N Forcadel

C Imbert

R Monneau

Steady state and long time convergence of spirals moving by forced mean curvature motion

Keywords: AMS Classification: 35K55, 35K65, 35A05, 35D40 spirals, steady state, mean curvature motion, Liouville theorem, long time convergence, motion of interfaces, viscosity solutions

In this paper, we prove the existence and uniqueness of a "steady" spiral moving with forced mean curvature motion. This spiral has a stationary shape and rotates with constant angular velocity. Under appropriate conditions on the initial data, we also show the long time convergence (up to some subsequence in time) of the solution of the Cauchy problem to the steady state. This result is based on a new Liouville result which is of independent interest.

Introduction

In this paper we are interested in curves in R 2 which are half-lines attached at the origin. These lines are assumed to move with normal velocity (1.1) V n = 1 + κ where κ is the curvature of the line. We assume that these curves Γ t can be parametrized in polar coordinates as follows Γ t = {(r cos θ, r sin θ), such that r ≥ 0, θ = -U (t, r)}.

On the one hand, the Geometric Law (1.1) holds true if U satisfies

U t = (1 + κ U )|∇U |.
On the other hand, it is known (see for instance [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF]) that the curvature of the parametrized curve Γ t has the following form (1.2) κ U (t, r) = U r 2 + (rU r ) 2

(1 + (rU r ) 2 )

3 2 + rU rr (1 + (rU r ) 2 ) 3 2
.

Hence, the function U has to satisfy the following quasi-linear parabolic equation in nondivergence form for (t, r) ∈ (0, +∞) × (0, +∞):

(1.3)

rU t = 1 + r 2 U 2 r + U r 2 + r 2 U 2 r 1 + r 2 U 2 r + rU rr 1 + r 2 U 2 r
supplemented with the following initial condition for r ∈ (0, +∞) (1.4) U (0, r) = U 0 (r).

Main results

In [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF], we were able to prove an existence and uniqueness result for equation (1.3)- (1.4). We improve it by proving in particular that solutions are regular up to the boundary r = 0.

Theorem 1.1 (Existence and uniqueness for the Cauchy problem). Assume that U 0 ∈ W 2,∞ loc (0, +∞) is globally Lipschitz continuous and satisfies

(U 0 ) r ∈ W 1,∞ (0, +∞) or κ U 0 ∈ L ∞ (0, +∞)
and that there exists a radius r 0 > 0 such that

|1 + κ U 0 | ≤ Cr for 0 ≤ r ≤ r 0 .
Then there exists a globally Lipschitz continuous (in space and time) solution U such that

U ∈ C 1+ 1 6 ,2+ 1 3 t,r
((0, +∞) × [0, +∞)) ∩ C ∞ ((0, +∞) × (0, +∞)).

Moreover, for every δ > 0, R > 0, there exists a constant C = C(δ, R) such that for every T ≥ δ > 0, U -U (T, 0)

C 1+ 1 6 ,2+ 1 3 t,r ([T,T +δ]×[0,R]) ≤ C.
Such a solution is unique in the class of continuous viscosity solutions of (1.3)- (1.4).

Remark 1.2. In view of (1.2) and (1.3), the regularity of U stated in the previous theorem implies in particular that (1.5) κ U + 1 = 0 at r = 0 holds for t > 0.

Remark 1.3. The assumption that U 0 is globally Lipschitz was missing in the statement of Theorem 1.7 in [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF]. We will recall below (see Theorem 2.1) the corrected version of this result.

Our second main result is about the existence of a spiral with stationary shape and rotating at constant speed.

Theorem 1.4 (A steady state). There exists a constant λ ∈ R and a globally Lipschitz function continuous Φ in [0, +∞), satisfying λ ≥ 0 and Φ r ≤ 0 on [0, +∞) such that U (t, r) = λt + Φ(r) is a solution of (1.3) in R × (0, +∞). Moreover such a λ is unique and such a function Φ is unique up to addition of a constant. Moreover, the following properties hold true:

i) we have 1 4 ≤ λ ≤ 1 2 .
ii) Φ ∈ C ∞ (0, +∞) ∩ C 2+ 1 3 ([0, +∞)) satisfies for all r ∈ [0, +∞)

- 1 2 ≤ Φ r ≤ -λ, (1.6) 0 ≤ 1 + κ Φ ≤ λr. (1.7)
Moreover Φ r and the curvature κ Φ are non-decreasing and

Φ r (0) = - 1 2 , Φ r (+∞) = -λ, κ Φ (0) = -1, κ Φ (+∞) = 0.
iii) There exist some constants a ∈ R and C > 0, such that Φ satisfies for all r ∈ [0, +∞) |Φ(r) + λr + λ ln(1 + r) -a| ≤ C 1 + r .

Remark 1.5. Notice that the value of the angular velocity λ have been estimated to be 0.315 by approximation in [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF], and computed to be 0.330958961 by a shooting method in [START_REF] Ohara | Modelling Crystal Growth Rates from Solution[END_REF].

Our third main result is concerned with the large time behaviour of solutions of the Cauchy problem for initial data that are "reasonably close" to the steady state. where (λ, Φ) are given in Theorem 1.4, then for any sequence t n → +∞, there exists a subsequence (still denoted by t n ) and a constant a ∈ R such that U (t + t n , r) -(λ(t + t n ) + Φ(r)) → a locally uniformly in R × [0, +∞).

Remark 1.7. The fact that convergence only happens along a subsequence of times is expected. Indeed a similar fact happens already for the linear heat equation on the real line. It is possible to cook up an initial data which stays between 0 and 1 such that the solution does not converges as times goes to infinity, but such that convergence to a constant (locally uniformly) still happens for subsequences in time (see in particular [START_REF] Collet | Space-time behaviour in problems of hydrodynamic type: a case study[END_REF]Lemma 8.6]). This happens here because we are working on the whole plane. On the contrary, when we work on the (compact) annulus (like in [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF]), there is a full convergence in time without taking a subsequence in time.

The proof of Theorem 1.6 is based on the following Liouville result of independent interest.

Theorem 1.8 (Liouville result). Let U (t, r) be a globally Lipschitz continuous function (in space and time) in R×[0, +∞). We assume that U is a global solution of (1.3) in R×(0, +∞) and that there exists a constant C > 0 such that the following holds:

(1.10)

|U (t, r) -λt -Φ(r)| ≤ C on R × [0, +∞)
where (λ, Φ) is given by Theorem 1.4. We also assume that there exists some δ > 0 such that

(1.11) U r ≤ -δ < 0 in R × [0, +∞).
Then U (t, r) = λt + Φ(r) + a for some constant a ∈ R.

Review of the literature

Spirals appear in several applications. Our main motivation comes from continuum mechanics. In a two dimensional space, the seminal paper of Burton, Cabrera and Frank [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF] studies the growth of crystals with vapor. When a screw dislocation line reaches the boundary of the material, atoms are adsorbed on the surface in such a way that a spiral is generated; moreover, under appropriate physical assumptions, these authors prove that the geometric law governing the dynamics of the growth of the spiral is precisely given by (1.1). We mention that there is an extensive literature in physics dealing with crystal growth in spiral patterns (see for instance [START_REF] Schulze | A geometric model for coarsening during spiral-mode growth of thin films[END_REF][START_REF] Ranganathan | Spiral Evolution in a Confined Geometry[END_REF]). We also want to point out that motion of spirals appear in other applications like in the modeling of the Belousov-Zhabotinsky reagent [START_REF] Murray | Mathematical biology[END_REF]. To model the appearence of such shapes, the reagent is modeled in [START_REF] Keener | A geometrical theory for spiral waves in excitable media[END_REF] by a system of semi-linear parabolic equations; so-called spiral wave fronts satisfying the geometric law (1.1) can be formally derived. The interested reader is also referred to e.g. [START_REF] Meron | Model for spiral wave formation in excitable media[END_REF][START_REF] Meron | Nonlocal effects in spiral waves[END_REF][START_REF] Ikota | On the structure of steady solutions for the kinematic model of spiral waves in excitable media[END_REF].

There exist different mathematical approaches to describe the motion of spirals. As far as we know, it appeared first in geometry in [START_REF] Altschuler | Singularities of the curve shrinking flow for space curves[END_REF]. It was also used in order to study singularity formation [START_REF] Angenent | On the formation of singularities in the curve shortening flow[END_REF][START_REF] Angenent | Asymptotic shape of cusp singularities in curve shortening[END_REF]. Other approaches have been used; for instance, a phase-field approach was proposed in [START_REF] Karma | Spiral surface growth without desorption[END_REF] and the reader is also referred to [START_REF] Fiedler | Rotating spirals of curvature flows: a center manifold approach[END_REF][START_REF] Ogiwara | Spiral traveling wave solutions of some parabolic equations on annuli[END_REF][START_REF] Ogiwara | Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth[END_REF]. In [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF], spirals moving in (compact) annuli with homogeneous Neumann boundary condition are constructed. From a technical point of view, the classical parabolic theory is used to construct smooth solutions of the associated partial differential equation; in particular, gradient estimates are derived. We point out that in [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF], the geometric law is anisotropic, and is thus more general than (1.1). In [START_REF] Smereka | Spiral crystal growth[END_REF][START_REF] Ohtsuka | A level set method for spiral crystal growth[END_REF][START_REF] Ohtsuka | A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces[END_REF][START_REF] Goto | Uniqueness and existence of generalized motion for spiral crystal growth[END_REF], the geometric flow is studied by using the level-set approach. As in [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF], the authors of [START_REF] Ohtsuka | A level set method for spiral crystal growth[END_REF][START_REF] Ohtsuka | A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces[END_REF] consider spirals that typically move inside a (compact) annulus and reaches the boundary perpendicularly.

Concerning the existence of "steady" spirals (in the case where the exterior stress is zero), we refer to [START_REF] Ishimura | Shape of spirals[END_REF] where the construction is done by studying an ordinary differential equation and to [START_REF] Chern | The self-similar expanding curve for the curvature flow equation[END_REF] where the authors consider a two-point free boundary problem for the curvature flow equation. We also refer to [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF] where they construct a steady state on an annulus using classical parabolic theory. In [START_REF] Ohara | Modelling Crystal Growth Rates from Solution[END_REF], a numerical computation of the angular velocity λ of the spirals is done. The authors find that the angular velocity is approximatively 0, 330958961 (recall that we find that 1 4 ≤ λ ≤ 1 2 ).

Organization of the article

In Section 2, we prove that the solution has a certain smoothness up to the boundary r = 0, namely Theorem 1.1. In Section 3, we construct the steady state, first on an annulus and then on the whole space. In Section 4, we prove some asymptotics of any profile, and then deduce the uniqueness of the profile (and of its angular velocity λ) as a consequence of the asymptotics. In Section 5, we provide some additional qualitative properties of the profile solution, including monotonicity of its gradient and of its curvature. We also give a bound from below on λ. In Section 6, we prove Liouville theorem 1.8. In Section 7, we prove the long time convergence of the solution to the steady state (up to addition of a constant), namely Theorem 1.6. This result follows from Liouville Theorem and a gradient bound on the solution (Proposition 7.1) that is proven in Section 7. Finally, Section A is an appendix where we recall standard materials, like strong maximum principle, Hopf lemma, Interior Schauder estimates. We also prove a technical lemma (Lemma A.4) which is used in Section 2, and also prove a result of independent interest which is not used in the rest of the paper: the equation satisfied by the curvature of the graph of the solution of the evolution problem.

Notation. For a real number a ∈ R, a + denotes max(a, 0) and a -denotes max(-a, 0). The ball of radius r centered at x are denoted B(x, r). If x = 0, we simply write B r .

Regular solutions up to the origin

This section is devoted to the proof of Theorem 1.1. This theorem improves [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF]Theorem 1.7] by establishing regularity of solutions up to the origin. As we pointed out previously, the assumption that U 0 is globally Lipschitz was missing in the statement of [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF]Theorem 1.7]. This is the reason why we first state a corrected version of this theorem.

Theorem 2.1 (Existence and uniqueness of smooth solutions for r > 0, [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF]). Assume that U 0 ∈ W 2,∞ loc (0, +∞) is globally Lipschitz continuous and satisfies

(U 0 ) r ∈ W 1,∞ (0, +∞) or κ U 0 ∈ L ∞ (0, +∞)
and that there exists a radius r 0 > 0 such that

|1 + κ U 0 | ≤ Cr for 0 ≤ r ≤ r 0 .
Then there exists a unique viscosity solution U of (1.3),(1.4) which is globally Lipschitz in space and time. Moreover this solution U belongs to C ∞ ((0, +∞) × (0, +∞)).

In view of this result, proving Theorem 1.1 amounts to prove the following proposition.

Proposition 2.2 (Space-time Lipschitz implies uniform regularity up to r = 0). Assume that U is a globally Lipschitz continuous (in space and time) solution of (1.3) in (0, +∞)×(0, +∞).

Then U (t, r) belongs to C

1+ 1 6 ,2+ 1 3 t,r
((0, +∞) × [0, +∞)). Moreover, for every δ > 0, R > 0, there exists a constant C = C(δ, R) such that we have the following uniform bound for every T ≥ δ > 0:

(2.1)

U -U (T, 0)

C 1+ 1 6 ,2+ 1 3 t,r ([T,T +δ]×[0,R]) ≤ C.
Before proving this proposition, we get some useful a priori estimates on the solution.

