R Danchin

P B Mucha

New maximal regularity results for the heat equation in exterior domains, and applications

This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of R n . We strive for time independent a priori estimates in regularity spaces of type L 1 (0, T ; X) where X stands for some homogeneous Besov space. In the case of bounded domains, the results that we get are similar to those of the whole space or of the half-space. For exterior domains, we need to use mixed Besov norms in order to get a control on the low frequencies. Those estimates are crucial for proving global-in-time results for nonlinear heat equations in a critical functional framework.

1 Without any support assumption, it is obvious that if s is positive then we have • Ḃs p,r (R n )

• Ḃs p,r (R n ) , and the opposite inequality holds true if s is negative.

Introduction

We are concerned with the proof of maximal regularity estimates for the heat equation with Dirichlet boundary conditions, namely,

u t -ν∆ u = f in (0, T ) × Ω , u = 0 at (0, T ) × ∂ Ω , u = u 0 on Ω (1)
in various domains Ω of R n (n ≥ 2).

We are interested in L 1 -in-time estimates for the solutions to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF] with a gain of two full spatial derivatives with respect to the data, that is Raphaël Danchin Université Paris-Est, LAMA, UMR 8050 and Institut Universitaire de France, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France, e-mail: raphael.danchin@u-pec.fr Piotr Bogusław Mucha Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland. e-mail: p.mucha@mimuw.edu.pl u t , ν∇ 2 u L 1 (0,T ;X) ≤ C u 0 X + f L 1 (0,T ;X) [START_REF] Bennett | Interpolation of operators[END_REF] with a constant C independent of T.

Such time independent estimates are of importance not only for the heat semigroup theory but also in the applications. Typically, they are crucial for proving global existence and uniqueness statements for nonlinear heat equations with small data in a critical functional framework. Moreover, the fact that two full derivatives may be gained with respect to the source term allows to consider not only the -∆ operator but also small perturbations of it. In addition, we shall see below that it is possible to choose X in such a way that the constructed solution u is L 1 -in-time with values in the set of Lipschitz functions. Hence, if the considered nonlinear heat equation determines the velocity field of some fluid then this velocity field admits a unique Lipschitzian flow for all time. The model may thus be reformulated equivalently in Lagrangian variables (see e.g. our recent work [START_REF] Danchin | A Lagrangian approach for solving the incompressible Navier-Stokes equations with variable density[END_REF] in the slightly different context of incompressible flows). This is obviously of interest to investigate free boundary problems.

Let us recall however that estimates such as [START_REF] Bennett | Interpolation of operators[END_REF] are false if X is any reflexive Banach space, hence in particular if X is a Lebesgue or Sobolev space (see e.g. [START_REF] Denk | Prüss: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type[END_REF]). On the other hand, it is well known that (2) holds true in the whole space R n if X is a homogeneous Besov space with third index 1. Let us be more specific. Let us fix some homogeneous Littlewood-Paley decomposition ( ∆ j ) j∈Z (see the definition in the next section) and denote by (e α∆ ) α>0 the heat semi-group over R n . Then it is well known (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]) that there exist two constants c and C such that for all j ∈ Z and α ∈ R + one has

e α∆ ∆ j h L p (R n ) ≤ Ce -cα2 2 j ∆ j h L p (R n ) . (3)
Hence if u satisfies (1) then one may write ∆ j u(t) = e νt∆ ∆ j u 0 + t 0 e ν(t-τ)∆ ∆ j f dτ.

Therefore, taking advantage of (3), we discover that

∆ j u(t) L p (R n ) ≤ C e -cνt2 2 j ∆ j u 0 L p (R n ) + t 0 e -cν(t-τ)2 2 j ∆ j f L p (R n ) dτ , whence ∆ j u L ∞ (0,T ;L p (R n )) + ν2 2 j ∆ j u L 1 (0,T ;L p (R n )) ≤ C ∆ j u 0 L p (R n ) + ∆ j f L 1 (0,T ;L p (R n )) .
Multiplying the inequality by 2 js and summing up over j, we thus eventually get for some absolute constant C independent of ν and T, u L ∞ (0,T ; Ḃs p,1 (R n )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs

p,1 (R n )) ≤ C( f L 1 (0,T ; Ḃs p,1 (R n )) + u 0 Ḃs p,1 (R n ) ), (4)
where the homogeneous Besov semi-norm that is used in the above inequality is defined by

u Ḃs p,1 (R n ) := ∑ j∈Z 2 s j ∆ j u L p (R n ) .
From this and the definition of homogeneous Besov space Ḃs p,1 (R n ) (see Section 1), we easily deduce the following classical result: Theorem 1. Let p ∈ [1, ∞] and s ∈ R. Let f ∈ L 1 (0, T ; Ḃs p,1 (R n )) and u 0 ∈ Ḃs p,1 (R n ). Then [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF] with Ω = R n has a unique solution u in

C ([0, T ); Ḃs p,1 (R n )) with ∂ t u, ∇ 2 u ∈ L 1 (0, T ; Ḃs p,1 (R n ))
and ( 4) is satisfied.

The present paper is mainly devoted to generalizations of Theorem 1 to the halfspace, bounded or exterior domains (that is the complement of bounded simply connected domains), and applications to the global solvability of nonlinear heat equations.

Proving maximal regularity estimates for general domains essentially relies on Theorem 1 and localization techniques. More precisely, after localizing the equation thanks to a suitable resolution of unity, one has to estimate "interior terms" with support that do not intersect the boundary of Ω and "boundary terms" the support of which meets ∂ Ω . In order to prove interior estimates that is bounds for the interior terms, it suffices to resort to the theorem in the whole space, Theorem 1, for those interior terms satisfy (1) (with suitable data) once extended by zero onto the whole space. In contrast, the extension of the boundary terms by zero does not satisfy (1) on R n . However, performing a change of variable reduces their study to that of (1) on the half-space R n + . Therefore, proving maximal regularity estimates in general domains mainly relies on such estimates on R n and on R n + . As a matter of fact, we shall see that the latter case stems from the former, by symmetrization, provided s is close enough to 0. In the case of a general domain, owing to change of variables and localization however, we shall obtain (4) either up to low order terms or with a timedependent constant C. In a bounded domain, it turns out that Poincaré inequality (or equivalently the fact that the Dirichlet Laplacian operator has eigenvalues bounded away from 0) allows to prove an exponential decay which is sufficient to cancel out those lower order terms. In the case of an exterior domain, that decay turns out to be only algebraic (at most t -n/2 in dimension n). As a consequence, absorbing the lower order terms will enforce us to use mixed Besov norms and to assume that n ≥ 3.

The paper unfolds as follows. The basic tools for our analysis (Besov spaces on domains, product estimates, embedding results) are presented in the next section. In Section 2 we prove maximal regularity estimates similar to those of Theorem 1 first in the half-space and next in exterior or bounded domains. As an application, in the last section, we establish global existence results for nonlinear heat equations with small data in a critical functional framework.

Tools

In this section, we introduce the main functional spaces and (harmonic analysis) tools that will be needed in this paper.

Besov spaces on the whole space

Throughout we fix a smooth nonincreasing radial function χ : R n → [0, 1] supported in B(0, 1) and such that χ ≡ 1 on B(0, 1/2), and set ϕ(ξ

) := χ(ξ /2) -χ(ξ ). Note that this implies that ϕ is valued in [0, 1], supported in {1/2 ≤ r ≤ 2} and that ∑ k∈Z ϕ(2 -k ξ ) = 1 for all ξ = 0. ( 5
)
Then we introduce the homogeneous Littlewood-Paley decomposition ( ∆k

) k∈Z over R n by setting ∆k u := ϕ(2 -k D)u = F -1 ϕ(2 -k •)F u .
Above F stands for the Fourier transform on R n . We also define the low frequency cut-off Ṡk := χ(2 -k D).

In order to define Besov spaces on R n , we first introduce the following homogeneous semi-norms and nonhomogeneous Besov norms (for all s ∈ R and

(p, r) ∈ [1, ∞] 2 ): u Ḃs p,r (R n ) := 2 sk ∆k u L p (R n ) ℓ r (Z) u B s p,r (R n ) := 2 sk ∆k u L p (R n ) ℓ r (N) + Ṡ0 u L p (R n ) .
The nonhomogeneous Besov space B s p,r (R n ) is the set of tempered distributions u such that u B s p,r (R n ) is finite. Following [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], we define the homogeneous Besov space Ḃs p,r (R n ) as

Ḃs p,r (R n ) = u ∈ S ′ h (R n ) : u Ḃs p,r (R n ) < ∞ ,
where S ′ h (R n ) stands for the set of tempered distributions u over R n such that for all smooth compactly supported function θ over R n , we have lim

λ →+∞ θ (λ D)u = 0 in L ∞ (R n ). Note that any distribution u ∈ S ′ h (R n ) satisfies u = ∑ k∈Z ∆k u in S ′ h (R n ).
We shall make an extensive use of the following result (see the proof in e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF]):

Proposition 1. Let b s p,r denote Ḃs p,r (R n ) or B s p,r (R n ).
Then the following a priori estimates hold true:

• For any s > 0, uv b s p,r u L ∞ v b s p,r + v L ∞ u b s p,r .
• For any s > 0 and t > 0,

uv b s p,r u L ∞ v b s p,r + v b -t ∞,r u b s+t p,∞ .
• For any t > 0 and s > -n/p ′ ,

uv b s p,r u L ∞ v b s p,r + u b n/p ′ p ′ ,∞ v b s p,r + v b -t ∞,r u b s+t p,∞ .
• For any q > 1 and 1n/q ≤ s ≤ 1,

uv b 0 q,1 u b s n,1 v b 1-s q,1
.

As obviously a smooth compactly supported function belongs to any space Ḃn/p p,1 (R n ) with 1 ≤ p ≤ ∞, and to any Besov space B σ p,1 (R n ), we deduce from the previous proposition and embedding that (see the proof in [START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF]):

Corollary 1. Let θ be in C ∞ c (R n ). Then u → θ u is a continuous mapping of b s p,r (R n ) • for any s ∈ R and 1 ≤ p, r ≤ ∞, if b s p,r (R n ) = B s p,r (R n ); • for any s ∈ R and 1 ≤ p, r ≤ ∞ satisfying -n/p ′ < s < n/p -n/p < s ≤ n/p if r = 1, -n/p ′ ≤ s < n/p if r = ∞ (6) if b s p,r (R n ) = Ḃs p,r (R n ).
The following proposition allows us to compare the spaces B s p,r (R n ) and Ḃs p,r (R n ) for compactly supported functions 1 (see the proof in [START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF]):

Proposition 2. Let 1 ≤ p, r ≤ ∞ and s > -n/p ′ (or s ≥ -n/p ′ if r = ∞). Then for any compactly supported distribution f we have f ∈ B s p,r (R n ) ⇐⇒ f ∈ Ḃs p,r (R n )
and there exists a constant C = C(s, p, r, n, K) (with K = Supp f ) such that

C -1 f Ḃs p,r (R n ) ≤ f B s p,r (R n ) ≤ C f Ḃs p,r (R n ) .
The following lemma will be useful for boundary estimates (see the proof in [START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF]):

Lemma 1. Let Z be a Lipschitz diffeomorphism on R n with DZ and DZ -1 bounded, (p, r) ∈ [1, ∞] 2 and s a real number satisfying [START_REF] Denk | Prüss: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type[END_REF].

