level. Ninety five percent confidence intervals are reported.

Results

A total of 4,195 individuals were recorded to have accessed clinical services in the study year: 2,820 (67%; 73% males) sought care from clinical services only, 1,375 individuals (33%; 76% males) attended both clinical and community services (table 1). Fifty three percent of all individuals were infected through sex between men (MSM) and 39% through heterosexual contact. Injecting drug use, blood/tissue and mother to child transmission was less common (2%, 1% and 2% respectively). The majority of individuals were aged between 25 and 44 years (69%). The profile of the subset attending only clinical services differed from those who also accessed community services (χ 2 =94, df=5, p<0.001; table 1). While MSM formed 62% of all clients seen at community services, HIV infected heterosexuals represented only 38%, and those infected through blood products, 1%. were less likely to attend a community service (AOR=0.78, 95% CI 0.66-0.94; p<0.001, table 1).

Attendance at community services was significantly higher in those living in the most deprived areas compared with those resident in the least deprived areas (48.3% vs 29.0%, AOR=2.40, 95% CI 1.99-2.91; ; p<0.001, table 1). Figure 1 displays the proportion of HIV positive individuals visiting community services as well as statutory services, by level of deprivation for the two main infection routes; MSM and heterosexual sex. For both groups, the proportion of persons visiting community services rose with increasing deprivation. The subcategory with the highest attendance at community services was MSM in the most deprived areas (54%) followed by heterosexuals in the most deprived areas (44%).

Discussion

We were able to explore a unique dataset that characterises HIV infected persons accessing HIV community services, in addition to clinical services, in a region of the UK representing a socio-economically and demographically diverse population.

Analysis of one year's figures, in this instance 2005, suggest community organisations in the study region specialise in care of MSM (61.8% of all clients seen), and are currently less involved with the care of HIV infected heterosexuals (31.9%) or persons infected through blood products (0.8%), especially when compared with those who solely visited clinical services (MSM: 49.4%; heterosexual: 41.8%; blood/tissue: 1.8%). This difference may be partly historical; highly specialist services were developed for haemophiliacs, for example Birchgrove (http://www.birchgrovegroup.org), while separate community services developed to support HIV positive gay men. Heterosexually infected individuals have tended to be mainly amongst black and minority ethnic groups (BME; HPA 2006) and can include asylum seekers or temporary visitors to the UK (Cook, Downing, Rimmer, & Bellis, 2006). Our data suggest the most vulnerable of these groups, such as refugees, migrant workers, and temporary visitors are now more likely than UK nationals to access community, as well as clinical, services. This increased attendance is despite stigma amongst the black African community [START_REF] Dodds | HIV-Related Stigma in England: Experiences of Gay Men and Heterosexual African Migrants Living with HIV[END_REF], and the slow adaptation of services to the needs of BME groups.