Lemma 2.3 (A priori estimates).

Assume that U is a globally Lipschitz continuous (in space and time) solution of (1.3) in (0, +∞) × (0, +∞), with Lipschitz constant L > 0. Then U ∈ C ∞ ((0, +∞) × (0, +∞)) and there exists a constant C = C(L) > 0 such that for every (t, r) ∈ (0, ∞) × (0, ∞), we have

(2.2) |U r (t, r) + 1 2 | ≤ Cr and |U rr (t, r)| ≤ C(1 + r 2 ).
Proof. We recall that we already proved in [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF]Theorem 1.7] that U ∈ C ∞ ((0, +∞) × (0, +∞)). We also recall that U t and U r are bounded, and that U solves

rU t = (1 + κ U ) 1 + r 2 U 2 r .
We deduce that

(2.3) |1 + κ U | ≤ Cr
for some constant C. Remarking that

1 + 2U r + rU rr = (1 + κ U )(1 + r 2 U 2 r ) 3 2 -r 2 U 3 r -(1 + r 2 U 2 r ) 3 2 -1 ,
and using the bound on U r and (2.3), we deduce that

|1 + 2U r + rU rr | ≤ C(r + r 2 + r 3 + r 4 ) ≤ C(r + r 4 ). (2.4)
For fixed t > 0, we set ψ(r) = U (t, r) + r/2 which satisfies (r 2 ψ r ) r = r(1 + 2U r + rU rr ), and deduce that

|(r 2 ψ r ) r | ≤ C(r 2 + r 5 ).
This implies |r 2 ψ r | ≤ C(r 3 + r 6 ) and we finally get

(2.5) |U r + 1 2 | = |ψ r | ≤ C(r + r 4 ).
Injecting this estimate in (2.4), we finally get for all r ∈ (0, +∞), t ∈ (0, +∞)

(2.6) |U rr (t, r)| ≤ C(1 + r 3 ).
Because U r and U t are bounded, we can use (1.3) to get for large r that |U rr | ≤ Cr 2 . We can then improve (2.5) and (2.6) to get (2.2). This ends the proof of the lemma.

Proof of Proposition 2.2. The idea of the proof is to see U as a radial solution of a partial differential equation in three dimensions and to use the interior regularity theory in 3D in order to deduce the boundary regularity up to r = 0. More precisely, we set

V (t, X) := U (t, |X|) + |X| 2 for X ∈ R 3 ,
where we see that V is smooth for X = 0. Here we have to add the term |X| 2 in the definition of V , in order to cancel the term ∇V (•, 0). Indeed, remember that U r (t, 0) = - 1 2 . If we do not add that term, this would make appear a bad term like 1 X in the coefficient of the PDE satisfied by V which would not allow us to control the regularity of the solution up to X = 0.

Step 1: Estimate on D 2 V . We make the following pointwise computation of the second derivatives

D 2 ji V =D j (D i V ) = D j X i |X| U r + 1 2 X i |X| =U rr X i X j |X| 2 + U r + 1 2 δ ij |X| - X i X j |X| 3 .
For R > 0 fixed and 0 < r ≤ R, we deduce from Lemma 2.3 that there exists a constant C R > 0 such that

|U rr | ≤ C R , |U r + 1 2 | ≤ rC R . This implies that D 2 V ∈ L ∞ ((0, +∞) × (B R \ {0})).
Moreover for all φ ∈ C ∞ c ((0, T ) × B R ), we have in the distribution sense

-D 2 ji V, φ = lim ε→0 (0,T )×(B R \Bε) (D j V )(D i φ) = lim ε→0 (0,T )×(B R \Bε) -(D 2 ji V )φ + (0,T )×∂Bε φ(n • e i )D j V
where n is the outward nomal to B R \B ε on the boundary ∂B ε , and e i is a unit vector of the canonical basis of R 3 . Since ∇V is bounded, we recover that

(D 2 V ) j,i = U rr X i X j |X| 2 + U r + 1 2 δ ij |X| - X i X j |X| 3
in the distribution sense on (0, +∞) × B R . This implies that the distribution

D 2 V satisfies D 2 V ∈ L ∞ ((0, +∞) × B R ).
Step 2: Estimate on ∇V . Moreover, since V r , V t ∈ L ∞ ((0, +∞) × B R ), we get that for every δ > 0 and for every 1 < p < +∞, there exists a constant C = C(δ, R, p) > 0 such that for every T ≥ δ, we have

V -V (T, 0) W 1,2;p ((T -δ,T +δ)×B R ) ≤ C.
Using parabolic Sobolev Embedding in parabolic Hölder spaces (see [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF]Lemma 3.3]), we get, for every 0 < α < 1 and a suitable constant C = C(δ, R, α) > 0, that (2.7)

V -V (T, 0)

C 1+α 2 ,1+α t,X ((T -δ,T +δ)×B R ) ≤ C which implies that (2.8) ∇V C α 2 ,α t,X ((T -δ,T +δ)×B R ) ≤ C.
Step 3: Equation satisfied by V . A computation gives that V is solution (at least in the distributional sense) of

V t = A(X, ∇V )∆V + B(X, ∇V ) for (t, x) ∈ (0, T ) × B R (0)
where

A(X, p) = 1 1 + (X • p -|X| 2 ) 2 , B(X, p) = (X • p) |X| 2 q 2 1 + q 2 + q 2 |X| G 0 (q) with q = X • p -|X| 2 and G 0 (q) = 1 q 2 1 + q 2 - 1 2 2 + q 2 1 + q 2 = 1 + O(q 2 ).
Let us set X = |X| α X |X| with α = 1/3.

In particular, we can easily check that the map X → X is in C α (see Lemma A.4). Then we can write

B(X, p) = ( X • p) ( X • p -| X| 2 ) 2 1 + q 2 + | X| X • p - | X| 2 2 G 0 (q) with q = X • p - |X| 2 .
Therefore on the set {|X| ≤ R, |p| ≤ R}, we see that the function B is Lipschitz continuous both in p and in X, i.e. satisfies

|B(X ′ , p ′ ) -B(X, p)| ≤ C R | X′ -X| + |p ′ -p| . Using Lemma A.4, this implies (increasing C R if necessary) that |B(X ′ , p ′ ) -B(X, p)| ≤ C R |X ′ -X| α + |p ′ -p| i.e. B is locally Lipschitz in p and C α in X. Similarly |A(X ′ , p ′ ) -A(X, p)| ≤ C R |X ′ -X| + |p ′ -p| i.e.
A is locally Lipschitz in p and X.

Denoting by Ã(t, X) = A(X, ∇V (t, X)) and B(t, X) = B(X, ∇V (t, X)), and using (2.8) for the regularity of ∇V , we get that there exists a constant C > 0 such that (2.9)

|| Ã|| C 1 6 , 1 3 t,X ((T -δ,T +δ)×B R ) , || B|| C 1 6 , 1 3 t,X ((T -δ,T +δ)×B R ) ≤ C. Because Ṽ (t, x) := V (t, x) -V (T, 0) solves Ṽt = Ã∆ Ṽ + B in (T -δ, T + δ) × B R ,
we can use interior Schauder estimates (see Proposition A.3 in the appendix), and deduce that

V -V (T, 0) C 2+ 1 3 ,1+ 1 6 x,t ([T,T +δ]×B R/2 ) ≤ C || B|| C α, α 2
x,t ((T -δ,T +δ)×B R ) + |V -V (T, 0)| L ∞ ((T -δ,T +δ)×B R ) ≤ C
where we have used (2.9) and (2.7) for the last inequality. This implies in particular (2.1) (changing R/2 in R), and ends the proof of the proposition.

Existence of a steady state

The main result of this section is the following proposition.

Proposition 3.1 (Existence of a steady state).

There exists a constant λ ≥ 0 and a function Φ ∈ C ∞ (0, +∞), satisfying

(3.1) -1/2 ≤ Φ r ≤ 0 on (0, +∞) such that U (t, r) = λt + Φ(r) is a solution of (1.3) on R × (0, +∞).
In a first subsection, we build a solution on an annulus R -1 < r < R, and in a second subsection we pass to the limit R → +∞.

Steady state in a annulus

In the following, we will frequently work in log coordinates with the function u(t, x) = U (t, e x ). The function U solves (1.3) if and only if u solves the following equation

(3.2) u t = F (x, u x , u xx ) := e -x 1 + u 2 x + e -2x u x + e -2x u xx 1 + u 2 x .
See for instance [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF].

For R > 1, we consider the annulus R -1 ≤ r ≤ R, we study the following problem with Neumann boundary condition on the boundary of the annulus:

(3.3)      rU t = 1 + r 2 U 2 r + U r 2 + r 2 U 2 r 1 + r 2 U 2 r + rU rr 1 + r 2 U 2 r on (0, +∞) × (R -1 , R), U r = 0 on (0, +∞) × R -1 , R , with initial data (3.4) U (0, r) = U 0 (r) for all r ∈ [R -1 , R]
Then we have the following result.

Lemma 3.2 (The Cauchy problem in an annulus). Let R > 1 and α ∈ (0, 1) and assume that

U 0 ∈ C 2+α ([R -1 , R]
) and that U 0 satisfies

(3.5) (U 0 ) r (R -1 ) = 0 = (U 0 ) r (R) -M ≤ (U 0 ) r (r) ≤ 0 for all r ∈ (R -1 , R).
Then there exists a unique solution

U ∈ C 1+ α 2 ,2+α ([0, +∞) × [R -1 , R]) of (3.3), (3.4). More- over U satisfies (3.6) -max(1/2, M ) ≤ U r (t, r) ≤ 0 for all (t, r) ∈ (0, +∞) × (R -1 , R).
Proof of Lemma 3.2. The proof proceeds in several steps.

Step 1: Existence of a smooth solution As it is explained in [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF], the classical theory allows to construct a unique solution

U ∈ C 2+α,1+ α 2 ([0, +∞) × [R -1 , R]) of (3.
3). Moreover, from the classical parabolic regularity theory, we can bootstrap and get that

U ∈ C ∞ ((0, +∞) × [R -1 , R]).
Step 2: Gradient bound from above. We first recall that u(t, x) = U (t, e x ) solves (3.2). Let

w = u x
Then by derivation of (3.2), we easily get that w solves in (0, +∞) × (-a, a) (with a = ln R),

(3.7) w t = -e -x 1 + w 2 + e -x ww x √ 1 + w 2 -2e -2x w + e -2x w x -2e -2x w x 1 + w 2 + e -2x w xx 1 + w 2 - 2w(w x ) 2 (1 + w 2 ) 2 ,
and w(t, ±a) = 0 for all t ∈ (0, +∞) and

(3.8) w(0, x) = e x (U 0 ) r (e x ) for all x ∈ [-a, a].
Notice that w = 0 is a supersolution of (3.7), (3.8), where we use (3.5) to check the initial condition inequality. Therefore the classical comparison principle implies that (3.9) w ≤ 0.

Step 3: Gradient bound from below. We now define the function

z(t, x) = e -x w(t, x) = U r (t, e x ).
It is easy to check that z satisfies

(3.10) z t = -e -2x 1 √ 1 + w 2 + z -e -3x w 1 + w 2 + 2w 3 (1 + w 2 ) 2 + e -2x z xx 1 + w 2 + O(z x ).
Because we already know that z ≤ 0, we deduce that:

(3.11) e 2x z t ≥ -g(x, z) + z xx 1 + w 2 + O(z x ) with g(x, z) = 1 √ 1 + e 2x z 2 + z + z 1 + e 2x z 2 . Let us set h(γ) = γ + z + γ 2 z.
Then we have

g(x, z) = h(γ) with γ = 1 √ 1 + e 2x z 2 ∈ (0, 1]. Remark that the maximum of h(γ) is reached at γ = -1 2z if z < 0. Therefore sup γ∈(0,1] h(γ) ≤ h - 1 2z = z - 1 4z ≤ 0 if z ≤ - 1 2 . Therefore g(x, z) ≤ 0 if z ≤ - 1 2 .
Remark now that z = -max(1/2, M ) is then a subsolution of the equation with equality in (3.11) (with zero boundary conditions). This implies that z is a subsolution of (3.10) with zero boundary condition. Again, the comparison principle for z implies that

(3.12) -max(1/2, M ) ≤ z.
Finally (3.12) and (3.9) implies (3.6) which ends the proof of the proposition.

Lemma 3.3 (Periodic solution in an annulus). For R > 1, there exists a solution U R of (3.3) in (0, +∞) × (R -1 , R) such that (3.13) U R (t + T R , r) = U R (t, r) + 2π
for some T R > 0.

Proof of Lemma 3.3. Let I denote the interval (R -1 , R). In view of [12, Remark 2.1] and the discussion preceding [12, Proposition 4.3], we know that for all U 0 ∈ C 2+α ( Ī) for some α ∈ (0, 1) such that (U 0 ) r ≤ 0, there exists a solution U R of (3.3) in (0, ∞) × I. Moreover, for all t > 0, we have U R (t, •) ∈ C ∞ ( Ī). We then choose U 0 ∈ C 2+α ( Ī) satisfying (3.5) with M = 1/2 and we denote by U R the corresponding solution. Thanks to Lemma 3.2, we know that

-1/2 ≤ (U R ) r (t, r) ≤ 0.
Moreover, by [START_REF] Giga | Spiral solutions for a weakly anisotropic curvature flow equation[END_REF]Proposition 4.3], there exists a period T R > 0 and U 0 such that (3.13) holds true. This achieves the proof of Lemma 3.3.