• If in addition s ∈ (-1, 1) and Z is measure preserving then the linear map u →

u • Z is continuous on Ḃs p,r (R n ). • In the general case, the map u → u • Z is continuous on Ḃs p,r (R n ) provided in addition J Z -1 ∈ Ḃn/p ′ p ′ ,∞ ∩ L ∞ with J Z -1 := | det DZ -1 |.

Besov spaces on domains

We aim at extending the definition of homogeneous Besov spaces to general domains. We proceed by restriction as follows 2 :

Definition 1. For s ∈ R and 1 ≤ p, q ≤ ∞, we define the homogeneous Besov space Ḃs p,q (Ω ) over Ω as the restriction (in the distributional sense) of Ḃs p,q (R n ) on Ω , that is φ ∈ Ḃs p,q (Ω ) ⇐⇒ φ = ψ |Ω for some ψ ∈ Ḃs p,q (R n ). We then set φ Ḃs p,q (Ω ) := inf

ψ |Ω =φ ψ Ḃs p,q (R n ) .
The embedding, duality and interpolation properties of these Besov spaces may be deduced from those on R n . As regards duality, we shall use repeatedly the following result:

Proposition 3. If -1 + 1/p < s < 1/p (with 1 ≤ p, r < ∞) then the space Ḃ-s
p ′ ,r ′ (Ω ) may be identified with the dual space of Ḃs p,r (Ω ); in the limit case r = ∞ then Ḃ-s p ′ ,1 (Ω ) may be identified with the dual space of the completion of C ∞ c (R n ) for • Ḃs p,∞ (Ω ) . Furthermore, without any condition over (s, p, r), we have

Ω uv dx ≤ C u Ḃs p,r (Ω ) v Ḃ-s p ′ ,r ′ (Ω ) .
Similarly, some product laws for Besov spaces on R n may be extended to the domain case. We shall use the last inequality of Proposition 1 and also the following result that is proved in [START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF]:

Proposition 4. Let b s p,r (Ω ) denote Ḃs p,r (Ω ) or B s p,r (Ω ),
and Ω be any domain of

R n . Then for any p ∈ [1, ∞], s such that -n/p ′ < s < n/p (or -n/p ′ < s ≤ n/p if r = 1, or -n/p ′ ≤ s < n/p if r = ∞), the following inequality holds true: uv b s p,r (Ω ) ≤ C u b n/q q,1 (Ω ) v b s p,r (Ω ) with q = min(p, p ′ ).
2 Nonhomogeneous Besov spaces on domains may be defined by the same token.

A very useful feature of Besov spaces is their interpolation properties. We refer to the books [START_REF] Bennett | Interpolation of operators[END_REF][START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF] for the proof of the following statement.

Proposition 5. Let b s p,q denote B s p,q or Ḃs p,q ; s ∈ R, p ∈ [1, ∞) and q ∈ [1, ∞]. The real interpolation of Besov spaces gives the following statement if s 1 = s 2 : b s 1 p,q 1 (Ω ), b s 2 p,q 2 (Ω ) θ ,q = b s p,q (Ω ) with s = θ s 2 + (1 -θ )s 1 , and 1 p = θ p 2 + 1-θ p 1 • Moreover, if s 1 = s 2 , t 1 = t 2 and if T : b s 1 p 1 ,q 1 (Ω ) + b s 2 p 2 ,q 2 (Ω ) → b t 1 k 1 ,l 1 (Ω ) + b t 2 k 2 ,l 2 (Ω ) is a linear map, bounded from b s 1 p 1 ,q 1 (Ω ) to b t 1 k 1 ,l 1 (Ω ) and from b s 2 p 2 ,q 2 (Ω ) to b t 2 k 2 ,l 2 (Ω ) then for any θ ∈ (0, 1), the map T is also bounded from b s p,q (Ω ) to b t k,q (Ω ) with s = θ s 2 + (1 -θ )s 1 , t = θt 2 + (1 -θ )t 1 , 1 p = θ p 2 + 1 -θ p 1 , 1 k = θ k 2 + 1 -θ k 1 •
The following composition estimate will be of constant use in the last section of this paper.

Proposition 6. Let f : R r → R be a C 1 function such that f (0) = 0 and, for some m ≥ 1 and K ≥ 0, |d f (u)| ≤ K|u| m-1 for all u ∈ R r . (7)
Then for all s ∈ (0, 1) and 1 ≤ p, q ≤ ∞ there exists a constant C so that

f (u) Ḃs p,q (Ω ) ≤ CK u m-1 L ∞ (Ω ) u Ḃs p,q (Ω ) . (8)
Proof. The proof relies on the characterization of the norm of Ḃs p,q (Ω ) by finite differences, namely3

f (u) Ḃs p,q (Ω ) = Ω Ω | f (u(y)) -f (u(x))| p |y -x| n+sp dy q p dx 1 q . ( 9
)
Now the mean value formula implies that

f (u(y)) -f (u(x)) = 1 0 d f (u(x) + t(u(y) -u(x))) dt • (u(y) -u(x)).
Hence using the growth assumption [START_REF] Friedman | Partial differential equations[END_REF],

| f (u(y)) -f (u(x))| ≤ K 1 0 |u(x) + t(u(y) -u(x))| m-1 dt |u(y) -u(x)|. ( 10
)
Therefore we get

| f (u(y)) -f (u(x))| ≤ K u m-1 L ∞ (Ω ) |u(y) -u(x)|.
Inserting this latter inequality in (9), we readily get [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF].

In [START_REF] Danchin | A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space[END_REF][START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF], we proved that:

Proposition 7.
Let Ω be the half-space, or a bounded or exterior domain with C 1 boundary. For all 1 ≤ p, q < ∞, and -1 + 1/p < s < 1/p, we have

B s p,q (Ω ) = C ∞ c (Ω ) • B s p,q (Ω) . ( 11
)
Remark 1. In any C 1 domain Ω and for 0 < s < n/p the space Ḃs p,q (Ω ) embeds in Ḃ0 m,q (Ω ) with 1/m = 1/ps/n. Therefore, if q ≤ min(2, m), it also embeds in the Lebesgue space L m (Ω ). So finally if s ∈ (0, 1 p ) and q ≤ min(2, m) with m as above then Proposition 7 allows us to redefine the space Ḃs p,q (Ω ) by

Ḃs p,q (Ω ) = C ∞ c (Ω ) • Ḃs p,q (Ω) . ( 12
)
Remark 2. In particular under the above hypotheses, both classes of Besov spaces admit trivial extension by zero onto the whole space. Combining with Proposition 2, we deduce that

B s p,q (Ω ) = Ḃs p,q (Ω ) if -1 + 1/p < s < 1/p and Ω is bounded.
Note also that, for obvious reasons, the above density result does not hold true if q = ∞, for the strong topology. However, it holds for the weak * topology.

A priori estimates for the heat equation

This section is the core of the paper. Here we prove generalizations of Theorem 1 to more general domains. First we consider the half-space case, then we consider the exterior and bounded cases. We shall mainly focus on the unbounded case which is more tricky and just indicate at the end of this section what has to be changed in the bounded domain case.

The heat equation in the half-space

The purpose of this paragraph is to extend Theorem 1 to the half-space case R n + , namely

u t -ν∆ u = f in (0, T ) × R n + , u| x n =0 = 0 on (0, T ) × ∂ R n + , u| t=0 = u 0 on R n + . (13)
Theorem 2. Let p ∈ [1, ∞) and s ∈ (-1 + 1/p, 1/p). Assume that f belongs to L 1 (0, T ; Ḃs p,1 (R n + )) and that u 0 is in Ḃs p,1 (R n + ). Then (13) has a unique solution u satisfying

u ∈ C ([0, T ); Ḃs p,1 (R n + )), u t , ∇ 2 u ∈ L 1 (0, T ; Ḃs p,1 (R n + ))
and the following estimate is valid:

u L ∞ (0,T ; Ḃs p,1 (R n + )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs p,1 (R n + )) ≤ C( f L 1 (0,T ; Ḃs p,1 (R n + )) + u 0 Ḃs p,1 (R n + ) ), ( 14
)
where C is an absolute constant with no dependence on ν and T .

Proof. We argue by symmetrization. Let u 0 and f be the antisymmetric extensions over R n to the data u 0 and f . Then, given our assumptions over s and Proposition 7, one may assert that

u 0 ∈ Ḃs p,1 (R n ), f ∈ L 1 (0, T ; Ḃs p,1 (R n ))
and that, in addition

u 0 Ḃs p,1 (R n ) ≈ u 0 Ḃs p,1 (R n + ) and f L 1 (0,T ; Ḃs p,1 (R n )) ≈ f L 1 (0,T ; Ḃs p,1 (R n + )) .
Let u be the solution given by Theorem 1. As this solution is unique in the corresponding functional framework, the symmetry properties of the data ensure that u is antisymmetric with respect to {x n = 0}. As a consequence, it vanishes over {x n = 0}. Hence the restriction u of u to the half-space satisfies [START_REF] Maremonti | On nonstationary Stokes problem in exterior domains[END_REF]. In addition,

• u t coincides with the antisymmetric extension of u t ,

• ∇ 2 x ′ u coincides with the antisymmetric extension of ∇ 2 x ′ u, • ∇ x ′ ∂ x n u coincides with the symmetric extension of ∇ x ′ ∂ x n u, • ∂ 2 x n ,x n u = (∆ -∆ x ′ ) u hence coincides with u t -f -∆ x ′ u. Hence one may conclude that u L ∞ (0,T ; Ḃs p,1 (R n + )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs p,1 (R n + )) ≤ u L ∞ (0,T ; Ḃs p,1 (R n )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs p,1 (R n )) .
This implies [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF].

Remark 3. The case of non-homogeneous boundary conditions where u equals some given h at the boundary, reduces to the homogeneous case : it is only a matter of assuming that h admits some extension h over (0,

T ) × R n + so that h t -ν∆ h ∈ L 1 (0, T ; Ḃs p,1 (R n + )).

The exterior domain case

Here we extend Theorem 1 to the case where Ω is an exterior domain (that is the complement of a bounded simply connected domain). Here is our main statement:

Theorem 3. Let Ω be a C 2 exterior domain of R n with n ≥ 3. Let 1 < q ≤ p < ∞ with q < n/2. Let -1 + 1/p < s < 1/p and -1 + 1/q < s ′ < 1/q -2/n. Let u 0 ∈ Ḃs p,1 ∩ Ḃs ′ q,1 (Ω ) and f ∈ L 1 (0, T ; Ḃs p,1 ∩ Ḃs ′ q,1 (Ω )).
Then there exists a unique solution u to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]

such that u ∈ C ([0, T ]; Ḃs p,1 ∩ Ḃs ′ q,1 (Ω )), u t , ∇ 2 u ∈ L 1 (0, T ; Ḃs p,1 ∩ B s ′ q,1 (Ω ))
and the following inequality is satisfied:

u L ∞ (0,T ; Ḃs p,1 ∩ Ḃs ′ q,1 (Ω )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs p,1 ∩ Ḃs ′ q,1 (Ω )) ≤ C u 0 Ḃs p,1 ∩ Ḃs ′ q,1 (Ω ) + f L 1 (0,T ; Ḃs p,1 ∩ Ḃs ′ q,1 (Ω )) , ( 15
)
where the constant C is independent of T and ν.