Despite the erosion of such barriers it is likely unmet need amongst the BME community continues. The relationship between deprivation and ill-health has been well documented (Townsend & Davidson, 1982;[START_REF] Acheson | ATCC: The Antiretroviral Therapy Cohort Collaboration[END_REF]. In terms of PLWHIV, international studies have shown a close correlation between socioeconomic status and antiretroviral use [START_REF] Joy | Impact of neighborhood-level socioeconomic status on HIV disease progression in a universal health care setting[END_REF], and with longevity (McFarland, Chen, Hsu, Schwarcz, & Katz, 2003;Borrel et al., 2006). In the UK, HIV prevalence is higher in more deprived areas (Cook & Hargreaves, 2007) and those in the most deprived areas are more likely to be admitted to hospital for HIVrelated care (Cook, Hargreaves, Tocque, & Bellis, 2004). Research into health inequalities reveals the most deprived areas have the highest levels of poor health indicators including binge drinking, smoking, births to lone mothers, mental health conditions, claimants of disability living allowance and mortality from all causes (Wood et al., 2006). Many socio-economic and cultural components collude towards this imbalance, such as unemployment, low wages, poor housing, xenophobia, and deserve more attention as an increasing proportion of new HIV infections are heterosexually acquired [START_REF] Hargreaves | HIV & AIDS in the North West of England Mid Year[END_REF]. The evidence that community organisations provide services for relatively more people from deprived areas highlights their ability, and the importance of, permeation of community Recorded cases of HIV in MSM are higher in poorer multicultural areas than in affluent areas (Cook & Hargreaves, 2007). Further, men from lower social classes and the poorly educated have been identified as being less accepting of homosexual behaviour (e.g. less likely to define their general opinion of sex between men as being 'not at all wrong'; Weatherburn Davies, Hickson, & Hartley, 1999). Those living in these poorer areas may experience negative responses and possible hostility to their sexuality. Those with HIV face further stigmatisation and barriers to disclosure of their status [START_REF] Dodds | HIV-Related Stigma in England: Experiences of Gay Men and Heterosexual African Migrants Living with HIV[END_REF]. Many deprived areas in this study area are located near city centres, and those residing there may have easier access to services, as shown by analysis of distance travelled to clinical HIV services [START_REF] Hargreaves | HIV & AIDS in the North West of England Mid Year[END_REF]. Community organisations are also usually located close to city centres. Those living in more deprived areas, those who are unemployed and poorer are more likely to progress faster with HIV and have worse health (Delpierre etc al 2008;[START_REF] Ezzy | Poverty, disease progression and employment among people living with HIV/AIDS in Australia[END_REF]Cook, Hargreaves, Tocque, & Bellis, 2004). This poorer health may cause them to seek more additional support from community organisations. This greater need may explain why the gay men living in the most deprived areas in our sample were also the group most likely to access the social support offered through community organisations (54%, figure 1). Further research in the UK thus needs to examine the relationship between attendance at community services and other indicators of deprivation including unemployment, income and ill-health allowances [START_REF] Delpierre | Unemployment as a risk factor for AIDS and death for HIV-infected patients in the era of highly active antiretroviral therapy[END_REF][START_REF] Ezzy | Poverty, disease progression and employment among people living with HIV/AIDS in Australia[END_REF][START_REF] Joy | Impact of neighborhood-level socioeconomic status on HIV disease progression in a universal health care setting[END_REF] Community organisations in the UK provide a range of services to PLWHIV, including counselling, adherence support, information services, advocacy, financial assistance, support groups and help lines. These services offer a variety of specialist support services for women, gay men, persons originating from Africa, young people, asylum seekers and also provide care and support to the friends and family of those affected by HIV. Despite this contribution and preferential use by vulnerable groups, funding is already being diverted from community and non-governmental organisations to clinical services (Terrance Higgins Trust, 2004). Since the withdrawal of ring-fenced HIV funding in England in 2004 HIV charities have coped in an uncertain financial climate and with even fewer resources (Weatherburn et al., 2007), yet have managed to provide invaluable services to many more people. Current economic restraints will further compromise community service finances. Despite this, community organisations provide value for money (ie an estimated £2 of services are gained for every £1 purchased by the clinical sector because of the value of volunteering and nonstatutory funding) (Cosgrove Lyons & Bellis, 2001).