Lemma 3.4 (Steady state in an annulus

). For R > 1, there exists λ R > 0 and Φ R ∈ C ∞ ([R -1 , R]) satisfying -1/2 ≤ (Φ R ) r ≤ 0 in [R -1 , R] such that λ R t + Φ R (r) is a solution of (3.3). Proof of Lemma 3.4. Remark first (with λ R = 2π T R ) that (3.14) v(t, r) = U R (t, r) -λ R t
is T R -periodic with respect to the time variable. We want to prove that it is constant. Consider ε, δ > 0 and define

U ε,δ R (t, r) = U R (t + δ, r) -ε. We have U ε,0 R < U 0,0 R and since U R is Lipschitz continuous, U ε,δ R < U 0,0
R for δ small enough. We then define for ε > 0

δ ε = sup{ δ > 0 : U ε,δ R < U 0,0 R ∀δ ∈ [0, δ)} > 0.
Since v is periodic, we deduce that δ ε < +∞. Remark that U ε,δ R and U 0,0 R are both solutions of (3.3) and the optimality of

δ ε implies that max t∈[0,T R ],r∈ Ī{U ε,δε R -U 0,0 R } = 0.
By Lemma A.2 and the Neumann boundary condition, we deduce that the maximum is attained for some inner point r 0 ∈ I. Since the function (inside the maximum) is T R -periodic with respect to the time variable, the strong maximum principle (Theorem A.1) written for the difference function

U ε,δε R -U 0,0 R implies that U ε,δε R ≡ U 0,0 R (note that w = U ε,δε R -U 0,0
R solves a linear locally uniformly parabolic equation and the coefficient of the linear equation for w are enough regular to apply the strong maximum principle, see the book by Gilbarg-Trudinger [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] for more details on this linearized argument). Then for all k ∈ N, we have

U R (t + kδ ε , r) = U R (t, r) + kε. The fact that U R -λ R T is T R -periodic implies in the limit k → +∞ that δε T R = ε λ R T R , i.e., λ R = ε δε .
Hence, for every ε > 0, we have

U R (t + ε λ R , r) = U R (t, r) + ε.
This implies that v defined in (3.14) is constant. The proof of Lemma 3.4 is now complete.

Steady state in the plane

In this subsection, we want to take the limit R → 0 to recover a steady state in the plane.

To this end, we first need the following estimate.

Lemma 3.5 (Bound on λ R ). There exists λ ≥ 0 such that for all R ≥ 2, we have 0 < λ R ≤ λ.

Proof of Lemma 3.5. We already know that λ R > 0. In order to exhibit λ ≥ 0 with the desired property, we are going to construct a super-solution of (1.3) of the type û(t, r) = λt + Ψ(r). Let θ = -γ(r) describe the circle (in polar coordinates) of equation 1 + κ = 0 which is tangent from above to the horizontal axis. From an analytical point of view, the reader can check that the right half circle (i.e. for θ ∈ [0, π/2] and 0 ≤ r ≤ 2) corresponds to γ(r) = -arcsin r 2 which satisfies F (r, γ r , γ rr ) = 0 for 0 < r < 2, where

F (r, γ r , γ rr ) := 1 r 1 + r 2 γ 2 r + γ r 2 + r 2 γ 2 r 1 + r 2 γ 2 r + rγ rr 1 + r 2 γ 2 r .
We choose Ψ as follows

Ψ(r) = ζ(r)γ(r)
where ζ is a smooth cut-off function which is equal to 1 in [0, 1 2 ] and equal to zero for r ≥ 1.

Now we choose λ such that λ ≥ sup r>0 F (r, Ψ r , Ψ rr ) = sup r∈[1/2,1] F (r, Ψ r , Ψ rr )
We also have for R ≥ 2:

Ψ r (R) = 0, and Ψ r (R -1 ) = γ r (R -1 ) < 0.
This implies that λt + Ψ(r) is a supersolution of (3.3), and the comparison principle with λ R t + Φ R (r) implies (for large times) that λ R ≤ λ which ends the proof of the lemma.

We now want to pass to the limit as R → +∞ and prove Proposition 3.1.

Proof of Proposition 3.1. Because the functions λ R t + Φ R (r) are uniformly Lipschitz continuous in space and time independently on R ≥ 2, we can pass to the limit R → ∞. We call the limit λt + Φ(r), which is then a viscosity solution of (1.3) and satisfies:

-1/2 ≤ Φ r ≤ 0 and λ ≥ λ ≥ 0.

Because λt+Φ(r) is globally Lipschitz continuous in space and time, we can apply Lemma 2.3 and deduce that Φ ∈ C ∞ (0, +∞). This ends the proof of the proposition.

Asymptotics of the steady state and uniqueness

The main result of this section is the following proposition.

Proposition 4.1 (Asymptotics of the steady state and uniqueness). Assume that λt + Φ(r) is a globally Lipschitz continuous solution of

(1.3) in R × (0, +∞) with Φ ∈ C ∞ (0, +∞) satisfying (4.1) λ ≥ 0 and Φ r ≤ 0.
Then such a λ is unique and such a Φ is unique up to an additive constant. Moreover we have λ > 0 and there exist constants a ∈ R and C > 0 such that

(4.2) |Φ(r) + λr + λ ln(1 + r) -a| ≤ C 1 + r .
We will do the proof of Proposition 4.1 using several lemmas and propositions.

Positivity of the angular velocity

We first prove that λ is positive. Proof of Lemma 4.2. Assume by contradiction that λ = 0. We look for a barrier solution that we will compare to Φ. To this end, let us consider the circle in R 2 of radius 1 (given by the equation 1 + κ = 0) and of center (0, -R) for some R > 1 in the Cartesian coordinates X = (x 1 , x 2 ). We can parametrize in polar coordinates, the right half circle as follows,

θ = -γ R (r) := arcsin (f (r)) for R -1 ≤ r ≤ R + 1 with f (r) = r 2 + R 2 -1 2Rr which satisfies f (R -1) = 1 = f (R + 1), and f ′ (r) = r 2 -(R 2 -1)
2Rr 2
with f ′ (R ± 1) = 0. This implies in particular that the graph of γ R has vertical tangents at r = R ± 1. Because γ R is a stationary solution of (1.3) on (R -1, R + 1), we can compare it on (R -1, R + 1) to the stationary solution Φ when λ = 0. We consider

min r∈[R-1,R+1] (Φ(r) -γ R (r)) .
Since lim r→R-1 γ ′ R (r) = +∞ and lim r→R+1 γ ′ R (r) = -∞ and using the fact that φ is Lipschitz continuous, we get that (φ -γ R )(R -1) and (φ -γ R )(R + 1) are local maximum. Hence, the minimum can not be achieved at r = R ± 1 and is therefore reached at some interior point. The strong minimum principle then implies that

Φ(r) -γ R (r) is constant in (R -1, R + 1).
By continuity, this is still true at r = R ± 1 which is again impossible. Finally, we conclude that λ = 0 and then λ > 0. This ends the proof of the lemma.

Asymptotics

In the following proposition, the asymptotics of the profile is stated in Log coordinates. It also contains the asymptotics of the derivative of the profile which will be used later.

(4.3) |ϕ(x) + λe x + λx -a| ≤ Ce -x for x ≥ x 1 and (4.4) ϕ x (x) = -λe x -λ + O(e -x ) for x ≥ x 1
for some constants a, x 1 ∈ R and C > 0.

Recalling (3.2), we see that ϕ is a solution of the following second order ODE (4.5) λ = e -x 1 + ϕ 2

x + e -2x ϕ x + e -2x ϕ xx 1 + ϕ 2

x for x ∈ R.

As we shall see it, Proposition 4.3 is a consequence of the study of the ODE satisfied by v := ϕ x ≤ 0, which is the following

(4.6) v x = f (v, x) for x ∈ R where f (w, x) = e 2x (1 + w 2 )ζ(w, x) with ζ(w, x) = λ -e -x 1 + w 2 -e -2x w.
We first need the following result.

Lemma 4.4 (Elementary estimates). Let λ > 0. Then there exists a real number x 0 ≥ 0 such that for x ≥ x 0 , the equation f (w, x) = 0 has a single root w = v 0 (x) which is non-positive. This function satisfies for x ≥ x 0

v 0 (x) = -λe x -λ + e -x 1 2λ -λ + O(e -2x ), (4.7) (v 0 ) x (x) = -λe x + O(1) ≤ 0. (4.8)
Moreover we have (4.9) ∂f ∂w (w, x) ≥ λ 2 2 e 3x for w ≤ v 0 (x) and x ≥ x 0 and for all w * , y * ∈ R, we have

(4.10) x ≥ y * ≥ x 0 v 0 (y * ) < w * ≤ w ≤ 0 =⇒ f (w, x) ≥ e 2y * min(ζ(w * , y * ), λ/2) > 0.
Proof of Lemma 4.4. The proof proceeds in several steps.

Step 1: Definition of v 0 . Remark that if f (w, x) = 0, then w solves the following second order polynomial equation

(4.11) (1 -e -2x )w 2 + 2λw + 1 -λ 2 e 2x = 0 .
For some x large enough, there is only one non-positive solution which is given by the following formula

v 0 (x) = -λ -λ 2 + (1 -e -2x )(e 2x λ 2 -1) 1 -e -2x = -λ -λe x 1 + 1 λ 2 e -4x -1 λ 2 e -2x 1 -e -2x = -λ -λe x 1 - 1 2λ 2 e -2x
1 + e -2x + O(e -2x ), which gives (4.7).

In order to recover (4.8), we take the x-derivative of equation (4.11) satisfied by v 0 , and we get

(v 0 ) x (v 0 (1 -e -2x ) + λ) + (v 0 ) 2 e -2x -λ 2 e 2x = 0.
This implies (using (4.7) in the second equality)

(v 0 ) x (x) = -(v 0 ) 2 e -2x + λ 2 e 2x v 0 (1 -e -2x ) + λ = λ 2 e 2x + O(1) -λe x + O(1) ,
which gives (4.8).

Step 2: Estimate on ∂f ∂w . Let us now compute

∂f ∂w (w, x) = 2we 2x ζ(w, x) + e 2x (1 + w 2 ) ∂ζ ∂w (w, x) and (4.12) ∂ζ ∂w (w, x) = - we -x √ 1 + w 2 -e -2x =: g(w, x).
Remark also that, increasing x 0 if necessary, we have for x ≥ x 0 both

v 0 (x) ≤ -1 and ∂ζ ∂w (w, x) ≥ 1 2 e -x for w(x) ≤ v 0 (x) ≤ -1.
But ζ(v 0 (x), x) = 0, and then the sign of ∂ζ ∂w implies

ζ(w(x), x) ≤ 0 for w(x) ≤ v 0 (x) and ∂f ∂w (w, x) ≥ e 2x (1 + w 2 ) ∂ζ ∂w (w, x).
Again, increasing x 0 if necessary, we can assume that v 0 (x) ≤ -λe x for x ≥ x 0 and then

∂f ∂w (w, x) ≥ λ 2 2 e 3x for w(x) ≤ v 0 (x).
Step 3: Estimate on f . Recall that the function g appears in (4.12). Remark that for x ≥ 0 we have g(w, x) = 0 with w ≤ 0 if and only if

w(x) = - 1 √ e 2x -1 =: w 0 (x).
Moreover we can then deduce that

g(w, x) ≥ 0 if w ≤ w 0 (x), g(w, x) ≤ 0 if w 0 (x) ≤ w ≤ 0.
Because of (4.12), we deduce that, increasing x 0 if necessary,

w 0 (x) ≤ w ≤ 0 =⇒ ζ(w, x) ≥ ζ(0, x) = λ -e -x ≥ λ/2 if x ≥ x 0
and then using the definition of f and a bound from below of ζ(w, x) for w ∈ [w * , 0], we get

v 0 (x) < w * ≤ w ≤ 0 =⇒ f (w, x) ≥ e 2x min(ζ(w * , x), λ/2) > 0 if x ≥ x 0
Let us notice that for w ≤ 0, we have up to increase x 0 if necessary,

∂ζ ∂x (w, x) = e -x 1 + w 2 + 2e -2x w ≥ 0 if x ≥ x 0
and then this implies (4.10). This ends the proof of the lemma.

We next prove the following estimate.

Lemma 4.5 (Asymptotics for v = ϕ x ). For any µ > 0, there exists a real number

x 1 ≥ x 0 such that v = ϕ x satisfies v 0 (x) ≥ v(x) ≥ v 0 (x) -µe -3 2 x for x ≥ x 1
where v 0 and x 0 are given by Lemma 4.4.

Proof of Lemma 4.5. Recall that λ > 0 and define v(x) := v 0 (x) -µe -3 2 x .

The proof proceeds in several steps.

Step 1: v is a super-solution. Remark that, thanks to (4.8),

vx (x) = (v 0 ) x (x) + 3 2 µe -3 2 x = -λe x + O(1).
We also remark that there exists w(x) ∈ [v(x), v 0 (x)] such that (4.13)

f (v(x), x) = f (v 0 (x), x) + ∂f ∂w (w(x), x)(v(x) -v 0 (x)) ≤ λ 2 2 e 3x (v(x) -v 0 (x)) ≤ -µ λ 2 2 e 3 2 x
where we used (4.9) in the second line. Therefore there exists

x 1 ≥ x 0 such that vx (x) ≥ f (v(x), x) for x ≥ x 1 .
Step 2: Comparison with v. Assume by contradiction that v(x * ) ≤ v(x * ) for some x * ≥ x 1 . Then, from the comparison principle, we deduce that v(x) ≤ v(x) for all x ≥ x * .