Proving this theorem relies on the following statement (that is of independent interest and holds in any dimension n ≥ 2), and on lower order estimates (see Lemma 2 below) which will enable us to remove the time dependency.

Theorem 4. Let Ω be a C 2 exterior domain of R n with n ≥ 2. Let 1 < p < ∞, -1 + 1/p < s < 1/p, f ∈ L 1 (0, T ; Ḃs p,1 (Ω )), and u 0 ∈ Ḃs p,1 (Ω ). Then equation (1) has a unique solution u such that u ∈ C ([0, T ]; Ḃs p,1 (Ω )), ∂ t u, ∇ 2 u ∈ L 1 (0, T ; Ḃs p,1 (Ω ))
and the following estimate is valid:

u L ∞ (0,T ; Ḃs p,1 (Ω )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs p,1 (Ω )) ≤ Ce CT ν u 0 Ḃs p,1 (Ω ) + f L 1 (0,T ; Ḃs p,1 (Ω )) , ( 16
)
where the constant C depends only on s, p, and Ω .

Additionally if K is a compact subset of Ω such that dist(∂ Ω , Ω \ K) > 0, there holds u L ∞ (0,T ; Ḃs p,1 (Ω )) + u t , ν∇ 2 u L 1 (0,T ; Ḃs p,1 (Ω )) ≤ C u 0 Ḃs p,1 (Ω ) + f L 1 (0,T ; Ḃs p,1 (Ω )) + ν u L 1 (0,T ; Ḃs p,1 (K)) , ( 17
)
where C is as above.

Proof. We suppose that we have a smooth enough solution and focus on the proof of the estimates. We shall do it in three steps: first we prove interior estimates, next boundary estimates and finally global estimates after summation. Note that performing the following change of unknown and data:

u new (t, x) = νu old (ν -1 t, x), u 0,new (x) = νu 0,old (x), f new (t, x) = f old (ν -1 t, x)
reduces the study to the case ν = 1. So we shall make this assumption in all that follows. Throughout we fix some covering (B(x ℓ , λ )) 1≤ℓ≤L of K by balls of radius λ and take some neighborhood

Ω 0 ⊂ Ω of R n \ K such that d(Ω 0 , ∂ Ω ) > 0.
We assume in addition that the first M balls do not intersect K while the last L -M balls are centered at some point of ∂ Ω .

Let η 0 : R n → [0, 1] be a smooth function supported in Ω 0 and with value 1 on a neighborhood of Ω \ K. Then we consider a subordinate partition of unity (η ℓ ) 1≤ℓ≤L such that:

1. ∑ 0≤l≤L η ℓ = 1 on Ω ; 2. ∇ k η ℓ L ∞ (R n ) ≤ C k λ -k for k ∈ N and 1 ≤ ℓ ≤ L; 3. Supp η ℓ ⊂ B(x ℓ , λ ).
We also introduce another smooth function η 0 supported in K and with value 1 on Supp ∇η 0 and smooth functions η 1 , • • • , η L with compact support in Ω ℓ and such that η ℓ ≡ 1 on Supp η ℓ .

Note that for ℓ ∈ {1, • • • , L}, the bounds for the derivatives of η ℓ together with the fact that Supp ∇η ℓ ≈ λ n and Proposition 5 implies that for k = 0, 1 and any q ∈ [1, ∞], we have ∇η ℓ

Ḃk+n/q q,1 (R n ) λ -1-k . (18)
The same holds for the functions η ℓ . Throughout, we set U ℓ := uη ℓ .

First step: the interior estimate

The vector-field U 0 satisfies the following modification of (1):

U 0 t -∆U 0 = η 0 f -2∇η 0 • ∇u -u∆ η 0 in (0, T ) × R n , U 0 | t=0 = u 0 η 0 on R n . (19)
Theorem 1 thus yields the following estimate:

U 0 L ∞ (0,T ; Ḃs p,1 (R n )) + U 0 t , ∇ 2 U 0 L 1 (0,T ; Ḃs p,1 (R n )) η 0 f L 1 (0,T ; Ḃs p,1 (R n )) + ∇η 0 • ∇u L 1 (0,T ; Ḃs p,1 (R n )) + u∆ η 0 L 1 (0,T ; Ḃs p,1 (R n )) + η 0 u 0 Ḃs p,1 (R n ) .
Let us emphasize that as ∇η 0 • ∇u and u∆ η 0 are compactly supported, we may replace the homogeneous norms by non-homogeneous ones in the first two terms.

As a consequence, because the function ∇η 0 is in C ∞ c (R n ) and η 0 ≡ 1 on Supp ∇η 0 , Corollary 1 ensures that

U 0 L ∞ (0,T ; Ḃs p,1 (R n )) + U 0 t , ∇ 2 U 0 L 1 (0,T ; Ḃs p,1 (R n )) η 0 u 0 Ḃs p,1 (R n )) + η 0 f L 1 (0,T ; Ḃs p,1 (R n )) + η 0 u L 1 (0,T ;B s+1 p,1 (R n )) . (20)
Now, by interpolation,

η 0 u B 1+s p,1 (Ω ) ≤ C η 0 u 1 2 B 2+s p,1 (Ω ) η 0 u 1 2 B s p,1 (Ω ) . (21)
As Supp η 0 ⊂ K and as homogeneous and nonhomogeneous norms are equivalent on K, one may thus conclude that

U 0 L ∞ (0,T ; Ḃs p,1 (R n )) + U 0 t , ∇ 2 U 0 L 1 (0,T ; Ḃs p,1 (R n )) f L 1 (0,T ; Ḃs p,1 (Ω )) + T 1/2 u L 1 (0,T ; Ḃ2+s p,1 (K))∩L ∞ (0,T ; Ḃs p,1 (K)) + u 0 Ḃs p,1 (Ω ) . ( 22
)
Note that starting from (21) and using Young's inequality also yields for all ε > 0:

U 0 L ∞ (0,T ; Ḃs p,1 (R n )) + U 0 t , ∇ 2 U 0 L 1 (0,T ; Ḃs p,1 (R n )) ≤ C u 0 Ḃs p,1 (Ω ) + f L 1 (0,T ; Ḃs p,1 (Ω )) + ε u L 1 (0,T ; Ḃ2+s p,1 (K)) + c(ε) u L 1 (0,T ; Ḃs p,1 (K)) . (23)
The terms U ℓ with 1 ≤ ℓ ≤ M may be bounded exactly along the same lines because their support do not meet ∂ Ω , hence their extension by 0 over R n satisfies

U ℓ t -∆U ℓ = f ℓ in (0, T ) × R n , U ℓ | t=0 = u 0 η ℓ on R n with f ℓ := -2∇η ℓ • ∇u -u∆ η ℓ + η ℓ f . ( 24
)
Arguing as above and taking advantage of the fact that the functions η ℓ are localized in balls of radius λ (that is we use (18)), we now get

f ℓ L 1 (0,T ; Ḃs p,1 (Ω )) η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) + λ -2 η ℓ u L 1 (0,T ; Ḃs p,1 (Ω )) + λ -1 η ℓ ∇u L 1 (0,T ; Ḃs p,1 (Ω )) . ( 25
)
Using again (21) (with η ℓ instead of η 0 ), we get

U ℓ L ∞ (0,T ; Ḃs p,1 (R n )) + U ℓ t , ∇ 2 U ℓ L 1 (0,T ; Ḃs p,1 (R n )) η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) + λ -1 T 1/2 + λ -2 T u L 1 (0,T ; Ḃ2+s p,1 (K))∩L ∞ (0,T ; Ḃs p,1 (K)) + u 0 η ℓ Ḃs p,1 (Ω ) , (26) U ℓ L ∞ (0,T ; Ḃs p,1 (R n )) + U ℓ t , ∇ 2 U ℓ L 1 (0,T ; Ḃs p,1 (R n )) ≤ C u 0 η ℓ Ḃs p,1 (Ω ) + η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) + λ -1 u 1/2 L 1 (0,T ; Ḃ2+s p,1 (K)) u 1/2 L 1 (0,T ; Ḃs p,1 (K)) . (27)
Second step: the boundary estimate

We now consider an index ℓ ∈ {L + 1, • • • , M} so that B(x ℓ , λ ) is centered at a point of ∂ Ω . The localization leads to the following problem:

U ℓ t -∆U ℓ = f ℓ in (0, T ) × Ω , U ℓ = 0 on (0, T ) × ∂ Ω , U ℓ t | t=0 = u 0 η ℓ on Ω , (28)
with f ℓ defined by (24), hence satisfying (25).

Let us now make a change of variables so as to recast (28) in the half-space. As ∂ Ω is C 2 , if λ has been chosen small enough then for fixed ℓ we are able to find a map Z ℓ so that

i) Z ℓ is a C 2 diffeomorphism from B(x ℓ , λ ) to Z ℓ (B(x ℓ , λ )); ii) Z ℓ (x ℓ ) = 0 and D x Z(x ℓ ) = Id ; iii) Z ℓ (Ω ∩ B(x ℓ , λ )) ⊂ R n + ; iv) Z ℓ (∂ Ω ∩ B(x ℓ , λ )) = ∂ R n + ∩ Z ℓ (B(x ℓ , λ )). Setting ∇ x Z ℓ = Id + A ℓ
then one may assume in addition that there exist constants C j depending only on Ω and on j ∈ {0, 1} such that

D j A ℓ L ∞ (B(x ℓ ,λ )) ≤ C j , (29)
a property which implies (by the mean value formula) that

A ℓ L ∞ (B(x ℓ ,λ )) ≤ C 1 λ , (30)
hence by interpolation between the spaces L q (B(x ℓ , λ )) and W r-1 q (B(x ℓ , λ )),

A ℓ B n q q,1 (B(x ℓ ,λ ))
≤ Cλ for all 1 ≤ q < ∞ such that n/q < r -1.

(31)

Let V ℓ := Z * ℓ U ℓ := U ℓ • Z -1 ℓ .
The system satisfied by V ℓ reads

V ℓ t -∆ z V ℓ = F ℓ in (0, T ) × R n + , V ℓ | z n =0 = 0 on (0, T ) × ∂ R n + , V ℓ | t=0 = Z * ℓ (U ℓ | t=0 ) on ∂ R n + , (32)
with

F ℓ := Z * ℓ f ℓ + (∆ x -∆ z )V ℓ .
According to Theorem 2, we thus get

V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) + V ℓ t , ∇ 2 z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) Z * ℓ f ℓ L 1 (0,T ; Ḃs p,1 (R n + )) + (∆ x -∆ z )V ℓ ) L 1 (0,T ; Ḃs p,1 (R n + )) + Z * ℓ (U ℓ | t=0 ) Ḃs p,1 (R n + ) .
Note that the first and last terms in the right-hand side may be dealt with thanks to Lemma 1: we have

Z * ℓ f ℓ L 1 (0,T ; Ḃs p,1 (R n + )) f ℓ L 1 (0,T ; Ḃs p,1 (Ω )) Z * ℓ (U ℓ | t=0 ) Ḃs p,1 (R n + ) U ℓ | t=0 Ḃs p,1 (Ω ) .
Compared to the first step, the only definitely new term is (∆ x -∆ z )V ℓ . Explicit computations (see e.g. [START_REF] Danchin | Critical functional framework and maximal regularity in action on systems of incompressible flows[END_REF]) show that

(∆ z -∆ x )V ℓ is a linear combination of com- ponents of ∇ 2 z A ℓ ⊗ V ℓ and ∇ z A ℓ ⊗ ∇ z V ℓ . Therefore (∆ x -∆ z )V ℓ ) L 1 (0,T ; Ḃs p,1 (R n + )) A ℓ ⊗ ∇ 2 z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) + ∇ z A ℓ ⊗ ∇ z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) .
Now, according to Proposition 4 and owing to the support properties of the terms involved in the inequalities,, we have

A ℓ ⊗∇ 2 z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) A ℓ Ḃ n q q,1 (B(x ℓ ,λ )) ∇ 2 z V ℓ Ḃs p,1 (R n + ) with q = min(p, p ′ ).
Therefore we have, thanks to (30) and to (31),

A ℓ ⊗ ∇ 2 z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) λ ∇ 2 z V ℓ Ḃs p,1 (R n + )
.