Limitations of study data

Of the known community organisations in the region eight out of the nine were able to provide data. While we could not assess any biases from this absence, the population served was relatively small and would have thus contributed marginally to our regional dataset. This study depends on data reported without obligation and therefore restricts robust analyses of all characteristics; inclusion of 95% confidence intervals are provided where possible to help readers understand the distribution of data. Some caution is required when interpreting the overlap between the community and clinical datasets since inaccurate reporting of pseudoanonymised unique identifiers can lead to some inaccuracies, preventing the complete matching between the two systems. This would result in an underestimate of the true overlap between community and clinical services. Bias may have been introduced as individuals with names that are unfamiliar to the English tongue (e.g. black Africans) are more likely to live in deprived areas (Tinsley & Jacobs, 2006), could thus be prone to miscoding. The higher rate of missing residence information of those seen only in the clinical sector (25% missing) would be expected to dilute the results and give weaker relationships. However, despite these caveats the relationship between service attended and deprivation was still highly significant (P<0.001). Thus, the finding that community services see people who, on average, live in more deprived areas appears robust despite possible non-matching and missing data. Comparison with other regions was not possible as such data are not collected at a national level. An ecological method was used for analyses by deprivation (i.e. socioeconomic characteristics are assumed based on an area of residence), and as such the data are subject to the 'ecological fallacy'. Clinical indicators could not be included in this model because the surveillance system lacks robust indicators of current stage of disease and has high levels of missing information for other clinical markers (e.g. viral load and CD4 counts). Evidence suggests BME groups and migrants are more likely to be diagnosed late with HIV [START_REF] Burns | Africans in London continue to present with advanced HIV disease in the era of highly active antiretroviral therapy[END_REF] and delayed diagnosis is often associated with worse prognosis (Chadborn Delpech, Sinka et al., 2005). A relationship between faster disease progression and increased poverty has also been shown [START_REF] Delpierre | Unemployment as a risk factor for AIDS and death for HIV-infected patients in the era of highly active antiretroviral therapy[END_REF][START_REF] Ezzy | Poverty, disease progression and employment among people living with HIV/AIDS in Australia[END_REF]

  epidemiology of HIV changes and access to effective life prolonging antiretroviral drugs increases (Joint United Nations Programme on HIV/AIDS [UNAIDS], 2009; ATCC 2008; Piot, Kazatchkine, Dybul & Lob-Levyt, 2009) the needs of people living with HIV (PLWHIV) change. In resource poor settings, where antiretroviral (ARV) drugs are now available, support services, which until recently had focused on prevention or palliative care, must now find ways to distribute life saving drugs to those in need. In countries where clinical care services are well established, and thus support services are not needed to provide ARV, they likewise need to adapt and develop new, more appropriate services. Support services that help with social issues such as immigration and poverty, and psychological issues, are needed. In the United Kingdom clinical care is provided free at the point of care to almost all people with HIV by the National Health Service (NHS) with additional support from the community services also being free, or nominal to users. Community organisations in the UK are charitable nongovernmental organisations utilising a workforce partly made up of individuals who volunteer a proportion of their time. They may also receive funding from local and national government and the NHS. Such organisations have provided additional support to PLWHIV since the first gay men in London died from acquired immune deficiency syndrome (AIDS) related illnesses in the early 1980s

  community organisations in the face of the changing epidemiology of HIV.

  community care settings between 1 st January and 31 st December 2005. The total was 4,195 records of individual people living with HIV.

FIGURE

  FIGURE 1 AROUND HERE

  services as well as clinical services lived in more deprived areas than those only accessing clinical treatment centres. This may be explained by the effectiveness of the referral systems which encourage the most vulnerable to access these additional services, as well as by the attractiveness of the services to those most in need. Many community services have set up support groups for people from BME populations, and these populations tend to reside in relatively deprived areas of the UK.

  those who most need additional support. Our analysis provides evidence of a relationship between deprivation and community service attendance for both of the main groups of people (MSM and heterosexual) affected by HIV.

  into stigma, both internalised and external, and how it impacts on use of community services would contribute towards our ability to improve services and encourage uptake.

  so those from more deprived areas may require additional support from community organisations. Therefore, we may have underestimated the support needed by these individualsConclusionsData from North West England that community services appeal to and reach disadvantaged people living with HIV, regardless of ethnicity, sexuality or gender. Attendance was highest in those living in the most deprived areas, as well as in migrants and those of uncertain immigration status. Consequent to their support of such populations, community services are critical for the management of HIV in BME groups. The British HIV Association recognises that community organisations are experts in providing patient education and support(British HIV Association, 2007). While the UK Government anticipates an increasing role for the community and independent sectors in HIV and sexual health care services(Department of Health, 2006) such views paradoxically coincide with increasing funding pressures on these services.