Then we have

(4.14) v x (x) = f (v(x), x) ≤ f (v(x), x) ≤ -µ λ 2 2 e 3 2 x
where we have used the fact that v ≤ v, the monotonicity of f (w, x) in w (see (4.9)) and estimate (4.13). Estimate (4.14) now gives a contradiction with the fact that Φ r (e x ) = e -x v(x) is bounded.

Step 3: v 0 is a sub-solution. The inequality (v 0 ) x (x) ≤ 0 = f (v 0 (x), x) for x ≥ x 0 follows from (4.8).

Step 4: Comparison with v 0 . We argue by contradiction. Let us assume that there exists a point y * ≥ x 0 such that v(y * ) > v 0 (y * ). Then from (4.10), we deduce that there exists a constant α > 0 such that

f (w, x) ≥ α > 0 for w ∈ [v(y * ), 0] and x ≥ y * . But recall that v x (x) = f (v(x), x).
This implies that v x (x) ≥ α for x ≥ y * while v(x) ≤ 0.

Therefore we conclude (using the continuity of f ) that there exists a point x 2 such that v(x 2 ) > 0, which is impossible because v = ϕ x ≤ 0. We thus get the desired contradiction. This ends the proof of the lemma.

Proof of Proposition 4.3. It follows from Lemma 4.5 and (4.7).

Uniqueness

Proposition 4.6 (Uniqueness). Under the assumptions of Proposition 4.1, λ is unique and Φ is unique up to addition of constants.

In order to prove Proposition 4.6, we will need the following space Liouville result which will be proven later in Section 6 as an independent result. Theorem 4.7 (Space Liouville theorem). Let Φ i for i = 1, 2 be two C 2 ([0, +∞)) functions such that for some λ > 0, the functions λt + Φ i (r) are solutions of (1.3) in R × (0, +∞) for i = 1, 2. Assume also that we have for i = 1, 2 and r ≥ 0:

(4.15) Φ i (r) + λr + λ ln(1 + r) ≤ C 1 + r and Φ i r (r) + λ ≤ C 1 + r . Then Φ 1 = Φ 2 .
Proof of Proposition 4.6. We already know that Φ satisfies (4.2). From Proposition 2.2, we deduce that Φ ∈ C 2 ([0, +∞)).

Uniqueness of λ. We argue by contradiction by assuming that there exist (Φ 1 , λ 1 ) and (Φ 2 , λ 2 ) two solutions such that λ 1 < λ 2 .

Because of (4.2), we deduce that there exists a constant K such that Φ 1 (r) ≥ Φ 2 (r) -K for r > 0.

From the comparison principle for (1.3) (see Theorem 1.3 in [START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF], with Lipschitz continuous initial data U 0 = Φ 1 ), we deduce

λ 1 t + Φ 1 (r) ≥ λ 2 t + Φ 2 (r) -K for all (t, r) ∈ (0, +∞) × (0, +∞)
which implies (for large times) that λ 1 ≥ λ 2 . This is the desired contradiction.

Uniqueness of Φ (up to an additive constant). We now consider two profiles Φ 1 , Φ 2 with the same λ = λ 1 = λ 2 . Recall that for i = 1, 2, each function Φ i satisfies (4.2) for some constant a i . Adding different constants to those two functions if necessary, we can asssume that a 1 = a 2 = a = 0, i.e.

|Φ i (r) + λr + λ ln(1 + r)| ≤ C 1 + r , for i = 1, 2.
We then apply Theorem 4.7 to conclude that Φ 1 = Φ 2 . The proof is now complete.

Proof of Proposition 4.1. It follows from Lemma 4.2 and Propositions 4.3 and 4.6.

5 Further properties of the steady state

Monotonicity properties

Proposition 5.1 (Monotonicity of the gradient of the profile). Let Φ be the profile given in Proposition 4.1. Then we have

(5.1) Φ rr ≥ 0 in [0, +∞) (5.2) - 1 2 ≤ Φ r ≤ -λ and
(5.3) Φ r (0) = - 1 2 and Φ r (+∞) = -λ < 0.
Proof of Proposition 5.1. For ϕ(x) = Φ(e x ), we recall from (3.10) that z(x) := e -x ϕ x (x) = Φ r (e x ) satisfies with w = ϕ x :

0 = z t = -e -2x 1 √ 1 + w 2 + z -e -3x w 1 + w 2 + 2w 3 (1 + w 2 ) 2 + e -2x z xx 1 + w 2 + O(z x ) .
Step 1: case of a local minimum of z. Assume that z has a local minimum at x 0 with value z 0 = z(x 0 ). Then z xx (x 0 ) ≥ 0 and z x (x 0 ) = 0 which implies,

1 1 + e 2x 0 z 2 0 + z 0 + z 0 1 + e 2x 0 z 2 0 + 2e 2x 0 z 3 0 (1 + e 2x 0 z 2 0 ) 2 ≥ 0. Setting γ = 1 1 + e 2x 0 z 2 0 ∈ (0, 1],
we see that this means γ + z 0 + γ 2 z 0 + 2γ 4 z 0 (1/γ 2 -1) ≥ 0 i.e.

(5.4)

γ + z 0 (1 + 3γ 2 -2γ 4 ) ≥ 0. Let g(y) := 1 + 3y -2y 2
Remark that g is maximum at y = 3/4 and then inf y∈(0,1] g(y) ≥ min (g(0), g(1)) = 1.

Therefore (5.4) means

z 0 ≥ - γ g(γ 2 ) =: -K(γ).
Step 2: Monotonicity of K. Let us compute with y = γ 2 :

K ′ (γ) = 1 g 2 (y) (g(y) -2yg ′ (y)) with g(y) -2yg ′ (y) = 1 + 3y -2y 2 -2y(3 -4y) = 1 -3y + 6y 2 =: h(y)
which is minimal at y * = 1/4 with value h(y * ) > 0. Therefore K is increasing.

Step 3: Monotonicity of z. Assume now that z has a local maximum at x with value z = z(x). Then we have

z ≤ -K(γ) with γ = 1 √ 1 + e 2x z 2 .
We already know (see (4.4)) that

z(x) = -λ -λe -x + o(e -2x )
which shows that z cannot be non-increasing in (x, +∞) (and satisfies z(+∞) = Φ r (+∞) = -λ). Therefore there exists x > x such that z has a local minimum at x with value z = z(x) that we can choose such that (5.5) z ≤ z ≤ 0.

Moreover we have z ≥ -K(γ) with γ = 1

1 + e 2x z 2 < γ.

The strict monotonicity of K implies

z ≤ -K(γ) < -K(γ) ≤ z,
which is in contradiction with (5.5). Therefore, we conclude that z has no local maximum.

Step 4: Behaviour at r = 0. We recall that Φ ∈ C 2 ([0, +∞)). From the fact that λt+Φ(r) is a solution of (1.3), we deduce that

rλ = 1 + r 2 Φ 2 r + Φ r 2 + r 2 Φ 2 r 1 + r 2 Φ 2 r + rΦ rr 1 + r 2 Φ 2 r .
At r = 0, we deduce that (5.6) 1 + 2Φ r (0) = 0.

Close to r = 0, we deduce (by Tayor expansion) that

Φ rr (r) = O(r) + λ - 1 r 1 + 2Φ r (r) + O(r 2 ) .
Using (5.6), we deduce that Φ rr (0) = λ 3 > 0.

Step 5: Conclusion. Using the fact that Φ rr (0) > 0 and the fact that Φ r has no local maximum (by Step 3), we deduce that Φ r is increasing, which in particular implies (5.1) and (5.2). This ends the proof of the proposition.

Proposition 5.2 (Sign and monotonicity of the curvature). Let Φ be the profile given in Proposition 4.1. Then the curvature κ Φ defined in (1.2) satisfies,

-1 ≤ κ Φ ≤ 0 and κ Φ (0) = -1, κ Φ (+∞) = 0.
Moreover we have (κ Φ ) r ≥ 0.

Proof of Proposition 5.2. We set κ(x) := κ Φ (e x ). Notice that we deduce from (1.2) and (5.3) that κ Φ (r = 0) = 2Φ r (0) = -1.

Step 1: κ ∈ [-1, 0]. Recall that for the profile, we have,

(5.7) λ = e -x 1 + u 2 x + e -2x u x + e -2x u xx 1 + u 2 x = e -x (1 + κ) 1 + u 2 x
where the curvature κ can be written as

(5.8) κ := e -x u x 1 + u 2 x + e -x u xx (1 + u 2 x ) 3 2
.

Equation (5.7) shows that we can find the following other expression for the curvature, (5.9)

κ = λe x 1 + u 2 x -1.
Using (4.4), we then deduce that (5.10) κ(x = +∞) = 0.

Moreover, using again (5.9), we have

κ x = λe x 1 + u 2 x - λe x (1 + u 2 x ) 3 2 u x u xx = λe x 1 + u 2 x -λe 2x u x κ -e -x u x 1 + u 2 x = λe x 1 + u 2 x -λe 2x u x κ.
Using the fact that u x ≤ 0, we conclude that

κ(x 0 ) > 0 =⇒ κ(x) ≥ κ(x 0 ) for x ≥ x 0
which is in contradiction with (5.10). Therefore κ ≤ 0. The fact that 1 + κ ≥ 0 comes directly from (5.9).

Step 2: κ is non-decreasing. Let us start again from (5.11)

κ x = λe x 1 + u 2 x -λe 2x u x κ.
Then

κ xx = λe x 1 + u 2 x -2λe 2x u x κ + λe x u x u xx 1 + u 2 x -λe 2x u xx κ -λe 2x u x κ x = 2κ x -λe x 1 + u 2 x -λe 2x u x κ x + u xx u x κ x - λe x 1 + u 2 x = κ x 2 -λe 2x u x + u xx u x - λe x 1 + u 2 x u x u x + u xx 1 + u 2 x = κ x 2 -λe 2x u x + u xx u x - λe x 1 + u 2 x u x e x 1 + u 2 x κ.
Recall that u x < 0, κ ≤ 0 and κ x = 0 implies in (5.11) that u x κ = e -x 1 + u 2 x > 0, which shows that κ < 0. Therefore we conclude from the above computation that κ xx < 0 at any point where κ x = 0. This implies that κ can not have local minima. Because -1 ≤ κ(x) ≤ 0 and κ(-∞) = -1, κ(+∞) = 0, we deduce that κ does not have local maxima neither (which would imply the existence of a local minimum). Therefore

κ x ≥ 0.
This ends the proof of the proposition.

Bound from below for the angular velocity

We next prove the following lemma.

Lemma 5.3 (Bound from below on λ). We have λ ≥ 1/4.

Proof of Lemma 5.3. The proof proceeds in several steps.

Step 1: comparison. The idea is to revisit the proof of the uniqueness of λ. For some µ > 0, we set ϕ 1 := ϕ and ϕ 2 := -µe x .

If µ > λ,
then a comparison of the behaviour at x = +∞ implies that

ϕ 2 ≤ ϕ 1 + K on R
for some suitable constant K. We recall that

λ = F (x, ϕ x , ϕ xx ),
with F defined in (3.2). We then define

h µ (x) := F (x, (ϕ 2 ) x , (ϕ 2 ) xx ) = e -2x + µ 2 -µe -x 1 + 1 1 + µ 2 e 2x . If (5.12) α ≤ inf x∈R h µ (x),
then we can take λ 2 = α and we see with

λ 1 := λ that λ 2 t + ϕ 2 (x) ≤ λ 1 t + ϕ 1 (x) + K
is true at t = 0 and then is true for every time t ≥ 0, because the left hand side is a subsolution and the right hand side is a solution. Then we conclude that

α = λ 2 ≤ λ 1 = λ i.e.
(5.13) µ > λ =⇒ λ ≥ α if α satisfies (5.12).

Step 2: estimate on α and conclusion. Remark that (5.12) is satisfied for α ≥ 0 if and only if

(5.14) α + µe -x 1 + 1 1 + µ 2 e 2x 2 ≤ e -2x + µ 2
Because we have

α + µe -x 1 + 1 1 + µ 2 e 2x 2 ≤ 2α 2 + 2e -2x µ 2 2 2
we see that inequality (5.14)

of 0 ≤ 1 + κ Φ = rλ 1 + r 2 Φ 2 r .
The proof of Theorem 1.4 is now complete.

A Liouville result

This section is devoted to the proof of a Liouville result (Theorem 1.8) for global solutions of (1.3). This Liouville result will be used in the next section. The Liouville Theorem 1.8 classifies global space-time solutions. Such kind of results have been for instance obtained for certain nonlinear heat equations in [START_REF] Giga | Asymptotically self-similar blow up of semilinear heat equations[END_REF][START_REF] Merle | type behavior of blow-up solutions of nonlinear heat equations[END_REF], where the nonlinearity comes from the source term. On the contrary, the nonlinearity in our problem comes from the geometry itself.