Similarly, we have

∇ z A ℓ ⊗ ∇ z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) ∇ z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) .
Therefore

(∆ x -∆ z )V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) λ ∇ 2 z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) + ∇ z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) .
Putting together the above inequalities and remembering of (25) and Lemma 1, we finally get, taking λ small enough

V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) + V ℓ t , ∇ 2 z V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) U ℓ | t=0 Ḃs p,1 (Ω ) + η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) +λ -2 η ℓ u L 1 (0,T ; Ḃs p,1 (Ω )) + λ -1 η ℓ ∇u L 1 (0,T ; Ḃs p,1 (Ω )) + ∇V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) .
By interpolation, we have

∇V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) ∇ 2 V ℓ 1/2 L 1 (0,T ; Ḃs p,1 (R n + )) V ℓ 1/2 L 1 (0,T ; Ḃs p,1 (R n + )) .
Therefore using Young's inequality enables us to reduce the above inequality to

V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) + V ℓ t , ∇ 2 V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) U ℓ | t=0 Ḃs p,1 (Ω ) + η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) + V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) +λ -2 η ℓ u L 1 (0,T ; Ḃs p,1 (Ω )) + λ -1 η ℓ ∇u L 1 (0,T ; Ḃs p,1 (Ω )) .
In order to handle the last term, there are two ways of proceeding depending on whether we want a time dependent constant or not. The first possibility is to write that, by interpolation and Hölder's inequality,

η ℓ ∇u L 1 (0,T ; Ḃs p,1 (Ω )) ≤ T 1/2 u L 1 (0,T ; Ḃs+2 p,1 (K))∩L ∞ (0,T ; Ḃs p,1 (K)) .
This yields

V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) + V ℓ t , ∇ 2 V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) η ℓ u 0 Ḃs p,1 (Ω ) + η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) + T V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) + λ -1 T 1/2 + λ -2 T u L 1 (0,T ; Ḃs+2 p,1 (K))∩L ∞ (0,T ; Ḃs p,1 (K)) . ( 33
)
The second possibility is to write that

η ℓ ∇u L 1 (0,T ; Ḃs p,1 (Ω )) ≤ u 1 2 L 1 (0,T ; Ḃs+2 p,1 (K)) u 1 2 L 1 (0,T ; Ḃs p,1 (K)) .
We eventually get

V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) + V ℓ t , ∇ 2 V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) η ℓ u 0 Ḃs p,1 (Ω ) + η ℓ f L 1 (0,T ; Ḃs p,1 (Ω )) + λ -1 u 1/2 L 1 (0,T ; Ḃ2+s p,1 (K)) u 1/2 L 1 (0,T ; Ḃs p,1 (K)) + λ -2 u L 1 (0,T ; Ḃs p,1 (K)) + ∇V ℓ L 1 (0,T ; Ḃs p,1 (R n + )) . ( 34
)
Third step: global a priori estimates Now, in view of Lemma 1, we may write

u L ∞ (0,T ; Ḃs p,1 (Ω )) ≤ ∑ ℓ U ℓ L ∞ (0,T ; Ḃs p,1 (Ω )) ∑ 0≤ℓ≤M U ℓ L ∞ (0,T ; Ḃs p,1 (R n )) + ∑ M<ℓ≤L V ℓ L ∞ (0,T ; Ḃs p,1 (R n + )) ,
similar inequalities for the other terms of the l.h.s of (33). Of course, Proposition 1 ensures that

u ℓ 0 Ḃs p,1 (Ω ) u 0 Ḃs p,1 (Ω ) and η ℓ f L 1 (0,T ; Ḃs p,1 (Ω ))
f L 1 (0,T ; Ḃs p,1 (Ω )) .

So using also ( 22) and ( 26) and assuming that T is small enough, we end up with

u L ∞ (0,T ; Ḃs p,1 (Ω )) + (u t , ∇ 2 u) L 1 (0,T ; Ḃs p,1 (Ω )) u 0 Ḃs p,1 (Ω ) + f L 1 (0,T ; Ḃs p,1 (Ω )) + (λ -1 T 1/2 + λ -2 T ) u L 1 (0,T ; Ḃs+2 p,1 (K))∩L ∞ (0,T ; Ḃs p,1 (K)) .
Hence if in addition λ -2 T is small enough,

u L ∞ (0,T ; Ḃs p,1 (Ω )) + u t , ∇ 2 u L 1 (0,T ; Ḃs p,1 (Ω )) ≤ C u 0 B s p,1 (R n ) + f L 1 (0,T ; Ḃs p,1 (Ω )) .
Repeating the argument over the interval [T, 2T ] and so on, we get exactly Inequality ( 16).

If we want to remove the time-dependency then it is just a matter of starting from (34) and ( 27) instead of ( 33) and ( 22). After a few computation and thanks to Young's inequality, we get for some constant C depending on λ ,

u t , ∇ 2 u L 1 (0,T ; Ḃs p,1 (Ω )) ≤ C( u 0 Ḃs p,1 (Ω ) + f L 1 (0,T ; Ḃs p,1 (R n + )) + u L 1 (0,T ; Ḃs p,1 (K)) ).
For completeness, let us say a few words about the existence, which is rather standard issue (see e.g. [START_REF] Ladyzhenskaja | Linear and quasilinear equations of parabolic type[END_REF]). If the domain is smooth then the easiest approach is via the L 2 -framework and Galerkin method. We may consider smooth approximations of data f and u 0 , such that to keep them in the space H m with sufficiently large m ∈ N. Then the energy method provides us with approximate solutions in Sobolev spaces H m with large m. In particular, the above a priori estimates ( 16) may be derived for such solutions. It is then easy to pass to the limit.

Remark 4. Let us emphasize that the term u L 1 (0,T ; Ḃs p,1 (K)) may be replaced by other lower order norms such as u L 1 (0,T ; Ḃs ′ p,1 (K)) with s ′ = s close to 0. In particular s ′ may be put to zero.

In order to complete the proof of Theorem 3, we now have to bound the last term of (17), namely u L 1 (0,T ; Ḃs p,1 (K)) , independently of T. This is the goal of the next lemma (where we keep the assumption that ν = 1). We here adapt to the heat equation an approach that has been proposed for the Stokes system in [START_REF] Maremonti | On nonstationary Stokes problem in exterior domains[END_REF].

Lemma 2. Assume that n ≥ 3 and that 1 < p < n/2. Then for any s ∈ (-1 +

1/p, 1/p -2/n) sufficiently smooth solutions to (1) fulfill u L 1 (0,T ; Ḃs p,1 (K)) ≤ C( f L 1 (0,T ; Ḃs p,1 (Ω )) + u 0 Ḃs p,1 (Ω ) ),
where C is independent of T .

Proof. Thanks to the linearity of the system, one may split the solution u into two parts, the first one u 1 being the solution of the system with zero initial data and source term f , and the second one u 2 , the solution of the system with no source term and initial data u 0 . In other words, u = u 1 + u 2 with u 1 and u 2 satisfying

u 1,t -∆ u 1 = f in (0, T ) × Ω , u 2,t -∆ u 2 = 0 in (0, T ) × Ω , u 1 = 0 on (0, T ) × ∂ Ω , u 2 = 0 on (0, T ) × ∂ Ω , u 1 | t=0 = 0 on Ω , u 2 | t=0 = u 0 on Ω . (35)
Let us first focus on u 1 . Recall that up to a constant we have (see Proposition 3):

u 1 (t) Ḃs p,1 (K) = sup K u 1 (t, x)η 0 (x) dx, ( 36
)
where the supremum is taken over all η 0 ∈ Ḃ-s p ′ ,∞ (K) such that η 0 Ḃ-s p ′ ,∞ (K) = 1. Of course, by virtue of Remark 2, any such function η 0 may be extended by 0 over R n , and its extension still has a norm of order 1. So we may assume that the supremum is taken over all

η 0 ∈ Ḃ-s p ′ ,∞ (R n ) with η 0 Ḃ-s p ′ ,∞ (R n ) = 1 and Supp η 0 ⊂ K. ( 37
)
Consider the solution η to the following problem:

η t -∆ η = 0 in (0, T ) × Ω , η = 0 on (0, T ) × ∂ Ω , η| t=0 = η 0 on Ω . (38)
Testing the equation for u 1 by η(t -•) we discover that

Ω u 1 (t, x)η 0 (x) dx = t 0 Ω f (τ, x)η(t -τ, x) dx dτ. ( 39
)
The general theory for the heat operator in exterior domains implies the following estimates:

η(t) L a (Ω ) ≤ C η 0 L b (Ω ) t -n 2 ( 1 b -1 a ) for 1 < b ≤ a < ∞, (40)
as well as

∆ η(t) L a (Ω ) ≤ C ∆ η 0 L b (Ω ) t -n 2 ( 1 b -1 a ) for 1 < b ≤ a < ∞. (41)
In the case Ω = R n , those two inequalities may be derived easily from the (explicit) heat kernel. To prove (40) in the case of an exterior domain, it is enough to look at solutions to (38) as subsolutions to the problem in the whole space. More precisely, if we assume that η 0 ≥ 0 (this is not restrictive for one may consider the positive and negative part of the initial data separately), we get a solution to (38) defined over (0, ∞) × Ω such that η ≥ 0. Then we consider an extension Eη : R n → R η, such that Eη = η for x ∈ Ω and Eη = 0 for x / ∈ Ω . We claim that Eη is a subsolution to the Cauchy problem ηt -∆ η = 0 in (0, T ) × R n with η| t=0 = Eη 0 .

(42)

It is sufficient to show that η ≤ η, since η is always nonnegative. It is clear that

(η -η) t -∆ (η -η) = 0 in (0, T ) × Ω . (43)
Consider (ηη) + := max{ηη, 0}. It is obvious that (ηη) + vanishes at the boundary, because η is zero and η is nonnegative there. Hence we conclude 1 2

d dt Ω (η -η) 2 + dx + Ω |∇(η -η) + | 2 dx = 0. ( 44
)
Thus, (η -η) + ≡ 0, since (η -η) + | t=0 = 0. So η is bounded by η.

To prove (41) we observe that for the smooth solutions the equation implies that ∆ η| ∂ Ω = 0, so we can consider the problem on ∆ η instead of η. Now, as η vanishes at the boundary, we have (see e.g. [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF])

∇ 2 η L c (Ω ) ≤ ∆ η L c (Ω ) for all 1 < c < ∞. (45)
Hence, interpolating between (40) and (41) yields for 0 < s < 1/b.