  .R., Delpech, V., Sinka, K.,Rice, B.D., & Evans, B. (2005). Late diagnosis and consequent short term mortality of individuals sexually infected with HIV: England and Wales 2002. Abstract 19, 11th Annual Conference of British HIV Association with the British Association for Sexual Health and HIV; Dublin, Ireland. Cook, P.A., Hargreaves, S.C., Tocque, K., & Bellis, M.A. (2004). Relationships between the use of hospital services and place of residence deprivation score among HIV positive individuals in the North West of England. Communicable Disease and Public Health, 7, 319-321. http://www.hpa.org.uk/cdph/index.html Cook, P.A., Downing, J., Rimmer, S.Q., & Bellis, M.A. (2006). Treatment and care of HIV positive asylum seekers. Journal of Epidemiology and Community Health, 60, 836-38. doi:10.1136/jech.2005.044776 Cook, P.A., & Hargreaves, S.C. (2007) Inequalities in HIV. In Downing, J, Cook PA & Bellis MA (Eds.), Ten Years of Monitoring HIV & AIDS in the North West of England. Liverpool: Liverpool John Moores University. Cook, P.A., Downing, J., Wheater, C.P., Bellis, M.A., Tocque, K., Syed, Q., & Phillips-Howard P.A. (2009). Influence of socio-demographic factors on distances travelled to access HIV services: enhanced surveillance of HIV patients in North West England. BMC Public Health, 69, 78-90, doi: 10.1186/1471-2458-9-78 Cosgrove, P., Lyons, M., & Bellis M.A. (2001). Economics of HIV and AIDS in the North West of England. Liverpool: Liverpool John Moores University.

  , San Francisco. Journal of Acquired Immune Deficiency Syndrome, 33, 96-103. http://journals.lww.com/jaids/pages/default.aspx NAM. (2006). HIV & AIDS Reference Manual. London: National AIDS Manual Office of National Statistics. (2008). Mid-2007 Population Estimates for Lower Layer Super Output Areas in England and Wales by Quinary Age and Sex. Crown Copyright. Piot, P., Kazatchkine, M., Dybul, M., & Lob-Levyt, J. (2009). AIDS: lessons learnt and myths dispelled. Lancet, 374, 260-263. doi:10.1016/S0140-6736(09)60321-4 Terrence Higgins Trust. (2004). Blueprint for the future, modernising HIV and sexual health service: a policy report. London: Terrence Higgins Trust. The Data Protection Act 1998. London: The Stationary Office. Retrieved from: http://opsi.gov.uk Tinsley, J., & Jacobs, M. (2006). Deprivation and Ethnicity in England: A Regional Perspective. Regional Trends 39: 2006 edition. Retrieved from: http://www.statistics.gov.uk/articles/RegionalTrends/Article3RT39.pdf Townsend, P., & Davidson, N. (1982). Inequalities in health: the Black report. Harmondsworth: Penguin.

  2009). AIDS epidemic update: November 2009. Geneva: UNAIDS Weatherburn, P.,Davies, P.D., Hickson, F., & Hartley, M. (1999). A Class Apart:The social stratification of HIV infection amongst homosexually active men. London: Sigma Research. Weatherburn, P., Keogh, P., Dodds, C., Hickson, F., Henderson, L., & Hartley, M. (2007). A strategic review of HIV social care, support and information services across the UK. London: Sigma Research. Weatherburn, P., Keogh, P., Reid, D., Dodds, C., Bourne, A., Owuor, J., Hammond, G. & Jessup, K. (2009) What do you need? 2007-2008 Findings from a national survey of people with diagnosed HIV. London: Sigma Research. Wood, J., Hennell, T., Jones, A., Hooper, J., Tocque, K., & Bellis, M.A. (2006).Where wealth means health: illustrating inequality in the North West. Liverpool: North West Public Health Observatory.