In order to prove Theorem 1.8, we first prove two comparison principles: one for small r's (i.e. in R × [0, r - 0 )), and one for large r's (i.e. in R × [r + 0 , +∞)).

Proposition 6.1 (Comparison principle for small r's). Given some constant C > 0, there exists some r - 0 = r - 0 (C) > 0 such that the following holds for every

r 0 ∈ (0, r - 0 ]. Let U ∈ C 2,1 (R × [0, r 0 ]) be a subsolution and V ∈ C 2,1 (R × [0, r 0 ]) be a supersolution of (1.3) in R × (0, r 0 ) satisfying (6.1) 1 + 2V r (t, 0) ≤ 0 ≤ 1 + 2U r (t, 0) for all t ∈ R.
Assume moreover that we have

(6.2)    |U r |, |V r | ≤ C, |rU rr | ≤ C, |U -V | ≤ C. If U ≤ V in R × {r 0 }, then U ≤ V in R × [0, r 0 ].
Remark 6.2 (The Neumann boundary condition). Notice that condition (6.1) can be seen as the evaluation on the boundary r = 0 of the inequalities in equation (1.3) associated to subsolutions U and supersolutions V .

Proof of Proposition 6.1. The proof proceeds in several steps.

Step 1: subsolution W = U -V . We set W = U -V . We write the difference of the two inequalities satisfied by U and V , which gives

rW t ≤ G(rU r ) -G(rV r ) + U r (K(rU r ) -K(rV r )) + (U r -V r )K(rV r ) + rU rr (H(rU r ) -H(rV r )) + rH(rV r )W rr with (6.3) G(p) = 1 + p 2 , K(p) = 2 + p 2 1 + p 2 , H(p) = 1 1 + p 2 .
This leads to (6.4)

W t ≤ AW r + H(rV r )W rr on R × (0, r 0 ) with (6.5) A = a + K(rV r ) r + b + c
where (6.6)

     a = 1 0 ds G ′ (r(U r -sW r )), b = U r 1 0 ds K ′ (r(U r -sW r )), c = rU rr 1 0 ds H ′ (r(U r -sW r )).
Using (6.2) and the fact that

|G ′ (p)| ≤ |p| and |K ′ (p)| = |H ′ (p)| ≤ 2|p|, this implies that A ≥ -r 0 C + 2 r 0 -2r 0 C 2 -2r 0 C 2 .
Choosing then r 0 = r 0 (C) > 0 small enough, we deduce that (6.7) A ≥ 0 and H(rV r ) ≥ 1 2 .

Step 2: supersolution Ψ. The goal is now to construct a non-negative supersolution (i.e. satisfying the reverse inequality in (6.4)) which explodes as |t| → +∞. We define for some

µ > 0 Ψ(r, t) = e -µt ζ(r) + f (t) with 0 ≤ f ∈ C ∞ (R) s.t.    f (t) = 0 if t < 0 f ′ ≥ 0 f (t) → +∞ as t → +∞ such that we have -µζ ≥ 1 2 ζ rr in (0, r 0 ), ζ r (0) = 0.
We can simply choose ζ(r) := cos π 4 r r 0 with 2µ := π 4r 0

2

. Because ζ r ≤ 0, ζ rr ≤ 0 on (0, r 0 ), we get, using (6.7), that

Ψ t ≥ 1 2 Ψ rr ≥ AΨ r + H(ru r )Ψ rr on R × (0, r 0 ).
Step 3: contact point. Notice that Ψ ≥ δ > 0 on R × [0, r 0 ]. Then for ε > 0 large enough, we have:

εΨ ≥ W on R × [0, r 0 ].
We can then decrease ε untill we get a contact point,

ε * = inf{ε ≥ 0, εΨ ≥ W on R × [0, r 0 ]}.
We now want to show that ε * = 0. By contradiction, assume that ε * > 0. We have

(6.8) inf (t,r)∈R×[0,r 0 ] {ε * Ψ -W } = 0.
Because W is bounded and lim inf

|t|→+∞ inf r∈[0,r 0 ] Ψ(t, r) = +∞
we deduce that the infimum in (6.8) is reached at some point (t * , r * ) ∈ R × [0, r 0 ]. Because ε * Ψ ≥ ε * δ > 0 and W ≤ 0 for r = r 0 we deduce that r * ∈ [0, r 0 ). Recall that

W = ε * Ψ -W solves        Wt ≥ A Wr + H(rV r ) Wrr , W ≥ 0 on R × (0, r 0 ), W (t * , r * ) = 0, Wr (t, 0) ≤ 0 for all t ∈ R,
and as a consequence of our assumptions, the functions A and H(rV r ) are continuous on R × (0, r 0 ].

Case 1: r * > 0. Then we can apply the strong maximum principle (see Theorem A.1) and deduce that (6.9)

ε * Ψ = W on (-∞, t * ] × [0, r 0 ],
which is absurd for r = r 0 .

Case 2: r * = 0. If the coefficient A would have been continuous up to r = 0, then we would have applied Hopf lemma (see Lemma A.2) to deduce again (6.9), in order to get the same contradiction.

The difficulty here is that the coefficient A blows-up as r goes to zero. We can easily circumvent this difficulty, if we replace Ψ with Ψ := Ψ -ηr for some η > 0 small enough. Now at the point (t * , 0) of minimum of W = ε * Ψ -W , we get in particular that 0 ≤ Wr (t

* , 0) = -ε * η -W r (t * , 0).
On the other hand, we have by assumption

W r (t * , 0) = (U r -V r )(t * , 0) ≥ 0
which gives a contradiction. Therefore, in all cases, we conclude that ε * = 0, which means that W ≤ 0. This ends the proof of the proposition.

Proposition 6.3 (Comparison principle for large r's). Given some constants λ > 0, δ > 0 and L 0 ≥ 1, there exists r + 0 = r + 0 (δ, L 0 , λ) > 0 such that the following holds for all

r 0 ∈ [r + 0 , +∞). Let U ∈ C 2,1 (R × [r 0 , +∞)) be a subsolution and V ∈ C 2,1 (R × [r 0 , +∞)) be a supersolution of (1.3) on R × (r 0 , +∞), satisfying in R × [r 0 , +∞), (6.10)        -L 0 ≤ U r , V r ≤ -δ, |U (t, r) -λt -Φ 0 (r)| ≤ C, |V (t, r) -λt -Φ 0 (r)| ≤ C, |(Φ 0 ) r (r)| ≤ L 0
for some function Φ 0 and some constant C > 0.

If U ≤ V on R × {r 0 }, then U ≤ V in R × [r 0 , +∞).
Proof of Proposition 6.3. We have:

U t ≤ 1 r G(rU r ) + U r r K(rU r ) + U rr σ 2 (rU r ) and V t ≥ 1 r G(rV r ) + V r r K(rV r ) + V rr σ 2 (rV r )
with

G(a) = 1 + a 2 , K(a) = 2 + a 2 1 + a 2 , σ(a) = 1 √ 1 + a 2 .
By contradiction, assume that

M = sup (t,r)∈R×[r 0 ,+∞) {U (t, r) -V (t, r)} > 0.
For α, η > 0, we set

M α,η = sup t∈R, r,ρ≥r 0 U (t, r) -V (t, ρ) - |r -ρ| 2 2α -α r 2 2 -η t 2 2 which satisfies (6.11) M α,η ≥ M 2 > 0 for α, η small enough.
Since U (t, r) -V (t, ρ) ≤ 2C + Φ 0 (r) -Φ 0 (ρ) ≤ 2C + L 0 |r -ρ| (using the L 0 -Lipschitz property of the profile Φ 0 ), we deduce that this supremum is reached at a point that we denote by (t, r, ρ). It satisfies

η t 2 2 + α r 2 2 ≤ 2C - M 2 + L 0 |r -ρ| - |r -ρ| 2 2α ≤ 2C - M 2 + αL 2 0 2
which in turn implies (for fixed α > 0) (6.12) lim η→0 ηt = 0.

We next distinguish two cases.

Case 1: r, ρ > r 0 . In that case, setting Ũ (t, r) = U (t, r) -α r 2 2 , we get with a = Ũt (t, r), b =

V t (t, ρ), A = Ũrr (t, r), B = V rr (t, ρ) that (6.13) a ≤ 1 r G(rp + αr 2 ) + p + αr r K(rp + αr 2 ) + (A + α)σ 2 (rp + αr 2 ) (6.14) b ≥ 1 ρ G(ρp) + p ρ K(ρp) + Bσ 2 (ρp) a -b = ηt (6.15) A 0 0 -B ≤ 1 α 1 -1 - 1
1
where p := r -ρ α satisfies (using equation (6.10) with p = V r (t, ρ) = Ũr (t, r))

-L 0 ≤ p, p + αr ≤ -δ.
Subtracting (6.14) to (6.13), we get that (6.16)

ηt ≤ I 1 + I 2 + I 3
where

I 1 := 1 r G(rp + αr 2 ) - 1 ρ G(ρp), I 2 := p + αr r K(rp + αr 2 ) - p ρ K(ρp)
and

I 3 := (A + α)σ 2 (rp + αr 2 ) -Bσ 2 (ρp).
Estimate on I 1 . We have

I 1 = 1 r G(rp + αr 2 ) - 1 r G(ρp) + 1 r G(ρp) - 1 ρ G(ρp) ≤G ′ (-r 0 δ) (r -ρ)p r + αr + ρ -r rρ G(ρp) ≤G ′ (-r 0 δ) αp 2 r + αr + ρ -r rρ (1 + ρ|p|) ≤G ′ (-r 0 δ) αp 2 r + αr + α|p| rρ + αp 2 r
where, for the second line, we have used that rp + αr 2 , ρp ≤ -r 0 δ and G ′ is non-decreasing on (-∞, -δr 0 ) and for the third line, we have used that G(a) ≤ 1 + |a|. Choosing r 0 such that G ′ (-r 0 δ) ≤ -1 2 , and such that r 0 ≥ 1 δ ≥ 1 we get that (6.17)

I 1 ≤ - 1 2 αr + 2αL 2 0 .
where we have used that |p| ≤ L 0 and L 0 ≥ 1.

Estimate on I 2 . Using that K is bounded by 2, we have

I 2 ≤ p r K(rp + αr 2 ) - p ρ K(ρp) + 2α ≤ p r K(rp + αr 2 ) -K(ρp) + 2α
where we have used the fact that p ≤ 0, ρ ≥ r. Using now the fact that K is non-decreasing on (-∞, 0) and that 0 ≥ rp + αr 2 ≥ ρp, we get that (6.18) I 2 ≤ 2α.

Estimate on I 3 . Using the matrix inequality (6.15), we have that for all ξ, ζ ∈ R

Aξ 2 ≤ Bζ 2 + (ξ -ζ) 2 α .
Using also that σ is bounded by 1, we get

I 3 ≤ α + 1 α σ(rp + αr 2 ) -σ(ρp) 2 ≤ α + 1 α σ ′ L ∞ (ρp,rp+αr 2 ) ((r -ρ)p + αr 2 ) 2 . Since |σ ′ (a)| ≤ 1 a 2 , we have σ ′ L ∞ (ρp,rp+αr 2 ) ≤ 1 (r(p+αr)) 2 ≤ 1 (rδ) 2 .
Hence we get (6.19)

I 3 ≤ α + 1 α (r -ρ)p + αr 2 (rδ) 2 2 = α + 1 δ 4 α αp 2 r 2 + α 2 ≤ α + 4αL 4 0 δ 4
where for the last inequality, we have used that r ≥ r 0 ≥ 1 and |p| ≤ L 0 with L 0 ≥ 1. Combining (6.16), (6.17), (6.18) and (6.19), we finally get

ηt ≤ - 1 2 αr + 5αL 2 0 + 4αL 4 0 δ 4 .
Taking the limit η → 0 and using (6.12), we get (using L 0 ≥ 1)

0 ≤ - 1 2 αr + 5αL 2 0 + 4αL 4 0 δ 4 which is absurd for r ≥ r 0 > 10L 2 0 + 8L 4 0 δ 4 .
Case 2: r = r 0 or ρ = r 0 . Assume for instance that r = r 0 (the case ρ = r 0 being similar).

Using that M α,η > 0 for α and η small enough, we get that

|r 0 -ρ| 2 2α + α 2 r 2 0 ≤ U (t, r 0 ) -V (t, ρ) ≤ V (t, r 0 ) -V (t, ρ) ≤ L 0 |r 0 -ρ|.
This implies in particular that |r 0 -ρ| ≤ 2αL 0 . Injecting this in the previous inequality, we obtain that α 2 r 2 0 ≤ 2αL 2 0 which is absurd for r 0 > 2L 0 . This ends the proof of the proposition.

Before proving Liouville Theorem 1.8, we first prove Theorem 4.7 that has been used in Subsection 4.3.

Proof of Theorem 4.7. For all ν ∈ R, we define

w ν = Φ 1 -Φ 2 + ν.
In view of (4.15), we can choose ν ≥ 0 big enough so that w ν ≥ 0. We then define ν * = inf{ν ≥ 0 : w ν ≥ 0 in [0, +∞), for all ν ≥ ν}.