η(t) Ḃs b,r (Ω ) ≤ C η 0 Ḃs a,r (Ω ) t -n 2 ( 1 p -1 q ) for 1 < a ≤ b < ∞ and 1 ≤ r ≤ ∞. (46)
In order to extend this inequality to negative indices s, we consider the following dual problem:

ζ t -∆ ζ = 0 in (0, T ) × Ω , ζ = 0 on (0, T ) × ∂ Ω , ζ | t=0 = ζ 0 on Ω , (47)
where ζ 0 ∈ B -s b ′ ,r ′ (Ω ). Now, testing (47) by η(t -•) yields

Ω η(t, x)ζ 0 (x) dx = Ω η 0 (x)ζ (t, x) dx. ( 48
)
Let us observe that

η(t) Ḃs b,r (Ω ) = sup ζ 0 Ω η(t, x)ζ 0 (x) dx, ( 49
)
where the supremum is taken over all

ζ 0 ∈ Ḃ-s b ′ ,r ′ (Ω ) such that ζ 0 Ḃ-s b ′ ,r ′ (Ω ) = 1.
Thus by virtue of (48), we get:

η(t) Ḃs b,r (Ω ) = sup ζ 0 Ω η 0 (x)ζ (t, x) dx ≤ sup ζ 0 η 0 Ḃs a,r (Ω ) ζ (t) Ḃ-s a ′ ,r ′ (Ω ) . (50)
Since -s is positive we can apply (46) and get if 0

< -s < 1/a ′ , η(t) Ḃs b,r (Ω ) ≤ C η 0 Ḃs a,r (Ω ) t -n 2 ( 1 a ′ -1 b ′ ) sup ζ 0 ζ 0 Ḃ-s b ′ ,r ′ (Ω ) . Since 1 a ′ -1 b ′ = 1 b -1 a , we conclude that η(t) Ḃs a,r (Ω ) ≤ C η 0 Ḃs b,r (Ω ) t -n 2 ( 1 b -1 a ) if s > -1 + 1/a. ( 51
)
In order to get the remaining case s = 0, it suffices to argue by interpolation between ( 46) and (51). One can thus conclude that for all 1 < b ≤ a < ∞, q ∈ [1, ∞] and -1 + 1/a < s < 1/b, we have

η(t) Ḃs a,r (Ω ) ≤ C η 0 Ḃs b,r (Ω ) t -n 2 ( 1 b -1 a ) . (52)
Now we return to the initial problem of bounding u 1 . Starting from (39) and using duality, one may write

Ω u 1 (t, x)η 0 (x) dx t 0 f (τ) Ḃs p,1 (Ω ) η(t -τ) Ḃ-s p ′ ,∞ (Ω ) dτ.
Hence splitting the interval (0,t) into (0, max(0,t -1)) and (max(0,t -1),t) and applying (52) yields for any ε ∈ (0, 1 + s), Ω ) . This may easily proved by introducing a suitable smooth cut-off function with value 1 over K and taking advantage of Proposition 1. A scaling argument yields the dependency of the norm of the embedding with respect to |K|. Hence we have for some constant C depending on K:

Ω u 1 (t, x)η 0 (x) dx t max(0,t-1) f (τ) Ḃs p,1 (Ω ) η 0 Ḃ-s p ′ ,∞ (Ω ) dτ + max(0,t-1) 0 f (τ) Ḃs p,1 (Ω ) η 0 Ḃ-s 1 1-ε ,∞ (Ω ) (t -τ) -n 2 ( 1 p -ε) dτ. Now, as η 0 is supported in K, one has η 0 Ḃ-s a,∞ (Ω ) ≤ C|K| 1 p + 1 a -1 η 0 Ḃ-s p ′ ,∞ (
η 0 Ḃ-s 1 1-ε ,∞ ≤ C η 0 Ḃ-s p ′ ,∞ (Ω ) .
So, keeping in mind (39) and the fact that the supremum is taken over all the functions η 0 satisfying (37), we deduce that

1 (t) Ḃs p,1 (K) ≤ C t max(0,t-1) f (τ) Ḃs p,1 (Ω ) dτ + max(0,t-1) 0 (t -τ) -n 2 ( 1 p -ε) f (τ) Ḃs p,1 (Ω ) dτ . Therefore, T 1 u 1 Ḃs p,1 (K) dt ≤ C 1 + T 1 τ -n 2 ( 1 p -ε) dτ T 0 f Ḃs p,1 (Ω ) dt. ( 53
)
For the time interval [0, 1], we merely have

1 0 u 1 Ḃs p,1 (K) dt ≤ C 1 0 f Ḃs p,1 (Ω ) dt.
Now, provided that one may find some ε > 0 such that

n 2 1 p -ε > 1, (54)
a condition which is equivalent to p < n/2, the constant in (53) may be made independent of T. Hence we conclude that

T 0 u 1 Ḃs p,1 (K) dt ≤ C T 0 f Ḃs p,1 (Ω ) dt (55)
with C independent of T .

Let us now bound u 2 . We first write that

u 2 (t) Ḃs p,1 (K) ≤ C u 0 Ḃs p,1 (Ω ) (56)
and, if -1

+ ε < s < 1/p, u 2 (t) Ḃs p,1 (K) ≤ C|K| 1 p -ε u 2 (t) Ḃs 1 ε ,1 (K) ≤ C|K| 1 p -ε u 0 Ḃs p,1 (Ω ) t -n 2 ( 1 p -ε) .
Then decomposing the integral over [0, T ] into an integral over [0, min(1, T )] and [min(1, T ), T ], we easily get

T 0 u 2 (t) Ḃs p,1 (K) dt ≤ C 1 + T min(1,T ) t -n 2 ( 1 p -ε) dt u 0 Ḃs p,1 (Ω ) . (57)
The integrant in the r.h.s. of (57) is finite whenever (54) holds. Hence,

T 0 u 2 (t) Ḃs p,1 (K) dt ≤ C u 0 Ḃs p,1 (Ω ) . (58)
Putting this together with (53) and (2.2) completes the proof of the lemma.

u 2 L 1 (0,T ; Ḃs (K)) u 0 Ḃs p,1 (Ω ) .
So one may conclude that Lemma 2 holds true for any 1 < p < ∞ and -1 + 1/p < s < 1/p. Consequently, we get:

Theorem 5. If 1 < p < ∞ and -1 + 1/p < s < 1/p then the statement of Theorem 2 remains true in any C 2 bounded domain.

Applications

In this last section, we give some application of the maximal regularity estimates that have been proved hitherto. As an example, we prove global stability results (in a critical functional framework) for trivial/constant solutions to the following system:

u t -ν∆ u + P • ∇ 2 u = f 0 (u) + f 1 (u) • ∇u in (0, T ) × Ω , u = 0 at (0, T ) × ∂ Ω , u| t=0 = u 0 on Ω . (60)
Above, ν is a positive parameter, u stands for a r-dimensional vector and P = (P 1 , • • • , P r ) where the P k 's are n × n matrices with suitably smooth coefficients. The nonlinearities f 0 : R r → R r and f 1 : R r → M r,n (R) are C 1 and satisfy

f 0 (0) = 0, d f 0 (0) = 0 and f 1 (0) = 0, (61)
together with some growths conditions that will be detailed below.

As we have in mind applications to Theorem 3, we focus on the case where Ω is a smooth exterior domain of R n with n ≥ 3. Of course, based on our other maximal regularity results, similar (and somewhat easier) statements may be proved for bounded domains, R n + or R n . Here are two important examples entering in the class of equations (60). The first one is the nonlinear heat transfer equation (see [START_REF] Xin | Front propagation in heterogeneous media[END_REF] and the references therein):

u t -ν∆ u = f (u). (62)
A classical form of the nonlinearity is f (u) = Ku 2 (uu * ). However one may consider more complex models describing a flame propagation like in [START_REF] Lewicka | On the existence of traveling waves in the 3D Boussinesq system[END_REF].

The second example is the viscous Burgers equation [9, 10]

u t + u∂ x 1 u -ν∆ u = 0. ( 63
)
which enters in the class of models like

u t -ν∆ u = B(u, ∇u). (64)
In the case where B(u, ∇u) = -u • ∇u, this is just the equation for pressureless viscous gases with constant density. Below, based on Theorem 3, we shall prove two global-in-time results concerning the stability of the trivial solution of System (60). In the first statement, to simplify the presentation, we only consider the case where the data belong to spaces with regularity index equals to 0. To simplify the notation, we omit the dependency with respect to the domain Ω in all that follows. Theorem 6. Let 1 < q < n/2 and Ω be an exterior domain of R n (n ≥ 3). There exist two positive constants η and c ν such that for all P :

[0, ∞)× Ω → R n × R n × R k satisfying 4 : P L ∞ (0,∞;M ( Ḃ0 n,1 ∩ Ḃ0 q,1 )) ≤ ην, (65)
for all nonlinearities f 0 and f 1 fulfilling (61) and

|d f 0 (w)| ≤ C|w|, |d f 1 (w)| ≤ C, ( 66
)
and for all u 0 ∈ Ḃ0 n,1 ∩ Ḃ0 q,1 such that

u 0 Ḃ0 n,1 ∩ Ḃ0 q,1 ≤ c ν , (67)
System (60) admits a unique global solution u in the space

C b (0, ∞; Ḃ0 n,1 ∩ Ḃ0 q,1 ) ∩ L 1 (0, ∞; Ḃ2 n,1 ∩ Ḃ2 q,1 ). (68)
Proof. Granted with Theorem 3, the result mainly relies on embedding, composition and and product estimates in Besov spaces. We focus on the proof of a priori estimates for a global solution u to (60). First, applying Theorem 3 yields u L ∞ (0,∞; Ḃ0 n,1 ∩ Ḃ0 q,1 )∩L 1 (0,∞; Ḃ2 n,1 ∩ Ḃ2 q,1 )

P L ∞ (0,∞;M ( Ḃ0 n,1 ∩ Ḃ0 q,1 )) u L 1 (0,∞; Ḃ2 n,1 ∩ Ḃ2 q,1 )

+ u 0 Ḃ0 n,1 ∩ Ḃ0 q,1 + f 0 (u) L 1 (0,∞; Ḃ0 n,1 ∩ Ḃ0 q,1 ) + f 1 (u) • ∇u L 1 (0,∞; Ḃ0 n,1 ∩ Ḃ0 q,1 ) . (69)

Bounding the last two terms follows from Propositions 1 and 6. More precisely, for p = q, n, we have , Therefore we have, f 0 L 1 (0,∞; Ḃ0 + q,1 ) u 2 L 1 (0,∞; Ḃ2 q,1 ∩ Ḃ2 n,1 )∩L ∞ (0,∞; Ḃ0 q,1 ∩ Ḃ0 n,1 ) .

f 1 (u) • ∇u Ḃ0 p,
(73)

On the other hand, using the fact that | f 0 (u)| ≤ C|u| 2 and Hölder inequality, we may write f 0 (u) L 1 (0,∞;L q -) ≤ u L ∞ (0,∞;L q ) u L 1 (0,∞;L ∞ -) .