TABLE 1

 1 Compared to those aged 35 to 44 years (the predominant age group of HIV positive persons in the study area), younger persons (those aged 25 to 34 years)

	AROUND HERE
	While 46% of individuals infected through injecting drug use accessed
	community services, this was the case for only 18% of those infected through
	blood/tissue. Heterosexuals were significantly less likely to attend community
	services than MSM (27.1% vs 37.9%, Adjusted Odds Ratio [AOR]= 0.41, 95% CI
	= 0.31-0.54; p<0.001 table 1).
	Whilst a larger proportion of white UK nationals than black African UK
	nationals (35% and 12%) were seen in the community services, of all those
	attending a clinical service, non-UK nationals were more likely than white UK

Table 1 .

 1 Univariate and Multivariate predictors of attendance at community services, for all those attending statutory services. Including those of uncertain residency status, failed asylum seekers and who do not fit into another category.2 Analysis repeated excluding those without postcode -odds ratios were similar and significance levels did not change Figure 1. Percentage of HIV positive persons seen in statutory services in 2005 who attend community services, by deprivation category.

			Univariate			Multivariate
			Number seen
			in
			community	Chi-
		n	services (%)	square	df	P	Adj OR (95%CI)	P
	sex			3.87	1	0.49		0.007
	Male	3102	1043 (33.6%)				ref
	Female	1093	332 (30.4%)				1.43 (1.10-1.85)	0.007
	Route			94.29	5	<0.001
	MSM	2242	850 (37.9%)				ref	<0.001
	Injecting drug use	101	46 (45.5%)				1.50 (0.97-2.31)	0.066
	Heterosexual	1618	438 (27.1%)				0.41 (0.31-0.54)	<0.001
	Blood/tissue	62	11 (17.7%)				0.37 (0.19-0.73)	0.004
	Mother to child	68	22 (32.4%)				0.08 (0.01-1.20)	0.068
	Other/not known	104	8 (7.7%)				0.20 (0.09-0.42)	<0.001
	Ethnicity/Residency			402.99	10	<0.001		<0.001
	White UK National	2691	953 (35.4%)				ref
	Black African UK
	National	274	34 (12.4%)				0.37 (0.24-0.56)	<0.001
	Other UK National	202	43 (21.3%)				0.64 (0.44-0.93)	0.020
	Asylum Seeker	391	120 (30.7%)				1.31 (0.95-1.81)	0.096
	Overseas Student	100	27 (27%)				0.81 (0.49-1.34)	0.420
	Migrant Worker	24	15 (62.5%)				5.48 (2.22-13.51)	<0.001
	Temp Visitor	38	19 (50.0%)				3.44 (1.68-7.05)	0.001
	Refugee	78	56 (71.8%)				5.75 (3.30-10.03)	<0.001
	Dependent	9	6 (66.7%)				4.30 (0.95-19.56)	0.059
	Other non UK
	national 1	98	89 (90.8%)				21.91 (10.48-45.83)	<0.001
	Unknown	290	13 (4.5%)				0.10 (0.06-0.18)	<0.001
	Age Group			32.44	5	<0.001		0.002
	0-14	64	22 (34.4%)				5.90 (0.40-86.65)	0.195
	15-24	208	58 (27.9%)				0.71 (0.5-1.02)	0.067
	25-34	1228	369 (30%)				0.78 (0.66-0.94)	0.008
	35-44	1682	606 (36%)				ref
	45-54	725	257 (35.4%)				0.99 (0.81-1.21)	0.923
	55+	288	63 (21.9%)				0.60 (0.43-0.83)	0.002
	Deprivation			344.18	3	<0.001		<0.001
	Least deprived	1144	332 (29.0%)				ref
	Average	1144	424 (37.1%)				1.48 (1.22-1.78)	<0.001
	Most deprived	1144	553 (48.3%)				2.40 (1.99-2.91)	<0.001
	No postcode 2	763	66 (8.7%)				0.26 (0.19-0.34)	<0.001
	Total	4195	1375 (32.8%)

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