We want to show that ν * = 0. By contradiction, let us assume that ν * > 0. Using (4.15), we then have

     w ν * ≥ 0 w ν * (r) > 0 for r large enough inf r∈[0,+∞) w ν * (r) = 0.
From Propositions 6.1 and 6.3, we deduce that we have inf

r∈[r - 0 ,r + 0 ] w ν * (r) = inf r∈[0,+∞)
w ν * (r) = 0 with 0 < r - 0 < r + 0 . Using again the Strong Maximum Principle (Theorem A.1), we deduce that w ν * ≡ 0. For r = +∞, this implies that ν * = 0. Contradition. Therefore ν * = 0 and Φ 1 ≥ Φ 2 . Exchanging Φ 1 and Φ 2 , we get the reverse inequality. This shows that Φ 1 = Φ 2 and ends the proof.

We now prove Theorem 1.8.

Proof of Theorem 1.8. The proof proceeds in several steps.

Step 0: regularity and condition at r = 0. Because U is globally Lipschitz continuous (in space and time), we can apply Proposition 2.2 to conclude that U ∈ C 1,2 (R × [0, +∞)). By continuity in equation (1.3) up to r = 0, we deduce that U satisfies U r (t, 0) = -1 2 for all t ∈ R.

Finally, from Lemma 2.3, we have

|U rr (t, r)| ≤ C(1 + r 2 ) for all (t, r) ∈ R × [0, +∞).
Step 1: preliminaries for the sliding method. We apply the sliding method (see [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF]).

For any h ∈ R, we set U h (t, r) = U (t + h, r).

Since U satisfies (1.10), one can choose b ≥ 0 large enough so that

U h + b ≥ U on R × [0, +∞). We now consider b * = inf{b ∈ R : U h + b ≥ U }
and we set

V := U h + b * ≥ U.
Notice that, using in particular Step 0, we can check that the assumptions of Propositions 6.1 and 6.3 are fulfilled with 0 < r - 0 < r + 0 < +∞ (decreasing r - 0 and increasing r + 0 if necessary). We claim that this implies (6.20) m := inf

(t,r)∈R×[r - 0 ,r + 0 ] (V -U ) = 0.
Indeed, if m > 0, applying Propositions 6.1 and 6.3, we deduce that

V -U ≥ m > 0 on R × [0, +∞)
which contradicts the definition of b * . Therefore (6.20) holds true.

Step 2: consequence. We distinguish two cases.

Case 1: the infimum in (6.20) is reached at (t 0 , r 0 ). We have

V ≥ U on R × [0, +∞), V = U at (t 0 , r 0 ) ∈ R × [r - 0 , r + 0 ]. Notice that W = V -U satisfies W t = AW r + H(rV r )W rr
with A and H defined in (6.5) and (6.3). Moreover A and H(rV r ) are continuous functions because U, V ∈ C 2,1 (R × [0, +∞)).

From the strong maximum principle (Theorem A.1) applied to W , we deduce that

V ≡ U which gives for all k ∈ Z U (t, r) = U (t + h, r) + b * = U (t + kh, x) + kb * .
In view of (1.10), this implies that b * = -λh, i.e.

U (t + h, r) = U (t, r) + λh.

Case 2: the infimum in (6.20) is reached at infinity. We now assume that there exists sequences (t n ) n and r n ∈ [r - 0 , r + 0 ] such that |t n | → +∞, r n → r ∞ ∈ [r - 0 , r + 0 ] and (V -U )(t n , r n ) → m. We define the functions U n (t, r) := U (t + t n , r) -λt n , V n (t, r) = V (t + t n , r) -λt n which have the same Lipschitz constant (in space and time) as the one of U . We can then apply Ascoli-Arzelà Theorem, to deduce that, up to a subsequence, we have

U n → U ∞ , V n → V ∞ , and V ∞ (t, r) = U ∞ (t + h, r) + b *
where U ∞ , V ∞ are two globally Lipschitz solutions of (1.3) on R × (0, +∞) satisfying again

V ∞ ≥ U ∞ on R × [0, +∞), V ∞ = U ∞ at (0, r ∞ ) ∈ R × [r - 0 , r + 0 ].
We can then repeat Step 0 and then case 1 for (U, V ) replaced by (U ∞ , V ∞ ) and get that b * = -λh, and then V ≥ U means (6.21) U (t + h, r) ≥ U (t, r) + λh.

Step 3: conclusion. Notice that (6.21) means that t → U (t, r) -λt is both nondecreasing (using h > 0) and nonincreasing (using h < 0). This implies that U (t, r) -λt = U (0, r).

From (1.11), we have in particular U r (0, r) ≤ 0 and by our assumptions U (0, r) is globally Lipschitz in the variable r. Then Theorem 1.4 i) implies that there exists a constant a ∈ R such that

U (0, r) = Φ(r) + a.
This ends the proof of the theorem.

Long time convergence

In order to prove Theorem 1.6 we need the following proposition, whose proof is postponed.

Proposition 7.1 (Gradient estimate from above). Let T > 0 and let U be a solution of (1.3)-(1.4) in (0, T ) × (0, +∞), such that U is globally Lipschitz continuous with respect to time. Assume that there exists a constant C such that for all (t, r) ∈ (0, T ) × (0, +∞),

(7.1) |U (t, r) -λt -Φ(r)| ≤ C.
If the initial datum U 0 satisfies

(7.2) (U 0 ) r ≤ Φ r in (0, +∞)
then we have U r ≤ Φ r in (0, T ) × (0, +∞). Finally using (1.9) and applying Proposition 7.1, we deduce that

(7.4) U r ≤ Φ r ≤ δ < 0.
Then for any sequence t n → +∞, by Ascoli-Arzelà theorem, we get the convergence (for a subsequence still denoted by (t n ) n ),

U (t + t n , r) -U (t n , 0) → U ∞ (t, r) locally uniformly on compact sets
where U ∞ is still globally Lipschitz continuous and still satisfies (7.3) and (7.4). Therefore the Liouville result (Theorem 1.8) implies that there exists a number a ∈ R such that

U ∞ (t, r) = λt + Φ(r) + a.
This ends the proof of the theorem.

Proof of Proposition 7.1. We have to prove that for r > ρ > 0

U (t, r) -U (t, ρ) ≤ Φ(r) -Φ(ρ).
Using log coordinates and setting u(t, x) = U (t, e x ) and φ(x) = Φ(e x ), this is equivalent to prove that for x > y > -∞ u(t, x) -u(t, y) ≤ φ(x) -φ(y).

Recall that u and λt + φ(x) are both solutions of the following equation

u t = F (x, u x , u xx ) = e -x 1 + u 2 x + e -2x u x + e -2x u xx 1 + u 2 x .
By contradiction, assume that

M = sup x>y, t∈[0,T ) {u(t, x) -u(t, y) -φ(x) + φ(y)} > 0.
For ε, α, η > 0, we consider the following approximate supremum,

(7.5) M ε,α,η = sup x>y, t,s∈[0,T ) u(t, x) -u(s, y) -φ(x) + φ(y) - |t -s| 2 2ε - α 2 x 2 - η T -t .
Remark that when the penalization parameters α and η are small enough, we have

M ε,α,η ≥ M/2 > 0.
From (7.1), we deduce that u(t, x) -u(s, y) -φ(x) + φ(y) is bounded by 2C + λT , and then the supremum in (7.5) is reached at a point that we denote by (t, x, s, y) which satisfies

|t -s| 2 2ε + α 2 x 2 + η T -t ≤ 2C + λT.
We deduce in particular that (7.6) lim α→0 αx = 0.

The proof is divided into two cases.

Case 1: there exists ε n → 0 such that t = 0 or s = 0. Assume for example that t = 0 (if s = 0, a similar reasoning provides the same contradiction). Then we have

η T < u(0, x) -u(s, y) -φ(x) + φ(y) - s 2 2ε ≤ u(0, y) -u(s, y) - s 2 2ε ≤ Ls - s 2 2ε ≤ εL 2 2
where in the second line we have used (7.2) and then used L, which denotes the Lipschitz constant in time of U . This is absurd for ε small enough.

Case 2: for all ε small enough we have t, s > 0. In that case, using that the function

(t ′ , x ′ ) → u(t ′ , x ′ ) -u(s, y) -φ(x ′ ) + φ(y) - |t ′ -s| 2 2ε - α 2 (x ′ ) 2 - η T -t ′
reaches a maximum at (t, x), we deduce that

t -s ε + η (T -t) 2 ≤ F (x, φ x (x) + αx, φ xx (x) + α).
Similarly, we have that t -s ε ≥ F (y, φ x (y), φ xx (y)).

Subtracting these two inequalities, we get

η T 2 ≤F (x, φ x (x) + αx, φ xx (x) + α) -F (y, φ x (y), φ xx (y)) ≤F (x, φ x (x) + αx, φ xx (x) + α) -F (x, φ x (x), φ xx (x)) + F (x, φ x (x), φ xx (x)) -F (y, φ x (y), φ xx (y)) ≤F (x, φ x (x) + αx, φ xx (x) + α) -F (x, φ x (x), φ xx (x)) + λ -λ which gives (7.7) η T 2 ≤ e -x α|x| + e -2x αx + e -2x α + I with I := e -2x φ xx (x) 1 1 + (φ x (x) + αx) 2 - 1 1 + (φ x (x)) 2 .
We write

I := e -2x φ xx (x) J, J := H(φ x (x) + αx) -H(φ x (x)), H(p) := 1 1 + p 2 .
Estimate on J. We observe that the function H is concave in -1 We now define some b > 0 such that

-1 2 √ 3 ≤ φ x (x) ≤ 0 for all x ≤ -b < 0, φ x (x) ≤ -2 √ 3
for all x ≥ b > 0,

We call L 1 the Lipschitz constant of H. Using (7.6), we can assume αx small enough. For instance, for |αx| ≤ 1 2 √ 3 , we deduce from the convexity/concavity property of H that (7.9)

     αx C |φ x (x)| 3 ≥ αxH ′ (φ x (x) + αx) ≥ J ≥ αxH ′ (φ x (x)) ≥ 0 for all x ≥ b > 0, αxL 1 ≤ J ≤ αxH ′ (φ x (x)) ≤ 0 for all x ≤ -b < 0,
where now C > 0 is generic constant that can change from line to line.

Estimate on I. Notice that λt + Φ(r) is a globally Lipschitz continuous solution of (1.3), and then Lemma 2.3 implies the bound (2.2), namely

|Φ rr (r)| ≤ C(1 + r 2 ).
Because Φ rr (e x ) = e -2x φ xx -e -2x φ x , we deduce that

I ≤ e -2x φ x J + C(1 + e 2x )|J|.
We deduce, using (7.8) and (7.9), that I ≤ Ce -x αx for all x ≥ b > 0, Ce -x α|x| + Cα|x| for all x ≤ -b < 0, which can be rewritten as

I ≤ Ce -x α|x| for all |x| ≥ b > 0.
Using (7.7), this leads to (7.10) η T 2 ≤ Ce -x α|x| + e -2x αx for all |x| ≥ b > 0

We now distinguish several cases. Assume first that there exists α → 0 such that x ≤ -b. Increasing b > 0 if necessary, we can assume that Ce -x α|x| + e -2x αx ≤ 0 for all x ≤ -b, which gives a contradiction.

Second, assume that there exists α → 0 such that x ≥ b. For x ≥ b > 0, sending α → 0 in (7.10), we get a contradiction.

Finally, assume that for all α small enough, we have -b ≤ x ≤ b. In that case, we have from (7.7) η T 2 ≤ e -x α|x| + e -2x αx + e -2x α + e -2x |φ xx (x)|L 1 α|x|. Again, sending α → 0, we get a contradiction. This ends the proof of the proposition.

A Appendix 1.1 Strong maximum principle and Hopf lemma

In this subsection, we recall the classical strong maximum principle and Hopf lemma. For -∞ ≤ t 1 < t 2 ≤ +∞ and 0 < R ≤ +∞, let us consider the following general linear parabolic equation (A.1) w t = a(t, r)w rr + b(t, r)w r + c(t, r)w on Q := (t 1 , t 2 ) × (0, R)

with the following assumptions on the coefficients (A.2) a, b, c ∈ C(Q), a ≥ δ > 0 on Q For A = Q or Q, we recall that we say that a function w ∈ C 2,1 (A) if and only if w, w r , w rr , w t ∈ C(A).

Then we have the following classical result.

Theorem A.1 (Strong maximum principle [START_REF] Nirenberg | A strong maximum principle for parabolic equations[END_REF]). Consider a function w ∈ C 2,1 (Q) which is a supersolution of (A.1). If w ≥ 0 on Q, w = 0 at (t 0 , r 0 ) ∈ Q then w ≡ 0 on Q ∩ {t ≤ t 0 }.

We also have (see [START_REF] Lieberman | Second order parabolic differential equations[END_REF]Lemma 2.8]).

Lemma A.2 (Hopf lemma). Consider a function w ∈ C 2,1 (Q) which is a supersolution of (A.1). If w ≥ 0 on Q, w = 0 at (t 0 , 0) ∈ ∂Q with t 0 ∈ (t 1 , t 2 )

then either w ≡ 0 on Q ∩ {t ≤ t 0 } or w r (t 0 , 0) > 0.

Interior Schauder estimate

The following result can be found in Krylov [START_REF] Krylov | Lectures on elliptic and parabolic equations in Hölder spaces[END_REF] (see also [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF][START_REF] Lieberman | Second order parabolic differential equations[END_REF]).

Proposition A.3 (Interior Schauder estimates). Let T > 0, δ > 0, R > 0 and α ∈ (0, 1) and N ≥ 1.