We obviously have Ḃ0 q,1 ֒→ L q and, because 2n/q < 0,

Ḃ2 q,1 ∩ Ḃ2 n,1 ֒→ L ∞ -.
Therefore f 0 (u) L 1 (0,∞;L q -) u L ∞ (0,∞; Ḃ0 q,1 ) u L 1 ( Ḃ2

q,1 ∩ Ḃ2 n,1 ) . ( 74
)
So putting (73) and (74) together and taking advantage of (72), we end up with

f 0 (u) L 1 (0,∞; Ḃ0 q,1 )
u 2

L ∞ (0,∞; Ḃ0 n,1 ∩ Ḃ0 q,1 )∩L 1 (0,∞; Ḃ2 n,1 ∩ Ḃ2 q,1 ) .

(75)

It is now time to plug (70), ( 71) and ( 75) in (69). We get u L ∞ (0,∞; Ḃ0 n,1 ∩ Ḃ0 q,1 )∩L 1 (0,∞; Ḃ2 n,1 ∩ Ḃ2 q,1 )

≤ C( P L ∞ (0,∞;M ( Ḃ0

n,1 ∩ Ḃ0 q,1 )) u L 1 (0,∞; Ḃ2 n,1 ∩ Ḃ2 q,1 ) + u 2 L ∞ (0,∞; Ḃ0 n,1 ∩ Ḃ0 q,1 )∩L 1 (0,∞; Ḃ2 n,1 ∩ Ḃ2 q,1 ) + u 0 Ḃ0 n,1 ∩ Ḃ0 q,1
). (76)

Obviously, the above estimate enables us to get a global-in-time control of the solution in the desired functional space whenever (65) and (67) are satisfied. Starting from this observation and using the existence part of Theorem 3, it is easy to prove Theorem 6 by means of Banach fixed point theorem as in [START_REF] Danchin | A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space[END_REF] for instance. The details are left to the reader. Theorem 7. Assume that P ≡ 0 and that f 1 ≡ 0. Suppose that f 0 satisfies (61) and |d f 0 (w)| ≤ C(|w| m-1 + |w|) for some m ≥ 2.

Let 1 < q < n 2 and q ≤ p < ∞. Assume that

s p := n p - 2 m -1 ∈ 0, 1 p and 0 < s q < 1 q - 2 n •
Then there exists a constant c ν such that if

u 0 Ḃsp p,1 ∩ Ḃsq q,1 ≤ c ν ( 77
)
then System (60) admits a unique global-in-time solution u such that

The smallness of the data in (69) enables to close the estimate for the left-hand side of the above inequality. The existence issue is just a consequence of Banach fixed point theorem. This completes the proof of the theorem.

Remark 5. Even though System (60) does not have any scaling invariance in general, our two statements are somewhat critical from the regularity point of view. Indeed, in the functional framework used in Theorem 6 and under the growth condition (66), the nonlinearity f 0 (u) is lower order compared to f 1 (u) • ∇u. Now, we notice that if f 0 ≡ 0 and P ≡ 0 then the initial value problem for System (60) (in the R n case) is invariant for all λ > 0 under the transform:

(u(t, x), u 0 (x)) -→ λ (u(λ 2 t, λ x), u 0 (λ x)).

At the same time, the norm • Ḃ0 n,1 (R n ) is invariant by the above rescaling for u 0 . As regards Theorem 7, the nonlinearity f 0 (w) is at most of order m. Now, if (the coefficients of) f 0 (w) are homogeneous polynomials of degree m then the system is invariant by (u(t, x), u 0 (x)) -→ λ 2 m-1 (u(λ 2 t, λ x), u 0 (λ x)).

Hence the regularity Ḃs p p,1 is critical.

Here we just consider the case q < ∞ to shorten the presentation.

Below M (X) denotes the multiplier space associated to the Banach space X, that is the set of those functions f such that f g ∈ X whenever g is in X endowed with the norm f M (X) := inf g f g X where the infimum is taken over all g ∈ X with norm 1.

Acknowledgements The second author has been supported by the MN grant IdP2011 000661.

We are now ready to prove Theorem 3. Granted with Theorem 4, it is enough to show that u L 1 (0,T ; Ḃs p,1 (K)∩ Ḃs ′ q,1 (K)) may be bounded by the right-hand side of [START_REF] Triebel | Theory of function spaces[END_REF]. As a matter of fact u L 1 (0,T ; Ḃs ′ q,1 (K)) may be directly bounded from Lemma 2, and the same holds for u L 1 (0,T ; Ḃs p,1 (K)) if p < n/2. If p ≥ n/2, then we use the fact so that Ḃs+2

q,1 (Ω ) ⊂ Ḃs q * ,1 (Ω ) with

Therefore, if q < n/2 ≤ p < q * then one may combine interpolation and Lemma 2 so as to absorb u L 1 (0,T ; Ḃs p,1 (K)) by the left-hand side of [START_REF] Triebel | Theory of function spaces[END_REF], changing the constant C if necessary.

If p ≥ q * then one may repeat the argument again and again until the all possible values of p in (n/2, ∞) are exhausted. Theorem 3 is proved.

The bounded domain case

We end this section with a few remarks concerning the case where Ω is a bounded domain of R n with n ≥ 2. Then the proof of Theorem 4 is similar : we still have to introduce some suitable resolution of unity (η ℓ ) 0≤ℓ≤L . The only difference is that, now, η 0 has compact support. Hence Theorem 4 holds true with K = Ω . In order to remove the time dependency in the estimates, we use the fact (see e.g. [START_REF] Friedman | Partial differential equations[END_REF]) that the solution η to (38) satisfies for some c > 0,

which also implies that

Hence we have for any 1 < p < ∞ and -1

Defining u 1 and u 2 as in (35), one may thus write

. Of course, we also have whence, using elementary interpolation,

Bounding f 0 (u) is slightly more involved. To handle the norm in L 1 (0, ∞; Ḃ0 n,1 (Ω )), we use the following critical embedding:

Hence Proposition 1 enables us to write that

,

.

The last inequality stems from the embedding Ḃ1 n,1 ֒→ L ∞ and from the fact that q < n/2 < n, whence Ḃ1 q,1 ∩ Ḃ1 n,1 ֒→ Ḃ1 n/2,1 . Therefore, using Hölder inequality and elementary interpolation, we deduce that

Finally we have to bound f 0 (u) in L 1 (0, ∞; Ḃ0 q,1 ). For that it suffices to estimate it in L 1 (0, ∞; Ḃ0 + q,1 ) and in L 1 (0, ∞; L q -). Indeed we observe that L q -֒→ Ḃ0 - q,∞ , and thus

Now, on the one hand, according to Proposition 6 and Hölder inequality we have

.

By interpolation, we easily get

and because q < n/2,

Proof. Once again, we start from Theorem 3 which implies the following inequality:

Now (a slight generalization of) Proposition 6 ensures that for s = s p , s q and for r = p, q, f 0 (u) Ḃs r,1

.

Therefore,

Hence it is only a matter of proving that the norm of u in L 1 (0, ∞; L ∞ ) and in L m-1 (0, ∞; L ∞ ) may be bounded by means of the norm in

). Now, we notice that Ḃs p +2/(m-1)

embeds continuously in L ∞ and that, by interpolation,

.

Hence we do have

.

Finally, we notice that Ḃ2 s q +2,1 ֒→ Ḃ2+s q -n/q ∞,1

and that 2 + s qn/q < 0. At the same time Ḃs p +2 p,1 ֒→ Ḃ1 ∞,1 , therefore

Hence we have u L 1 (0,∞;L ∞ ) u L 1 (0,∞; Ḃsp+2 p,1 )∩L 1 (0,∞; Ḃsq+2 q,1 )

.

Putting ( 81) and (82) into (80) and then into (79) we get u L ∞ (0,∞; Ḃsp p,1 ∩ Ḃsq q,1 )∩L 1 (0,∞; Ḃ2+sp p,1 ∩ Ḃ2+sq q,1 )

.


Romuald Lepers
email: romualdlepers<rlepers@u-bourgogne.fr

Paul Stapley

Relative in performance with age: evidence from 25 years of Hawaii Ironman racing

Keywords: Master athletes, Endurance, Gender differences. Triathlon, Swimming, Cycling, Running, Aging. '

Despite of the growth of ultra-endurance sports events (of duration > 6h) over the previous few decades, the age-related declines in ultra-endurance performance have drawn little attention. The aim of the study was to analyze the changes in participation and performance trends of older (> 40 yrs of age) triathletes between 1986 and 2010 at the Hawaii Ironman triathlon consisting of 3.8 km swimming, 180 km cycling and 42 km running. Swimming, cycling, running and total times of the best male and female triathletes between 18 and 69 yrs of age who competed in the Hawaii Ironman triathlon were analyzed. The relative participation of master triathletes increased during the 1986 2010 period, while the participation of triathletes younger than 40 yrs of age decreased. Linear regression showed that males older than 44 yrs and females older than 40 yrs significantly improved their performances in the three disciplines and in the total time taken to complete the race. Gender differences in total time performance significantly decreased in the same time period for all age groups between the 40 44 yrs and 55 59 yrs ones. The reasons for these relative improvements of --Ironman athlete performances in older age groups remain, however, unknown. Further studies investigating training regimes, competition experience or socio-demographic factors are needed to gain better insights into the phenomenon of increasing participation and improvement of ultra-endurance performance with advancing age.

Introduction

Anecdotal reports show that some older athletes nowadays can achieve very high levels of ultra-endurance performance (defined as performance exceeding 6 h, Zaryski & Smith 2005). For example, an 81-year-old male athlete was able to finish a 100-km ultra-marathon in a time of 19 h 44 min [START_REF] Knechtle | Change of body composition in an 81 year old runner in a 100 km run[END_REF]). In the field of multi-sport events, an 80 year old male triathlete also finished the 2010 Hawaii Ironman Triathlon consisting of 3.8 km swimming, 180 km cycling and 42 km running, in less than 16 h ( ). That same year, and at the same event, a 75 year old female triathlete finished the race in http://ironmanworldchampionship.com/results/ 16h 20min. These unique older athletes have begun to push the limits of the interaction between aging and human endurance. They represent a positive example of healthy aging and provide unique insights into a person s ability to maintain physical performance and ' physiological function with advancing age [START_REF] Tanaka | Endurance exercise performance in Masters athletes: age-associated changes and underlying physiological mechanisms[END_REF].

Age-related declines in endurance performance have been well described in the literature for running [START_REF] Trappe | Marathon runners. How do they age ?[END_REF][START_REF] Leyk | Performance, training and lifestyle parameters of marathon runners aged 20 80 years: results of the PACE-study[END_REF][START_REF] Lepers | Do older athletes reach limits in their performance during marathon running?[END_REF], swimming [START_REF] Tanaka | Endurance exercise performance in Masters athletes: age-associated changes and underlying physiological mechanisms[END_REF] and triathlon [START_REF] Bernard | Age related decline in Olympic triathlon performance: effect of locomotion mode[END_REF][START_REF] Baker | Aging performance for masters records in athletics, swimming, rowing, cycling, triathlon, and weightlifting[END_REF]. Changes in performance across the ages have also been examined for ultra-endurance events, including running events of 50-km to 161-km [START_REF] Hoffman | The Western States 100-Mile Endurance Run: participation and performance trends[END_REF][START_REF] Hoffman | Historical Analysis of Participation in 161 km Ultramarathons in North America[END_REF][START_REF] Knechtle | Age-related changes in 100-km ultra-marathon running performances[END_REF] and Ironman distance triathlons [START_REF] Lepers | Age-related changes in triathlon performances[END_REF][START_REF] Lepers | Age and gender interactions in ultra-endurance performance: insight from triathlon[END_REF].