Assume that w solves (in the sense of distributions) with

w t =

A technical lemma

T 1 = (|X ′ | α -|X| α ) X ′ |X ′ | and T 2 = |X| α X ′ |X ′ | - X |X| .
Step 1: estimate on T 1 . We have

(A.3) ||X ′ | α -|X| α | = |X| α |r α -1| with r = |X ′ | |X| ≥ 1.
Case A: 1 ≤ r ≤ 2. We write r = 1 + δ with 0 ≤ δ ≤ 1. Then we have

|r α -1| = αδ + O(δ 2 ) ≤ Cδ α = C|r -1| α .
Case B: r ≥ 2. Then we have (A.4) |r α -1| ≤ C|r -1| α .

Putting together cases A and B, we see that (A.4) holds true for any r ≥ 1. Using (A.3), we get for some C ≥ 1:

(A.5) |T 1 | = ||X ′ | α -|X| α | ≤ C||X ′ | -|X|| α ≤ C|X ′ -X| α .
Step 2: estimate on T Step 3: conclusion. From Steps 1 and 2, we deduce that there exists a constant C > 0 such that

|ζ(X ′ ) -ζ(X)| ≤ C|X ′ -X| α
This last estimate is also true if X = 0. By symmetry between X ′ and X, we see that it is finally true for any X, X ′ ∈ R N . This ends the proof of the lemma.

Equation satisfied by the curvature

The following result is not used in the rest of the paper. We give it as an interesting result of independent interest. which shows the result. This ends the proof of the lemma.

Theorem 1 . 6 (

 16 Long time convergence). Under assumptions of Theorem 1.1, if the initial data U 0 further satisfies(1.8) |U 0 -Φ| ≤ C and (1.9) (U 0 ) r ≤ Φ r ≤ -λ < 0,

Lemma 4 . 2 (

 42 Positivity of λ). Under the assumptions of Proposition 4.1, we have λ > 0.

Proposition 4 . 3 (

 43 Asymptotics near r = +∞). Under the assumptions of Proposition 4.1, the function ϕ(x) = Φ(e x ) satisfies

√ 3 , 1 √ 3

 313 and convex outside. Recalling (1.6), we also see that(7.8) e -x φ x (x) = Φ r (e x ) ∈ [-1, -λ].

  a∆w + b on (T -δ, T + δ) × B R with B R the ball of radius R in R N . Assume that a, b ∈ C α 2 ,α t,x ((T -δ, T + δ) × B R ) with for some η > 0: 0 < η ≤ a ≤ 1/η on (T -δ, T + δ) × B R ((T -δ,T +δ)×BR) ≤ C 0 .Then there exists a constant C = (δ, R, α, N, η, C 0 ) > 0 such thatw ([T,T +δ)×B R/2 ) ((T -δ,T +δ)×BR)+ |w| L ∞ ((T -δ,T +δ)×BR) .

Lemma A. 4 (

 4 A Hölder estimate). Let α ∈ (0, 1) and N ≥ 1. For X ∈ R N , let us define the function Then there exists a constant C = C(α) > 0 such that for all X ′ , X ∈ R N , we have|ζ(X ′ ) -ζ(X)| ≤ C|X ′ -X| α . Proof of Lemma A.4. Let us assume that |X ′ | ≥ |X| > 0. We write ζ(X ′ ) -ζ(X) = T 1 + T 2

2 .(

 2 Writing e = X |X| , Y = X ′ |X| with |Y | ≥ 1, and using the fact that the map Z → Z |Z| is 1-Lipschitz (for the euclidean norm) on R N \B(0, 1), we get that e| = |X ′ -X| |X| .Case A: |X| ≤ |X ′ | ≤ 2|X|. Using (A.6), we deduce that|T 2 | ≤ |X| α |X ′ -X| |X| = |X ′ -X| 1-α |X| 1-α |X ′ -X| α ≤ 2 1-α |X ′ -X| |X ′ | 1-α |X ′ -X| α which implies (A.7) |T 2 | ≤ 4 1-α |X ′ -X| α .Case B: |X ′ | ≥ 2|X|. We have|T 2 | ≤ |X| α ≤ ||X ′ | -|X|| α ≤ |X ′ -X| αPutting together cases A and B, we see that (A.7) holds true for any |X ′ | ≥ |X| > 0.

Lemma A. 5 (+ u 2 x+ κ 2 ( 1 + 2 x 1 + u 2 x+ e x u x 1 + u 2 x. 2 . 1 ( 1 + a 2 ) 3 2 .-x 1 + u 2 x( 1 +x u x 1 + u 2 x - 1 + κ x 1 + u 2 x= 1 + u 2 x= 2 + κ xx 1 + u 2 x+ ( 1 ++ u 2 x+ ( 1 + κ)κ 2 +e

 522121222112121211212221212 Equation satisfied by the curvature). Let Φ be the profile given by Theorem 1.4. The curvature κ(x) = κ Φ (e x ) solves the following equation(A.8) κ t = e -2x κ xx 1 κ) + e -2x κ x -1 + 2uProof of Lemma A.5. We start fromu t = e -x 1 + u 2 x (1 + κ) Let us define M (a) = a √ 1 + a 2 with M ′ (a) = Then we can write κ = e -2x (e x M (u x )) x and κ t = e -2x (e x M ′ (u x )u xt ) x .We now computeu xt = e -x 1 + u 2 x (κ x -(1 + κ)) + e κ)u x u xx = e -x 1 + u 2 x (κ x -(1 + κ)) + e -x 1 + u 2 x (1 + κ)u x u xx 1 + u 2 x + u x -u x = e -x 1 + u 2 x κ x -e -x 1 + u 2 x (1 + κ)(1 + u 2 x ) + (1 + u 2 x )(1 + κ)κu x = (1 + κ) (1 + u 2 x )κu x -e -x (1 + u 2 x )This givese 2x κ t = ∂ x (1 + κ) κ e ∂ x (1 + κ) {κe x M (u x ) -1} + κ x κ x κe x M (u x ) -1 + (1 + κ)e x M (u x ) -2u x u xx (1 + u 2 x ) κ)κe 2x κ.Thereforeκ t = e -2x κ xx 1 -2x κ x {-1 + (1 + 2κ)e x M (u x ) -2e x M (u x )(κ -e -x M (u x ))} = e -2x κ xx 1 + u 2x + (1 + κ)κ 2 + e -2x κ x -1 + e x M (u x ) + 2(M (u x )) 2

  is satisfied in particular if 2α 2 ≤ µ 2 and 8µ 2 ≤ 1.

	For instance for	µ = 1/(2 √	2) and α = 1/4,

we conclude from (5.13) that λ ≥ 1/4. This ends the proof of the lemma. Proof of Theorem 1.4. Apart from (1.7), Theorem 1.4 is then a consequence of Propositions 3.1, 4.1, 5.1, 5.2 and Lemma 5.3. As far as (1.7) is concerned, it is a simple consequence

  Proof of Theorem 1.6. By Theorem 2.1, there exists a unique solution U to (1.3), (1.4) which is globally Lipschitz continuous (in space and time). Notice that λt+Φ(r) is a global solution. Therefore, using (1.8) and applying the comparison principe (see[START_REF] Forcadel | Uniqueness and existence of spirals moving by forced mean curvature motion[END_REF] Theorem 1.3]), we deduce the following estimate for all times,

	(7.3)	|U (t, r) -λt -Φ(r)| ≤ C

2 + e -x 1 + u 2 x κ x .

Acknowledgements. The authors would like to thank the referee for the careful reading of the paper, his/her corrections of some mistakes and for some indications on the literature. This work is partially supported by the ANR projects HJnet ANR-12-BS01-0008-01, AMAM ANR 10-JCJC 0106 and IDEE ANR-2010-0112-01. R.M. thanks R.V. Kohn for indications on the literature. N.F. and R.M. thank the conference center of Oberwolfach for providing them excellent research conditions during the preparation of this work. They also thank the organizers of the meeting "Interfaces and Free Boundaries: Analysis, Control and Simulation" in 2013, for the invitation to participate and present their works. R.M. also thank Y. Giga for an invitation to a meeting in Sapporo in 2010 and stimulating and enlighting discussions about the problem studied in this paper.


Julien Petillon
email: julien.petillon@univ-rennes1.fr

Simon Potier

Alexandre Carpentier

Angus Garbutt

Julien Pétillon

Evaluating the success of managed realignment for the restoration of salt marshes: Lessons from invertebrate communities

Keywords: restoration, intertidal systems, macro-arthropods, Essex

come   L'archive ouverte pluridisciplinaire

Introduction

Salt marshes are intertidal ecosystems, located at the interface between marine and terrestrial systems. They are among the most surface-restricted habitats in the world, dramatically declining in Europe where they currently have a high interest in terms of nature conservation, especially in relation to global change and sea-level rise. Salt marshes make excellent systems to demonstrate both fundamental and applied ecological principles. They have strong directional and predictable physical and ecological gradients and a limited species pool of specialist fauna and flora. Salt-marsh dynamics are driven by the frequency of tidal inundation and sediment supply. When there is net accumulation of sediment, vegetation communities respond driving succession, defined as a non-seasonal, directional pattern of plant species change (e.g. [START_REF] Adam | Saltmarsh ecology[END_REF]). Succession mechanisms have been extensively studied for plants. Without the impact of herbivores, salt marshes move towards typical late successional stages over time [START_REF] Kuijper | Herbivory and competition slow down invasion of a tall grass along a productivity gradient[END_REF]). Grazing by both small herbivores (mainly hare and geese) and large herbivores (i.e., cattle, sheep) retards succession. This general scheme is in reality much more complex and factors acting on plant succession are numerous and produce many feedbacks [START_REF] Olff | Vegetation succession and herbivory in a salt marsh: changes induced by sea level rise and silt deposition along an elevational gradient[END_REF]. Although complex, there has been much research carried out on plant assemblages along successional schemes, and this is in contrast with work on invertebrates.

Self-sustaining plant communities are often a primary goal of habitat creation efforts as they can perform biological and economically desirable functions within wetland ecosystems.

There is little evidence however that shows similar trends in assembly rule between plants and other groups. Dispersal is generally not considered to be a limiting factor for plant reassembly, due to the general connectivity of the system through tidal flooding [START_REF] Wolters | Plant colonization after managed realignment: the relative importance of diaspore dispersal[END_REF][START_REF] Erfanzadeh | Factors affecting the success of early salt-marsh colonizers: seed availability rather than site suitability and dispersal traits[END_REF]. In contrast, dispersal has been proved to be a critical element in arthropod assemblage structuring (e.g. [START_REF] Varet | Age-dependent colonization of urban habitats: a diachronic approach using carabid beetles and spiders[END_REF], and few comparisons between plants and arthropod succession can be made in European salt marshes. Late successional stages of vegetation development are characterised by low diversity Elytrigia atherica swards [START_REF] Kuijper | Herbivory and competition slow down invasion of a tall grass along a productivity gradient[END_REF]. Conversely, late successional spider communities generally show an increase in diversity and in density [START_REF] Pétillon | Impact of an invasive plant (Elymus athericus) on the conservation value of tidal salt marshes in western France and implications for management: responses of spider populations[END_REF]. Mechanisms and processes of succession in salt marshes thus largely differ between plants and other groups of animals.

Current intertidal habitat creation schemes and sites where, in the past, large storm events led to accidental breaches in embankments resulted in the re-instatement of tidal inundation and salt-marsh development have created unique experimental conditions for studying salt-marsh successions, and critical study sites for restoration ecology (e.g. [START_REF] Teal | Ecological engineering, adaptive management, and restoration management in Delaware Bay salt marsh restoration[END_REF].

"Historic" sites can be used as analogues for modern day managed realignment and, along with current managed realignment sites, give space for time examples of saltmarsh development on former agricultural land [START_REF] Hinkle | Salt marsh vegetation recovery at salt hay farm wetland restoration sites on Delaware Bay[END_REF].

One recent study, based on vegetation analysis, showed that restoration of tidal inundation to former agricultural land remains insufficient measures to ensure "equivalent biological characteristics" to natural reference sites [START_REF] Mossman | Does managed coastal realignment create saltmarshes with "equivalent biological characteristics" to natural reference sites?[END_REF]. We conducted a similar survey at a subset of the same sites (pair-matched natural and recreated, either accidentally or by managed realignment, salt marshes) on invertebrate communities in a field experiment in UK, and assessed both the conservation (condition at a given time) and functional (proxy of process over time) equivalencies (sensu [START_REF] Zedler | Functional equivalency of restored and natural salt marshes[END_REF] between recreated and natural salt marshes.

The conservation value of salt marshes was estimated through two classical conservation criteria, i.e. abundance of halophilic species and species richness. In salt marshes spiders are the most abundant and diversified of terrestrial macro-arthropods, and host some specialist, halophilic, species, with behavioural and physiological adaptations to flood and saline stress (e.g. [START_REF] Foucreau | Effects of diet and salinity on the survival, egg laying and metabolic fingerprints of the ground-dwelling spider Arctosa fulvolineata (Araneae, Lycosidae)[END_REF]. Trophic guilds were used to evaluate the functional restoration of invertebrate communities (e.g. [START_REF] Gratton | Restoration of arthropod assemblages in a Spartina salt marsh following removal of the invasive plant Phragmites australis[END_REF]. One more particular focus was done on amphipods because of their role as decomposers (thereby controlling the magnitude of organic matter outwelling toward marine systems) and as main food item for juveniles of several fish species (nursery function of salt marshes: e.g. [START_REF] Laffaille | Does the invasive plant Elymus athericus modify fish diet in tidal salt marshes?[END_REF].