Endurance and ultra-endurance performance, in terms of the overall time taken, appears to be maintained until approximately 35 40 years of age, followed by modest decreases until 50 years of age, with a progressive decrease in performance thereafter with the greatest declines occurring after the age of 70 years. Physiological factors, both central (maximum heart rate; maximum stroke volume; blood volume) and peripheral (muscle mass; muscle fiber composition, size and capillarization; muscle enzyme activity), contribute to age-related declines in endurance performance in older athletes. However, these physiological factors can be regulated by changing the intensity and volume of the older endurance athletes training [START_REF] Reaburn | Endurance performance in masters athletes[END_REF][START_REF] Tanaka | Endurance exercise performance in Masters athletes: age-associated changes and underlying physiological mechanisms[END_REF] and there is obviously an interaction between last three decades [START_REF] Jokl | Masters s performance in the New York City marathon 1983 1999[END_REF][START_REF] Lepers | Do older athletes reach limits in their performance during marathon running?[END_REF]. To our knowledge, only two studies have focussed upon the ultra-endurance participation and performance trends of masters athletes. An increase in participation of both male and female athletes older than 40 yrs has been reported for 161-km ultra-marathons in North America over the past 30 years [START_REF] Hoffman | Historical Analysis of Participation in 161 km Ultramarathons in North America[END_REF]) and for 100-km running in Switzerland [START_REF] Knechtle | Age-related changes in 100-km ultra-marathon running performances[END_REF] during the last ten years, but no data are available for a multi-sport ultra-endurance event, such the Ironman triathlon. An increased participation of masters athletes in such ultra-endurance events may inevitably be linked to an increase in training volume, and could be interpreted as a possible strategy for maintaining good health during old age; these older athletes being conscious that regular physical activity may increase life expectancy.

Ironman triathlon represents an intriguing model to analyse age-related declines in ultra-endurance performance. Indeed, the age-related declines in performance can be analyzed in the same subjects collectively, and also for three modes of locomotion (swimming, cycling and running) separately [START_REF] Bernard | Age related decline in Olympic triathlon performance: effect of locomotion mode[END_REF], Lepers et al. 2010[START_REF] Lepers | Age and gender interactions in ultra-endurance performance: insight from triathlon[END_REF]. It has been shown for example, that age-related declines in cycling performance were less pronounced compared to running, presumably due to the non-weight bearing aspect and the concentric type of muscle action during cycling. According to previous investigations on the participation trends conducted on ultra-endurance running events [START_REF] Hoffman | Historical Analysis of Participation in 161 km Ultramarathons in North America[END_REF][START_REF] Knechtle | Age-related changes in 100-km ultra-marathon running performances[END_REF], we would also expect an increase in the participation of masters triathletes in an ultra-endurance competitions such Ironman triathlon since the emergence of this sport in the 1980 s [START_REF] Lepers | Analysis of Hawaii ironman performances in elite triathletes from 1981 to 2007[END_REF].

'

In accordance with this expected increase in participation, it may also be interesting to see if older Ironman triathletes have shown improvements in their swimming, cycling and running performance over the past 3 decades.

A previous study conducted on the Hawaii Ironman triathlon showed that gender differences in total performance time were stable until 55 yrs of age but increased thereafter [START_REF] Lepers | Age and gender interactions in ultra-endurance performance: insight from triathlon[END_REF]. However, this study focused only on a recent three-year period (2006 2008) and did not examine the possible differences between the genders with age since the beginning of the event in the early 1980

- '
s. If the changes in age-related decline in performance differed between males and females over the past three decades, we would also expect some change in the difference between gender performances across the ages.

Previous studies have focussed mainly upon declines in overall performance with age by comparing older age group performances with younger ones, but have not really investigated if performances within each older age group have increased relatively over the years.

To date, no data exist regarding the participation and performance trends of masters athletes in a multi-sport ultra-endurance event such as the Ironman triathlon in the last 25 years. Accordingly, the first purpose of this large cross-sectional study was to examine the changes in

Methods

Approval for this study was obtained from the Burgundy University Committee on Human Research. It involved the analysis of publicly available data so content was waived. Age and time performance data for all triathletes completing the Hawaii Ironman Triathlon from 1986 to 2010 were obtained through the Hawaii Ironman Triathlon web site:

. In the http://ironmanworldchampionship.com/results/ present paper, we focus our attention on the Hawaii Ironman Triathlon, which has become the Ironman World Championship i.e. the premier race in the field of long distance triathlon [START_REF] Lepers | Analysis of Hawaii ironman performances in elite triathletes from 1981 to 2007[END_REF]. In order to participate at the Hawaii Ironman Triathlon, all triathletes must earn a qualifying slot at one of the qualifying Ironman triathlons held worldwide during the previous year (http://ironman.com/events

). The number of slot allocations for each age-group at qualifying events is distributed according to the number of age group starters in each category. For example, if 5 of the age group starters are female 40 44 yrs, then 5 of the total slots are allocated to the best female

% - %
of the 40 44 yrs category. The number of total slots attributed depends on the qualifying events and ranges from 50 to 100. This mode of qualification makes the Hawaii Ironman Triathlon the most competitive long distance triathlon in the world.

Although there is no consensus in the literature about the definition of a masters athlete, in the present study we defined masters triathletes as those equal to or older than 40 yrs. For male and female athletes younger than 40 yrs including elite and non-elite triathletes, we considered only one age group 18 39 yrs. To focus more on changes in participation and performance in masters triathletes, we To analyze the changes in participation across the years, we quantified the number of female and male finishers in each age-group category annually from 1986 to 2010. We considered only the age groups with at least 10 finishers per annum for the analysis of performances. Averaged swimming, cycling, running and total time performances of the top 10 finishers of each age group for both females and males were analyzed from 1986 to 2010. Taking the top 10 performers in the master categories gave a better comparison with higher performing younger athletes.

In order to limit the influence of environmental conditions and small changes in the courses in the comparison of performance, the swimming, cycling, running and overall time performances of each triathlete finishing in the top 10 of each age group were divided by the mean time performance of the top 10 overall finishers. We considered the mean performance of the top 10 overall finishers (whose age was always comprised between 18 and 39 years old) as the reference for calculating the relative performance ratio for the master age-groups. Thus, the age-related declines in performance were expressed using a performance ratio calculated between the individual and the mean time performances of the top 10 overall finishers for each mode of locomotion and total event time [START_REF] Lepers | Age-related changes in triathlon performances[END_REF]. This normalization procedure assisted the comparison of age-related declines in performance across the years for both males and females.

The magnitude of gender differences in total time performance of each age group was examined by calculating the difference in total times between the top 10 males versus females as a percentage of the average time for the top 10 males for each year from 1986 to 2010.

Statistical analysis

Data is reported as means SD in the text. Linear regressions analyses were used to estimate changes in the selected variables in each ± year of the race. Pearson s correlation coefficients were used to assess the association between various variables using Statsoft (Version ' 6.1, Statistica, Tulsa, OK, USA). Linear regression analyses were used to assess changes in the percentage of finishers in the different age-groups for both sexes across the years, the mean performance ratios for swimming, cycling, running and total event times, for males and females within the different age-groups, and gender differences in total time performances. A significance level of P < 0.05 was used to identify statistical significance.

Results

Participation trends

From 1986 to 2010, there were a total of 35,293 finishers (7,825 females and 27,468 males) at the Hawaii Ironman triathlon. The number of finishers during the studied period is shown in Figure 1. During these years, the average number of finishers per year was 1,099 93 range: 950 1,299 for males and 313 90 range: 218 469 for females. Females accounted on average for 22 3 range: 18 28

± [ - ] ± [ - ] ± % [ -%]
of the field over the 25-year period.

In 1986, masters triathletes (i.e. > 40 yrs) represented 31 of the total field for the males and 23 of the total field for females, while % % in 2010 master triathletes represented 56 of the total field for males and 47 of the total field for females. The changes in the percentage

% %
of finishers in different age-groups for both sexes are presented in Figure 2. For both females and males, there was a significant rise in relative participation among age-groups 40 yrs. In contrast, relative participation significantly decreased among both males and females ≥ < 40 yrs, especially for the youngest age-groups 18 29 years.

-

Performances trends

Total time records and corresponding split times for male and female age-groups at the Hawaii Ironman Triathlon are presented in Table 1. The results clearly show that the best master age-group performances have been set during the last decade, and eight of them have been set in 2010. An exceptional performance can be seen in the male 40 44 yrs total time record that was been set by the 6 times Hawaii -Ironman triathlon winner, Dave Scott, who returned to competition in 1994 at 40 years of age, after several years break from the race.

Figure 3 shows the trends in swimming, cycling, running and total performance ratios for the top 10 male finishers subdivided into the different age groups across the years 1986 through 2010. There were no significant changes across years within age group 40 44 yrs, for any mode of locomotion and total time. In contrast, the average performance ratios improved between 1986 and 2010 in age groups 45 49 yrs and older.

Figure 4 shows the trends in performance ratios for the top 10 female finishers subdivided into the different age groups across the studied period. The average performance ratios for each mode of locomotion and total time improved between 1986 and 2010 in all age groups from 40 44 yrs to 60 64 yrs. The r , P and slopes values of the linear regressions corresponding to the Figure 3 and 4 are presented --2 in the Table 2.

Gender differences trends

The changes across the years in total time gender differences between the top 10 female and top 10 male subdivided by different age groups are presented in Figure 5. There was no significant change in gender difference across the years within the age group 18 39 yrs (r -2 0.09, P 0.13). In contrast, the gender difference significantly (P < 0.001) decreased between 1986 and 2010 in age groups comprised

= =
between 40 44 yrs and 55 59 yrs. Surprisingly, the gender difference for age group 60 64 yrs did not significantly change between 1999 ---ultramarathon running [START_REF] Jokl | Masters s performance in the New York City marathon 1983 1999[END_REF][START_REF] Lepers | Do older athletes reach limits in their performance during marathon running?[END_REF][START_REF] Hoffman | Historical Analysis of Participation in 161 km Ultramarathons in North America[END_REF]. The growth of female and masters age groups in worldwide marathon and ultra-marathon racing has been particularly startling over the last 20 years [START_REF] Burfoot | The history of the marathon[END_REF], Knecthle et al. 2011).

Our data suggest that similar trends have occurred in Ironman triathlon racing over the last 10 years and we can expect that the participation of female master will continue to increase in the future to soon reach half of the total female field.

Different factors could be used to explain the increase in participation of masters Ironman triathletes. With the increase in life expectancy and training facilities, such as specific masters programs, the increased participation of older athletes described may be a reflection that these athletes, who are perhaps retired, have more available time and resources at their disposal to train and therefore to compete. The relative increase in participation of older Ironman triathletes could result from a relative decrease in Ironman triathletes in the younger age groups (18 39 years). Younger triathletes maybe more and more attracted by shorter distances such as short distance and naturally move towards longer distances where endurance capacities play a greater role in performance than maximal oxygen consumption. It has been suggested that enjoyment, health, fitness benefits, and social factors are the primary drivers for sport involvement with advancing age [START_REF] Shaw | Motivation and psychological skills in the senior athlete[END_REF]. For masters triathletes that are more achievement-oriented, there is no doubt that competition, personal challenge and skill development may also be strong reasons for their participation in ultra-endurance events. However, among older triathletes, the distinction between people who started training in triathlon later in life (i.e. after 40 yrs age) and people who participated in triathlon at a younger age is not known.

Performances of masters triathletes

The present results show that swimming, cycling, running and total performance times decreased over the 25-year periods studied for male triathletes older than 44 yrs age and female triathletes older than 40 yrs age. Several reasons may explain the improved performance of masters triathletes, such as a greater participation of the older age groups, which would increase the probability of finding better triathletes in these age groups, increased training facilities for older people, and increases in the competitive spirit in these older groups.

These last years, the age-related declines in total Ironman triathlon performance with advancing age until 70 yrs of age was about 12-13 per decade for males and 14 15 per decade for females, respectively. These values are in accordance with those found by Lepers & % -% Cattagni ( 2011) for the New-York marathon running, but appear greater compared to those found in 100-km ultra-marathon running (7 8

-% per decade for males and 9 10 per decade for females), by [START_REF] Knechtle | Age-related changes in 100-km ultra-marathon running performances[END_REF]. These findings may be explained by the shorter -% duration of 100-km running compared to Ironman triathlon, and the different level of competitiveness between the two events: regional one for the 100-km running World Championship for Ironman triathlon. Our data also confirm previous observations (Bernard et al. vs. 2010, Lepers et al. 2010) showing a lesser age-related decline in cycling performance compared with running and swimming performances in triathlon. Physiological (e.g. concentric muscle contractions) and mechanical (e.g. relation between mechanical power output and velocity, non-weight bearing aspect) specificities of cycling compared with the two other disciplines may be explained by the lower age-related decline in cycling performance [START_REF] Bernard | Age related decline in Olympic triathlon performance: effect of locomotion mode[END_REF], Lepers et al. 2010). In the context of health promotion, this finding suggests that cycling may represent an optimal activity with advancing age.

The question of whether the masters triathletes have reached their limits in Ironman triathlon performance can therefore be raised, at least for the age-group categories studied (i.e. between 40 69 yrs for males and 40 64 yrs for females). For marathon running, it has been -recently suggested that male (> 65 years) and female (> 45 years) master runners should still improve their performances in the future [START_REF] Lepers | Do older athletes reach limits in their performance during marathon running?[END_REF]. All these data suggest that older athletes have probably not yet reached their limits in endurance and ultra-endurance performance. The new generation of well-trained masters triathletes will move progressively into the older age group competitions. In addition to better performances for older triathletes, an increase in density of the top 10 spreads of each age group may also be expected in the future. Moreover, the expected arrival of previous elite triathletes to the youngest masters age-group category i.e.

40 44 yrs (The Hawaii Ironman triathlon male winner was 37 years old in 2010) in the near future will result in a need to redefine the notion of masters triathletes, regarding the decline in their performance.

Gender differences in total time

The current Hawaii Ironman triathlon record is 8:54:202 for females and 8:04:08 for males (see Table 2), corresponding to a gender gap of 10.3 . The present results showed that for masters triathletes, the gender differences in Ironman triathlon performance times % decreased over the last 25 years, suggesting that masters females have reduced the gap to males. During these last years, gender differences in total event performance times remained slightly greater for age groups > 54 yrs compared to youngest age groups. This finding corroborates the results of [START_REF] Lepers | Age and gender interactions in ultra-endurance performance: insight from triathlon[END_REF] who showed that gender differences in Ironman triathlon performance times analyzed from 2006 to 2008 increased with advancing age from 55 yrs. In contrast, it has been shown that gender difference in New-York marathon running performances did not significantly change across this age group over the past decade [START_REF] Lepers | Do older athletes reach limits in their performance during marathon running?[END_REF]. Ironman triathlon is a newer ultra-endurance event with less participants compared to the NYC Marathon, which may explain why gender differences across the ages groups in Ironman triathlon are currently similar to those that were observed for marathon 30 yrs ago.

There is no doubt that gender differences for older Ironman triathletes would still diminish in the future as it has been observed for marathon running because there is no real evidence that the age-related declines in physiological function differ between males and females. Age-related changes in the physiological determinants of endurance and ultra-endurance performance (e.g. maximal oxygen uptake, lactate threshold, and running economy) between males and females have not received considerable attention. For example, [START_REF] Holloszy | Exercise . Bethesda[END_REF] suggested that the relative rates of decline in maximal oxygen consumption with age are similar between the sexes. However, further investigations are required in order to analyse the age-related declines in other physiological parameters such lactate threshold and running economy in males and females.

Practical applications

Most healthcare providers would agree that racing an Ironman triathlon after 50 years of age is not good for the body. But most older Ironman triathletes will say that training for an Ironman triathlon is one of the healthiest things they can do [START_REF] Wortley | Training principles and issues for ultra-endurance athletes[END_REF]. Where should the limits be set? If masters athletes performed at such a high-level for so long, it is reasonable to expect that those destined to maintain that intensity could do so because they remained largely injury-free [START_REF] Knobloch | Acute and overuse injuries correlated to hours of training in master running athletes[END_REF]. The exponential increase in participation of masters athletes especially in the female field in ultra-endurance sporting events such as Ironman triathlon represents an immense opportunity to learn how the aging process is related to physical performance. However, the remarkable growth in ultra-endurance older athletes and varying levels of fitness also represent unique challenges to sports medicine professionals because these individuals are particularly exposed to overuse injuries [START_REF] Egermann | Analysis of injuries in long-distance triathletes[END_REF]. A framework for pre-participation evaluation, training program and injury prevention is required to help older athletes reach their participation and performance goals injury free i.e. to maximise the benefits and minimize the risks [START_REF] Marr | Care of the multisport athlete: lessons from Goldilocks[END_REF]. We have much to learn from these ultra-endurance masters athletes that can pursue their passion with greater health and safety once we gain a greater understanding of them [START_REF] Wortley | Training principles and issues for ultra-endurance athletes[END_REF].

Conclusion

The participation of older triathletes at the World Ironman triathlon championship has increased over the 1986 2010 period and it will probably still grow in the future in Hawaii and other Ironman triathlons worldwide. During that period master triathletes have shown relative improvements of their performances in the three disciplines and in total event times. The gender differences in Ironman triathlon performance have progressively decreased over the last 25 years for older age-groups suggesting that female masters triathletes have reduced the gap to males. The question whether older triathletes have yet reached limits in their performance during Ironman triathlon should therefore be raised. Further studies investigating training regimes, competition experience or socio-demographic factors are needed to gain better insights into the phenomenon of the relative improvement in ultra-endurance performance with advancing age.

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  considered age groups 5 years in length for males and females older than 40, as follows: 40 44 yrs, 45 49 yrs, 50 54 yrs, 55 59 yrs, 65 69 yrs. Due to the very small number of finishers in the age-groups 70 74 yrs and 75 79 yrs, we did not consider these age --groups.

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  half-Ironman distance triathlons. Indeed, since 2000 short distance triathlon (1.5km swim, 40 km cycle and 40 run) became an Olympic sport and it may have increased the popularity of the short distance especially in the young triathlete field. With advancing age, athletes

Fig. 1

 1 Fig. 1 Total number of finishers at the Hawaii Ironman Triathlon and number of finishers of each sex from 1986 to 2010.

Fig. 2

 2 Fig. 2Linear regressions defining the percentages of finishers within different age groups across the period of 1986 through 2010, for females (Panel A) and for males (Panel B). All the slopes of the linear regressions were significantly different (P < 0.01) from zero.

Fig. 3

 3 Fig. 3Changes in performance ratios for swimming, cycling, running and total times across the years for top 10 finishers in each male age group. A performance ratio equal to 1 corresponded to the performance of the top 10 finishers of the 18 39 year age group. Solid lines represent conditions where the slopes of the linear regressions were significantly different (P < 0.01) from zero. Dashed lines indicate the slopes of the linear regressions were not significantly different from zero. Lines cross only the range of years for which data were considered.

Fig. 4

 4 Fig. 4

Fig. 5

 5 Fig. 5 Linear regressions defining gender differences in total times within different age groups across the period of 1986 through 2010. Solid lines represent conditions where the slopes of the linear regressions were significantly different (P < 0.01) from zero. Dashed lines indicate the slopes of the linear regressions were not significantly different from zero. Lines traverse only the range of years for which data were considered.

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Acknowledgements:

No funding has been received for this work from the NIH, Wellcome Trust, HHMI or others.

4 11 and 2010 (r 0.06, P 0.42). In 2010, the gender difference in total time was 11.8 , 11.9 , 11.3 , 14.4 , 12.7 and 17.8

Discussion

The main findings of the present study were first that the participation of masters triathletes at the Hawaii Ironman triathlon increased during the 1986 2010 period. Second, during that period, female triathletes older than 40 years and male triathletes older than 44 years significantly improved their swimming, cycling, running and total time performances in the most recent years, relative to performances recorded at the beginning of the time period studied. Third, gender differences in total Ironman triathlon time decreased over the last 25 years for masters age-groups while it remained stable for the young age-group, 18 39 yrs.

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Increase in participation of masters Ironman triathletes

Despite the rapid growth of triathlon since the early 1980 s (The first Hawaii Ironman triathlon took place in 1978), the number of total ' male and female finishers has slightly increased (~34 ) over the last 25 years. This could be explained by the fact that, for safety reasons, % the race organisation of the Hawaii Ironman has imposed limits on the number of entries accepted since 1985, by creating a qualification system that requires all triathletes wanting to participate to earn a qualifying slot at one of the qualifying Ironman triathlons held worldwide (except for some limited slots offered by lottery). In 2010, it was estimated that around 50,000 triathletes had finished an Ironman distance triathlon in the world, either at one of the 23 qualifying races ( ) or at another non-qualifying http://ironman.com/events/ event (

). Because the number of slot allocations for each age-group at qualifying events is distributed according to http://www.k226.com/ the relative number of age group starters in each category, the percent of finishers in each age group at the Hawaii Ironman triathlon represents a good picture of the age and gender of Ironman triathletes in the world.

Independently of the age, the female field has increased from 19 to 27 of total finishers over the past 25 years. Female % % participation at such an ultra-endurance event remains slightly lower compared to traditional endurance events such as marathon running, but greater than ultra-endurance events such as a 161-km ultramarathon. Indeed, [START_REF] Lepers | Do older athletes reach limits in their performance during marathon running?[END_REF] showed that the relative participation of females at the New-York City Marathon increased over the last 30 years from 17 to 33 of the total field. In contrast,

females accounted for 20 of the finishers in the same time period at 161-km ultramarathons in North America [START_REF] Hoffman | Historical Analysis of Participation in 161 km Ultramarathons in North America[END_REF]).

%

In addition, the relative proportion of masters triathletes in the total field has also increased for both males and females over the last 25 years while those of male triathletes younger than 40 yrs of age decreased during the same period. In 2010, master triathletes represented 56 of male finishers and 47 of female finishers, respectively. This finding corroborates previous observations for marathon and % %

Footnotes:

The authors have no conflict of interests to report.