Material and Methods

Study sites

We selected 20 study sites varying in their restoration age (including 4 managed realigned sites coded MR and aged 3 to 15 years, and 6 accidentally realigned sites coded AR and aged from 52 to 107 years), allowing 11 comparisons natural (coded Reference) /recreated salt marshes (for a full description of the sites, see [START_REF] Garbutt | The natural regeneration of salt marsh on formerly reclaimed land[END_REF]. We were not interested here in the comparison AR vs MR per se, but instead in assessing the influence of age since restoration on the recolonization process of salt-marsh invertebrates. Sites were therefore considered as forming a continuum over time (chrono-sequence approach). Natural and recreated sites had 100m-study transect at the same elevation, and located along the Blackwater Estuary (Essex, UK). All MR/AR sites were under agricultural use prior to reexposure to tidal flooding. The Essex marshes are amongst the oldest in the UK, probably developed from the Holocene period after the last ice age.

Sampling design

Pitfall traps (80mm in diameter and 105mm deep), containing a solution of 50:50 water/ethylene-glycol, were inserted into the ground 10 meters apart to avoid interference and pseudo-replication [START_REF] Topping | Limitations to the use of pitfall traps in ecological studies exemplified by a study of spiders in a field of winter wheat[END_REF], ensuring that the rim of the trap was at ground level. Ten pitfall traps were set along each transect and were active 2 times 3-days (i.e. when tides permitted sampling) during June 2005. We used a large number of traps (220*2 sessions in total) to increase spatially the sampling effort. Thirty sweep net samples were taken along each of the three 30m lengths of the sampling transect (i.e. approximately 1 sweep/m) to give three samples per transect. Four 20 second suction samples by Vortis (diameter: 16cm) were taken within a 1m² area, centred on the pitfall traps. The three 1m² sampling areas were selected by randomisation of the number of pitfall traps (from1 to 10).

Both sweep netting and vortexing were carried out during August 2005 (at low tide).

Percentage cover of each plant species (and bare soil) and litter depth (to the nearest mm) were finally estimated within 1m² around each trap at the same time.

Data analysis

Data from pitfall traps were used to study ground-dwelling spider assemblages. The target species were halophilic species, defined by their preference or exclusive presence in salt marsh habitats, which can be assessed using distribution maps and habitat preferences [START_REF] Pétillon | Impact of an invasive plant (Elymus athericus) on the conservation value of tidal salt marshes in western France and implications for management: responses of spider populations[END_REF]. The fraction of spider species pool was calculated for the total number of individuals and species richness (of all species and that of halophilic species, for the two parameters), and analysed over time using Spearman's rank correlation. The fraction of the population pool was then calculated for the three most abundant spider species (here more than 80 individuals) and analysed over time using Spearman's rank correlation. ANOSIM (using Bray-Curtis similarity index) and Non-metric Multi-dimensional Scaling (NMDS) were also used to test for differences in spider assemblages among salt-marsh types.

Samples from individual sites were grouped by marsh types into MR, AR and reference marsh data sets. All invertebrates collected by the three sampling methods were assigned to trophic guilds. For amphipods, only data from pitfall traps were used because the activity-densities of Orchestia sp. obtained through this method can be used a proxy of biomasses of that arthropod [START_REF] Mantzouki | Accuracy of pitfall traps for monitoring populations of the amphipod Orchestia gammarella (Pallas 1766) in a salt marsh[END_REF]). Abundances of amphipods, number of individuals per guild and habitat characteristics were compared among classes of habitat with GLMM (Generalized Linear Mixed Model). Fixed effect was the marsh type (MR, AR, Reference) and the random effect was the study sites. In the absence of significant random effect, the statistic used was a GLM (Generelized Linear Model). Indval was calculated to estimate the fidelity of plant species to the different types of habitats (i.e. natural salt marshes vs polders, AR vs MR; Dufrêne and Legendre, 1997; 1000 permutations).

All analyses were performed in R 2.14.2 (R Development Core Team 2012, see the full list of packages in ESM: Table A.1). Throughout the paper, means are presented ± se and significance was assumed for P<0.05.

Results

A total of 27180 invertebrates was sampled, of which 6418 were collected by Vortis, 5915 by sweep net and 14847 by pitfall trap. Amphipods and spiders constituted the most abundant and diversified groups of macro-arthropods respectively, with 9666 individuals (Orchestia sp.) and 42 spider species (see taxonomic list in ESM: Table A.2).

Conservation equivalency

Spider species richness significantly decreased over time since reintroduction of tidal flooding (Spearman: P<0.001; Rho=-0.34; Fig. 1a). No correlation was found between time since restoration and fraction of halophilic species pool (Spearman: P=0.401; Fig. 1b), fraction of activity-density of all species (Spearman: P=0.183; Fig. 1c) and fraction of activity-density of halophilic species (Spearman: P=0.205; Fig. 1d). Spider species composition of MR sites was significantly different from that of reference and accidentally realigned sites (ANOSIM, R=0.265, P<0.001 and R=0.227, P<0.001, respectively; see also the NMDS in Fig. A.1 of ESM). The difference in spider species composition between AR and Reference sites was less obvious (R=0.029, P<0.001). Abundance of Pardosa prativaga (Fig. 2b; Spearman: P=0.033, Rho=-0.20) and Pardosa purbeckensis (Fig. 2c; Spearman: P<0.001, Rho=-0.31) decreased over time while that of Pirata piraticus increased in the meantime (Fig. 2a; Spearman: P<0.001, Rho=0.37).

Functional equivalency

Abundance of predator and phytophagous species decreased from managed realignment to reference salt marshes (Fig. 3a,c). The number of polyphagous arthropods was higher in managed realignment salt marsh compared to accidental one, with an intermediate value for reference salt marsh (Fig. 3b). Abundance of detritivores was significantly higher in reference sites, with a significantly lower mean in MR compared to AR sites (Fig. 3d). Vegetation height decreased from managed realignment to reference salt marsh (Fig. 3e). Litter depth significantly differed between the three classes of salt marsh, with a maximum mean in AR sites and a minimum in MR sites (Fig. 3f). Abundance of Orchestia gammarella increased from managed realignements to reference sites (Fig. 4). Several plant species, Atriplex portulacoides, Bostrychia scorpioides, Cochlearia officinalis, Limonium vulgare, Puccinellia maritima, Spergularia media and Triglochin maritima, were indicator of reference salt marshes (IndVal=64. 75,IndVal=19.13,IndVal=28.00,IndVal=29.08,IndVal=56.41,IndVal=18.84,and IndVal=21.99, respectively; P=0.017) compared to MR and AR sites.

Only Spartina anglica was characteristic of all realigned sites (IndVal=37.40, P=0.017), but it was significant in MR sites (IndVal=32.82, P=0.017) only (i.e. not in AR sites alone).

Discussion

The conservation equivalency between managed realigned, accidentally realigned and reference sites was reached for spiders, after only a few (3-15) years of succession. All conservation parameters were similar between sites, except the diversity of spider species which decreased over time. This is in contrast with plants, for which managed realignment did not lead to the restoration of equivalent halophytic communities [START_REF] Mossman | Does managed coastal realignment create saltmarshes with "equivalent biological characteristics" to natural reference sites?[END_REF], with high percentages of target species lacking [START_REF] Garbutt | The natural regeneration of salt marsh on formerly reclaimed land[END_REF]. The same pattern is observed in this study with the only characteristic plant species of restored (either by managed realignment or accidentally, the pioneer and invasive Spartina anglica) sites different from typical, target, species from natural salt marshes (mainly Atriplex portulacoides). The great long-distance dispersal abilities of spiders likely explain the early colonization of realigned sites by halophilic species, with a possible fast colonization of most sites from the regional species pool [START_REF] Pétillon | Success of managed realignment for the restoration of salt-marsh biodiversity: preliminary results on ground-active spiders[END_REF][START_REF] Cristofoli | Spider communities as evaluation tools for wet heathland restoration[END_REF]. These dispersal capacities are probably higher than those of ground beetles [START_REF] Varet | Age-dependent colonization of urban habitats: a diachronic approach using carabid beetles and spiders[END_REF], group in which incomplete restoration has been shown after dozen of years in littoral habitats [START_REF] Desender | Ground beetles as "early warning-indicators" in restored salt marshes and dune slacks[END_REF]. Studying recolonization of restored sites by ground beetles could therefore be interesting in the future, although they are much less abundant and diversified than spiders there (number of adults and species 5 and 4 times lower respectively: Pétillon and Garbutt, unpublished data). In salt marshes, long-distance dispersal of certain ground beetles can be reduced in old and large sites due to an increase in the percentage of apterous specimens [START_REF] Desender | Age and size of European saltmarshes and the population genetic consequences for ground beetles[END_REF]. Spiders are also less dependent on soil properties than plants. The role of habitat structure in itself has indeed been proved repetitively in determining species richness of spiders (more than age of habitats for example: [START_REF] Hurd | Cursorial spiders and succession: age or habitat structure?[END_REF], and this parameter is overall constant in the sites we studied [START_REF] Garbutt | Monitoring the development of intertidal habitats on former agricultural land after the managed realignment of coastal defences at Tollesbury, Essex, UK[END_REF]. In contrast, enhanced pH and salinity and reduced oxygenation and porosity are likely to delay plant succession [START_REF] Mossman | Does managed coastal realignment create saltmarshes with "equivalent biological characteristics" to natural reference sites?[END_REF], and also to reduce litter decomposition (as suggested by both the low number of detritivores and a deep litter layer in restored sites). Although the species richness and total number of individuals were overall constant over time, assemblage composition significantly varied through years, tending to be closer to assemblages from natural salt marshes over time from restoration (results from Anosim). Exclusive competition probably occurred between spider species, as suggested here by the opposite population patterns over time of the three dominant lycosid species (also stressed in [START_REF] Pétillon | Success of managed realignment for the restoration of salt-marsh biodiversity: preliminary results on ground-active spiders[END_REF].

Using a space-for-time approach to saltmarsh development upon re-exposure to tidal flooding, we deduce a clear shift in functional equivalency from managed realigned to natural salt marshes through accidentally realigned sites, and most of the parameters studied exhibited a graduate, either increasing or decreasing response to time. Our main results fit well to the assembly rule found by [START_REF] Schrama | Ecosystem assembly rules: the interplay of green and brown webs during salt marsh succession[END_REF] in an European salt-marsh, through a chronosequence of 100 years. They indeed described that, at the beginning of the succession, most incoming fluxes of organic matter and decomposers are mostly marine, whereas the importance of terrestrial matter and invertebrates increases over time. Yet, we did not observe an increase of vegetation height over time, but the inverse, mainly due to the dominance of Spartina anglica in early stages. The number of amphipods, even accounting for the increase between managed realigned and accidentally realigned sites, was much lower (more than 3 times) in restored than in natural salt marshes. This can be explained by reduced content of organic matter [START_REF] Garbutt | Monitoring the development of intertidal habitats on former agricultural land after the managed realignment of coastal defences at Tollesbury, Essex, UK[END_REF]) and could have strong implications for the food provisioning ability of regenerated marshes for fish species, and also a likely reduced export of organic matter toward marine adjacent ecosystems (as amphipods represent key-component in the decomposition of organic matter: [START_REF] Graça | Grazing rates of organic matter and living fungal biomass of decaying Spartina alteniflora by three species of salt-marsh invertebrates[END_REF].

As a conclusion, this study underlines the necessity to study both conservation and functional traits of restored systems, and to multiply monitoring with different model groups even when they are time-costly to sample and identify. Indeed, conservation equivalency can be quickly achieved (which is shown here with spider assemblages), but that does not translate into a complete functional equivalency (which was also stressed by [START_REF] Zedler | Functional equivalency of restored and natural salt marshes[END_REF] for the restoration of salt marshes in USA). In this study, neither the salt-marsh trophic structure, nor the potential role of marine enrichment and fish nursery (estimated through the population size of amphipods) were achieved by managed realignments. We finally urge the scientific community to consider managed realignments as excellent study systems for applied and fundamental purpose (assuming a long chrono-sequence), with a particular attention to less-studied taxa like terrestrial arthropods.

Fig. A. 1 .

 1 Fig. A.1. Non-metric multi-dimensional scaling ordination of spider assemblages from

Figure captions Fig. 1 .

 captions1 Figure captions

Fig. 2 .

 2 Fig. 2. Fraction of spider population pool that have colonized realigned saltmarshes in relation

Fig. 3 .

 3 Fig. 3. Comparisons of total number of individuals per guild and habitat characteristics among

Fig. 4 .

 4 Fig. 4. Comparisons of total number of amphipods among classes of habitat. Different

Figure 1 .

 1 Figure 1. Pétillon et al.

FigureFigure 4 .

 4 Figure 2. Pétillon et al.

Acknowledgments

We thank Peter Merett for the identification of problematic specimens, Pete Nuttall and Sarah Hulmes for sorting the samples for further identification, Boris Leroy, Maxime Hervé and two anonymous referees for constructive comments.

Supporting information

Electronic Supplementary Material may be found in the online version of this article under the DOI reference: