Nicolas Champagnat
email: nicolas.champagnat@sophia.inria.fr

Pierre-Emmanuel Jabin
email: jabin@unice.fr

Well posedness in any dimension for Hamiltonian flows with non BV force terms

Keywords: MSC 2000 subject classifications: 34C11, 34A12, 34A36, 35L45, 37C10 Flows for ordinary differential equations, Kinetic equations, Stability estimates

We study existence and uniqueness for the classical dynamics of a particle in a force field in the phase space. Through an explicit control on the regularity of the trajectories, we show that this is well posed if the force belongs to the Sobolev space H 3/4 .

Introduction

This paper studies existence and uniqueness of a flow for the equation

∂ t X(t, x, v) = V (t, x, v), X(0, x, v) = x, ∂ t V (t, x, v) = F (X(t, x, v)), V (0, x, v) = v, (1.1)
where x and v are in the whole R d and F is a given function from R d to R d . Those are of course Newton's equations for a particle moving in a force field F . For many applications the force field is in fact a potential

F (x) = -∇φ(x),
even though we will not use the additional Hamiltonian structure that this is providing. This is a particular case of a system of differential equations

∂ t Ξ(t, ξ) = Φ(Ξ), (1.2)
with Ξ = (X, V ), ξ = (x, v), Φ(ξ) = (v, F (x)). Cauchy-Lipschitz' Theorem applies to (1.1) and gives maximal solutions if F is Lipschitz. Those solutions are in particular global in time if for instance F ∈ L ∞ . Moreover because of the particular structure of Eq. (1.1), this solution has the additional Property 1 For any t ∈ R the application

(x, v) ∈ R d × R d → (X(t, x, v), V (t, x, v)) ∈ R d × R d (1.3)
is globally invertible and has Jacobian 1 at any (x, v) ∈ R d × R d . It also defines a semi-group ∀s, t ∈ R, X(t + s, x, v) = X(s, X(t, x, v), V (t, x, v)), and V (t + s, x, v) = V (s, X(t, x, v), V (t, x, v)).

(1.4)

In many cases this Lipschitz regularity is too demanding and one would like to have a well posedness theory with a less stringent assumption on F . That is the aim of this paper. More precisely, we prove Theorem 1.1 Assume that F ∈ H 3/4 ∩ L ∞ . Then, there exists a solution to (1.1), satisfying Property 1. Moreover this solution is unique among all limits of solutions to any regularization of (1.1).

Many works have already studied the well posedness of Eq. (1.2) under weak conditions for Φ. The first one was essentially due to DiPerna and Lions [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], using the connection between (1.2) and the transport equation

∂ t u + Φ(ξ) • ∇ ξ u = 0. (1.5)
The notion of renormalized solutions for Eq. (1.5) provided a well posedness theory for (1.2) under the conditions Φ ∈ W 1,1 and div ξ Φ ∈ L ∞ . This theory was generalized in [START_REF] Lions | Sur les équations différentielles ordinaires et les équations de transport[END_REF], [START_REF] Bris | Renormalized solutions of some transport equations with partially W 1,1 velocities and applications[END_REF] and [START_REF] Hauray | Deux remarques sur les flots généralisés d'équations différentielles ordinaires[END_REF]. Using a slightly different notion of renormalization, Ambrosio [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF] obtained well posedness with only Φ ∈ BV and div ξ Φ ∈ L ∞ (see also the papers by Colombini and Lerner [START_REF] Colombini | Uniqueness of continuous solutions for BV vector fields[END_REF], [START_REF] Colombini | Uniqueness of L ∞ solutions for a class of conormal BV vector fields[END_REF] for the BV case). The bounded divergence condition was then slightly relaxed by Ambrosio, De Lellis and Malỳ in [START_REF] Ambrosio | On the chain rule for the divergence of vector fields: applications, partial results, open problems[END_REF] with only Φ ∈ SBV (see also [START_REF] Lellis | Notes on hyperbolic systems of conservation laws and transport equations[END_REF]).

Of course there is certainly a limit to how singular Φ may be and still provide uniqueness, as shown by the counterexamples of Aizenman [START_REF] Aizenman | On vector fields as generators of flows: A counterexample to Nelson's conjecture[END_REF] and Bressan [START_REF] Bressan | An ill posed Cauchy problem for a hyperbolic system in two space dimensions[END_REF]. The example by De Pauw [START_REF] Pauw | Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d'un hyperplan[END_REF] even suggests that for the general setting (1.2), BV is probably close to optimal.

But as (1.1) is a very special case of (1.2), it should be easier to deal with. And for instance Bouchut [START_REF] Bouchut | Renormalized solutions to the Vlasov equation with coefficients of bounded variation[END_REF] got existence and uniqueness to (1.1) with F ∈ BV in a simpler way than [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF]. Hauray [START_REF] Hauray | On Liouville transport equation with force field in BV loc[END_REF] handled a slightly less than BV case (BV loc ).

In dimension d = 1 of physical space (dimension 2 in phase space), Bouchut and Desvillettes proved well posedness for Hamiltonian systems (thus including (1.1) as F is always a derivative in dimension 1) without any additional derivative for F (only continuity). This was extended to Hamiltonian systems in dimension 2 in phase space with only L p coefficients in [START_REF] Hauray | On two-dimensional Hamiltonian transport equations with L p loc coefficients[END_REF] and even to any system (non necessarily Hamiltonian) with bounded divergence and continuous coefficient by Colombini, Crippa and Rauch [START_REF] Colombini | A note on two-dimensional transport with bounded divergence[END_REF] (see also [START_REF] Colombini | Uniqueness in the Cauchy problem for transport in R 2 and R 1+2[END_REF] for low dimensional settings and [START_REF] Bouchut | One dimensional transport equation with discontinuous coefficients[END_REF] with a very different goal in mind).

Unfortunately in large dimensions (more than 1 of physical space or 2 in the phase space), the Hamiltonian or bounded divergence structure does not help so much. To our knowledge, Th. 1.1 is the first result to require less than one derivative on the force field F in any dimension. Note that the comparison between H 3/4 and BV is not clear as BV ⊂ H 3/4 and H 3/4 ⊂ BV . Even if one considers the stronger assumption that the force field be in L ∞ ∩ BV , that space contains by interpolation H s for s < 1/2 and not H 3/4 . As the proof of Th. 1.1 uses orthogonality arguments, we do not know how to work in spaces non based on L 2 norms (W 3/4,1 for example). Therefore strictly speaking Th. 1.1 is neither stronger nor weaker than previous results.

We have no idea whether this H 3/4 is optimal or in which sense. It is striking because it already appears in a question concerning the related Vlasov equation

∂ t f + v • ∇ x f + F • ∇ v f = 0. (1.6)
Note that this is the transport equation corresponding to Eq. (1.1), just as Eq. (1.5) corresponds to (1.2). As a kinetic equation, it has some regularization property namely that the average

ρ(t, x) = R d f (t, x, v) ψ(v) dv, with ψ ∈ C ∞ c (R d ),
is more regular than f . And precisely if f ∈ L 2 and F ∈ L ∞ then ρ ∈ H 3/4 ; we refer to Golse, Lions, Perthame and Sentis [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] for this result, DiPerna, Lions, Meyer for a more general one [START_REF] Diperna | L p regularity of velocity averages[END_REF], or [START_REF] Jabin | Averaging Lemmas and Dispersion Estimates for kinetic equations[END_REF] for a survey of averaging lemmas. Of course we do not know how to use this kind of result for the uniqueness of (1.6) or even what is the connection between the H 3/4 of averaging lemmas and the one found here. It could just be a scaling property of these equations. Note in addition that the method chosen for the proof may in fact be itself a limitation. Indeed it relies on an explicit control on the trajectories : for instance, we show that

|X(t, x, v)-X δ (t, x, v)| and |V (t, x, v)-V δ (t, x, v)| remain approximately of order |δ| if δ = (δ 1 , δ 2 ) and X δ (t, x, v) = X(t, x + δ 1 , v + δ 2 ), V δ (t, x, v) = V (t, x + δ 1 , v + δ 2 ).
However the example given in Section 3 demonstrates that such a control is not always possible: Even in the physical space is one-dimensional, it requires at least 1/2 derivative on the force term (F ∈ W 1/2,1 loc ) whereas well posedness is known with essentially F ∈ L p (see the references above). Note moreover that if X, V satisfies (1.3) and (1.4), then X δ and V δ satisfy (1.3) but not (1.4). However, the functions

X δ (t, x -δ 1 , v -δ 2 ) and V δ (t, x - δ 1 , v -δ 2 ) of course satisfy (1.4).
This kind of control is obviously connected with regularity properties of the flow (differentiability for instance), which were studied in [START_REF] Ambrosio | Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow[END_REF] (see also [START_REF] Ambrosio | Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields[END_REF]). The idea to prove them directly and then use them for well posedness is quite recent, first by Crippa and De Lellis in [START_REF] Crippa | Estimates and regularity results for the DiPerna-Lions flow[END_REF] with the introduction and subsequent bound on the functional

Ω sup r |δ|≤r log 1 + |Ξ(t, ξ) -Ξ(t, ξ + δ)| |δ| dδ dx. (1.7)
This gave existence/uniqueness for Eq. (1.2) with Φ ∈ W 1,p loc for any p > 1 and a weaker version of the bounded divergence condition. This was extended in [START_REF] Bouchut | Uniqueness, renormalization, and smooth approximations for linear transport equations[END_REF] and [START_REF] Jabin | Differential Equations with singular fields[END_REF].

We use here a modified version of (1.7) which takes the different roles of x and v into account. The way of bounding it is also quite different as we essentially try to integrate the oscillations of F along a trajectory.

The paper is organized as follows: The next section introduces the functional that is studied, states the bounds that are to be proved and briefly explains the relation with the well posedness result Th. 1.1. The section after that presents the example in 1d and the last and longer section the proof of the bound.

Notation

• u • v denotes the usual scalar product of u ∈ R d and v ∈ R d .

• S d-1 denotes the (d -1)-dimensional unit sphere in R d .

• B(x, r) is the closed ball of R d for the standard Euclidean norm with center x ∈ R d and radius r ≥ 0.

• C denotes a positive constant that may change from line to line.

2 Preliminary results

Reduction of the problem

In the sequel, we give estimates on the flow of Eq. (1.1) for initial values

(x, v) in a compact subset Ω = Ω 1 × Ω 2 of R 2d and for time t ∈ [0, T ]. Fix some A > 0 and consider any F ∈ L ∞ with F L ∞ ≤ A.
Then for any solution to Eq. (1.1)

|V (t, x, v) -v| ≤ F L ∞ t ≤ A t and |X(t, x, v) -x| ≤ vt + F L ∞ t 2 /2 ≤ vt + A t 2 /2.
Therefore, for any t ∈ [0, T ] and for any (x, v) at a distance smaller than 1 from Ω, (X(t,

x, v), V (t, x, v)) ∈ Ω ′ = Ω ′ 1 × Ω ′ 2 for some compact subset Ω ′ of R 2d .
Moreover Ω ′ depends only on Ω and A. Similarly, we introduce Ω ′′ a compact subset of R 2d such that the couple (X(-t, x, v), V (-t, x, v)) belongs to Ω ′′ for any t ∈ [0, T ] and any (x, v) at a distance smaller than 1 from Ω ′ .

For T > 0 and δ ∈ R 2 \ {0}, define the quantity

Q δ (T ) = Ω log 1 + 1 |δ| 2 sup 0≤t≤T |X(t, x, v) -X δ (t, x, v)| 2 + T 0 |V (t, x, v) -V δ (t, x, v)| 2 dt dx dv,
where X, V and X δ , V δ are two solutions to (1.1), satisfying

           X(0, x, v) = x, V (0, x, v) = v, (X, V ) satisfies (1.3), (1.4)
and either

X δ (0, x, v) = x, V δ (0, x, v) = v, (X, V ) satisfies (1.3), (1.4) or X δ (0, x, v) = x + δ 1 , V δ (0, x, v) = v + δ 2 , (X, V )(t, x -δ 1 , v -δ 2 ) satisfies (1.3), (1.4). (2.1)
We prove the following result Proposition 2.1 Fix T > 0, any A > 0 and Ω ∈ R 2d compact. Define Ω ′ and Ω ′′ as above. There exists a constant C > 0 depending only of diam(Ω ′ ), |Ω ′′ |, T and A, such that, for any a ∈ (0, 1/4), F ∈ H 3/4+a with F L ∞ ≤ A and any solutions (X, V ) and (X δ , V δ ) to (1.1) such that (X, V ) and (X δ , V δ ) satisfy (2.1), one has for any |δ| < 1/e,

Q δ (T ) ≤ C log 1 |δ| 1-2a 1 + F H 3/4+a (Ω ′′ ) .
This result can be actually extended without difficulty to any F ∈ L ∞ such that

R d |k| 3/2 |α(k)| 2 f (k) dk < ∞, (2.2)
for some function f ≥ 1 such that f (k) → +∞ when |k| → +∞, where α(k) is the Fourier transform of F . By de la Vallée Poussin, such an f may be found for any F ∈ H 3/4 . The modified proposition hence holds

Proposition 2.2 Fix T > 0, A > 0, Ω ∈ R 2d compact and any f ≥ 1 such that f (k) → +∞ when |k| → +∞.
Define Ω ′ and Ω ′′ as above. There exists a continuous, increasing function ε(δ) with ε(0) = 0 s.t. for any

F ∈ H 3/4 ∩ L ∞ satisfying (2.
2), with F L ∞ ≤ A, for any solutions (X, V ) and (X δ , V δ ) to (1.1) such that (X, V ) and (X δ , V δ ) satisfy (2.1), one has for any |δ| < 1/e,

Q δ (T ) ≤ log 1 |δ| ε(|δ|) 1 + R d |k| 3/2 |α(k)| 2 f (k) dk 1/2
, with α the Fourier transform of F .

For reasons of simplicity we essentially give the proof of Prop. v) dk ds dx dv dt, with

Q δ (T ) ≤C T 0 Ω V (t, x, v) -V δ (t, x, v) A δ (t, x, v) t 0 R d α(k) e ik•X(s,x,v) -e ik•X δ (s,x,
A δ = |δ| 2 + sup 0≤s≤t |X(s, x, v) -X δ (s, x, v)| 2 + t 0 |V (s, x, v) -V δ (s, x, v)| 2 ds.
The estimation is linear because after that point, we never use the relation between X, V, X δ , V δ and the Fourier transform α of F involved in the previous expression. The only information on F that is used is the fact that F L ∞ is finite (when bounding |V(t, x, v)| in the proof of Lemma 4.3). So in fact for any given X, V, X δ , V δ , we may define

Z( F ) = T 0 Ω V (t, x, v) -V δ (t, x, v) A δ (t, x, v) t 0 R d α(k) e ik•X(s,x,v) -e ik•X δ (s,x,v) dk ds dx dv dt
where α is the Fourier transform of F . What we prove is that if (X, V ) and (X δ , V δ ) satisfy (2.1), then for any A > 0, there exists a constant C depending only on F and A such that for any

F ∈ H 3/4+a ∩ L ∞ with F L ∞ ≤ A, one has Z( F ) ≤ C (log 1/|δ|) 1-2a 1 + R d |k| 3/2+2a |α(k)| 2 dk 1/2 . ( 2.3)
The key point is now to notice that the numerical constant C in (2.3) may be taken independently of a. It is of course not obvious and has to be checked carefully in the proof of Prop 2.1. With that, one deduces the limit inequality by letting a → 0

Z( F ) ≤ C log(1/|δ|) 1 + R d |k| 3/2 |α(k)| 2 dk 1/2 .
(2.4)

In itself this inequality is useless as Z(F ) is trivially dominated by a constant times log(1/|δ|). Nevertheless this allows us to interpolate with (2.3).

For any

F ∈ H 3/4 ∩ L ∞ satisfying (2.
2), fix some M to be precised later and write

F = F 1 + F 2 , R d e -ik•x F 1 (x) dx = α(k) I |k|≤M , R d e -ik•x F 2 (x) dx = α(k) I |k|>M . Now apply (2.3) to F 1 to find Z(F 1 ) ≤ C (log 1/|δ|) 1-2a 1 + F 1 2 H 3/4+a 1/2 ≤ C M a (log 1/|δ|) 1-2a 1 + F 2 H 3/4 1/2 . Apply (2.4) to F 2 Z(F 2 ) ≤ C log(1/|δ|) (1 + F 2 2 H 3/4 ) 1/2 ≤ C inf |k|≥M f (k) log(1/|δ|) 1 + R d |k| 3/2 |α(k)| 2 f (k) dk 1/2 .
It simply remains to take for example M = log(1/|δ|) 1/4 to deduce

Z(F ) ≤ log(1/|δ|) ε(|δ|) 1 + R d |k| 3/2 |α(k)| 2 f (k) dk 1/2
, for an ε which depends only on f , which ends the proof of Prop 2.2.

2.2 From Prop. 2.1 and 2.2 to Th. 1.1

It is well known how to pass from an estimate like the one provided by Prop. 2.1 to a well posedness theory (see [START_REF] Crippa | Estimates and regularity results for the DiPerna-Lions flow[END_REF] for example) and therefore we only recall the main steps. Take any

F ∈ H 3/4+a ∩ L ∞ .
We start by the existence of a solution. For that, define F n a regularizing sequence of F ; F n is hence uniformly bounded in

H 3/4+a ∩ L ∞ . Denote X n , V n the solution to (1.1) with F n instead of F and (X n , V n )(t = 0) = (x, v). For any δ = (δ 1 , δ 2 ) in R 2d , put (X δ n , V δ n )(t, x, v) = (X n , V n )(t, x + δ 1 , v + δ 2 ).
The function F n and the solutions (X n , V n ), (X δ n , V δ n ) satisfy all the assumptions of Prop. 2.1, as

F n ∈ W 1,∞ , using Property 1. Since F n is uniformly bounded in L ∞ ∩ H 3/4+a , the proposition then shows that Q δ,n (T ) = Ω log 1 + 1 |δ| 2 sup 0≤t≤T |X n (t, x, v) -X δ n (t, x, v)| 2 + T 0 |V n (t, x, v) -V δ n (t, x, v)| 2 dt dx dv is uniformly bounded in n by C log 1 |δ| 1-2a
. This is enough to prove that the sequence (

X n , V n ) is compact in L 1 loc (R + × R 2d ). Indeed, by Cauchy-Schwartz inequality, T 0 Ω (|X n (t, x, v) -X δ n (t, x, v)| + |V n (t, x, v) -V δ n (t, x, v)|) dx dv dt ≤ C|Ω| 1/2 Ω A δ (x, v) dx dv 1/2 , with A δ (x, v) = sup t≤T |X n (t, x, v) -X δ n (t, x, v)| 2 + T 0 |V n (t, x, v) -V δ n (t, x, v)| 2 dt. Now A δ (x, v) is uniformly bounded by a finite constant A. Hence T 0 Ω (|X n (t, x, v) -X δ n (t, x, v)| + |V n (t, x, v) -V δ n (t, x, v)|) dx dv dt ≤ C|Ω| 1/2 Ω A log 1 + A δ (x, v)/|δ| 2 log(1 + A|δ| -2 ) dx dv 1/2 ≤ C |Ω| Q δ,n (T ) log(1 + |δ| -1 ) 1/2 ≤ C |Ω| 1/2 log 1 |δ| -a
if |δ| is small enough, by Prop. 2.1. This shows that the first integral converges to 0 as |δ| tends 0, uniformly in n, which directly implies the compactness of the sequence (X n , V n ) in L 1 loc . Denoting by (X, V ) an extracted limit, one directly checks that (X, V ) is a solution to (1.1) by taking the limit in each term. In addition, as (X n , V n ) satisfies (1.3) uniformly in n then so does (X, V ) and finally passing to the limit in (1.4), one concludes that (X, V ) verifies this last property. Thus existence is proved.

For uniqueness, consider another solution (X δ , V δ ) to (1.1), which is also the limit of solutions to a regularized equation (such as the one given by F n or by another regularizing sequence of F ). Then with the same argument, (X δ , V δ ) also satisfies (1.3) and (1.4). Moreover

X(0, x, v) -X δ (0, x, v) = x -x = 0, V (0, x, v) -V δ (0, x, v) = v -v = 0,
so that (X, V ) and (X δ , V δ ) also verify (2.1) for any δ = 0. Applying again Prop. 2.1, one finds that

Q δ,n (T ) = Ω log 1 + 1 |δ| 2 sup 0≤t≤T |X(t, x, v) -X δ (t, x, v)| 2 + T 0 |V (t, x, v) -V δ (t, x, v)| 2 dt dx dv is bounded by C log 1 |δ| 1-2a
.

Letting δ go to 0, one concludes that X = X δ and V = V δ , a.e. in t, x, v. Note from this sketch that one has uniqueness among all solutions to (1.1) satisfying (1.3) and (1.4) and not only those which are limit of a regularized problem. However not all solutions to (1.1) necessarily satisfy those two conditions so that the uniqueness among all solutions to (1.1) is unknown. Indeed in many cases, it is not true, as there is a hidden selection principle in (1.3) (see the discussion in [START_REF] Ambrosio | On the chain rule for the divergence of vector fields: applications, partial results, open problems[END_REF], [START_REF] Crippa | The ordinary differential equation with non-Lipschitz vector fields[END_REF] or [START_REF] Lellis | Notes on hyperbolic systems of conservation laws and transport equations[END_REF]).

Finally if F ∈ H 3/4 only, then one first applies the De La Vallée Poussin's lemma to find a function f s.t. f (k) → +∞ when |k| → +∞ and

R d |k| 3/2 f (k) |α(k)| 2 dk < +∞.
(2.5)

One proceeds as before with a regularizing sequence F n which now has to satisfy uniformly the previous estimate. Using Prop. 2.2 instead of Prop. 2.1, the rest of the proof is identical.

The question of optimality : An example

It is hard to know whether the condition F ∈ H 3/4 is optimal and in which sense (see the short discussion in the introduction). Instead the purpose of this section is to give a simple example showing that F ∈ W 1/2,1 is a necessary condition in order to use the method followed in this paper; namely a quantitative estimate on X -X δ and V -V δ . More precisely, for any α < 1/2, we are going to construct a sequence of force fields (F N ) N ≥1 uniformly bounded in W α,1 ∩ L ∞ and a sequence (δ N ) N ≥1 in R 2 converging to 0 such that functionals like Q δ (T ) cannot be uniformly bounded in N . This example is one dimensional (two in phase space) where it is known that much less is required to have uniqueness of the flow (F continuous by [START_REF] Bouchut | On two-dimensional Hamiltonian transport equations with continuous coefficients[END_REF] and even F ∈ L 2 by [START_REF] Hauray | On two-dimensional Hamiltonian transport equations with L p loc coefficients[END_REF]). So this indicates in a sense that the method itself is surely not optimal. Moreover what this should imply in higher dimensions is not clear. . . Through all this section we use the notation

f = O(g) if there exists a constant C s.t. |f | ≤ C |g| a.e.
In dimension 1 all functions F derive from a potential so take

φ(x) = x + h(N x) N α+1 , F = -φ ′ (x)
with h a periodic and regular function (C 2 at least) with h(0) = 0. As φ is regular, we know that the solution (X, V ) with initial condition (x, v) and the shifted one (X δ , V δ ) corresponding to the initial condition (x, v) + δ with δ = (0, δ 2 ), δ 2 > 0, satisfy the conservation of energy or

V 2 + 2φ(X) = v 2 + 2φ(x), |V δ | 2 + 2φ(X δ ) = |v + δ 2 | 2 + 2φ(x).
As φ is defined up to a constant, we do not need to look at all the trajectories and may instead restrict ourselves to the one starting at x s.t. v 2 +2 φ(x) = 0. By symmetry, we may assume v > 0.

Let t 0 and t δ 0 be the first times when the trajectories stop increasing: V (t 0 ) = 0 and V δ (t δ 0 ) = 0. As both velocities are initially positive, they stay so until t 0 or t δ 0 . So for instance

Ẋ = V = -2φ(X).
Hence t 0 is obtained by

t 0 = t 0 0 Ẋ -2φ(X) dt = x 0 x dy -2φ(y) = x 0 x dy -2y -2h(N y) N -1-α , if x 0 = X(t 0 )
. Of course by energy conservation φ(x 0 ) = 0 and again as we are in dimension 1 we can assume for convenience that x 0 = 0. We have the equivalent formula for t δ 0 with x δ 0 (which we may not assume equal to 0). Put

K δ = |v + δ 2 | 2 + 2φ(x) = δ 2 2 + 2vδ 2 , η = N (x δ 0 -x 0 ) = N x δ 0 φ x 0 = 0 x δ 0 x x v 2 (v + δ 2 ) 2
) -2φ(x 0 ) = K δ , so that |x δ 0 -x 0 | = |x δ 0 | ≤ Cδ 2 since φ ′ ≥ 1/2 for N large enough. Then t δ 0 = x δ 0 x dy K δ -2φ(y) = x 0 x-η/N dy K δ -2y -2η/N -2 N -1-α h(N y + η) = O(δ 2 ) + x 0 x dy -2y -2 N -1-α (h(N y + η) -h(η)) ,
as the integral between x and x -η/N is bounded by O(δ 2 ) (the integrand is bounded here) and

K δ = 2φ(x δ 0 ) = 2x δ 0 + 2 N 1+α h(N x δ 0 ) = 2 η N + 2 N 1+α h(η).
Note that as h is Lipschitz regular

|h(N x + η) -h(η)| N 1+α = O x N α , |h(N x)| N 1+α = |h(N x) -h(0)| N 1+α = O x N α .
So subtracting the two formula and making an asymptotic expansion

t 0 -t δ 0 = O(δ 2 ) + x 0 x dy (-2y) 3/2 - 2 N 1+α (h(N y) -h(N y + η) + h(η)) +O h(N y) N 1+α √ y 2 .
Making the change of variable N y = z in the dominant term in the integral, one finds

t 0 -t δ 0 =O(δ 2 ) -2 0 N x N 1/2-1-α h(z) -h(z + η) + h(η) (-2z) 3/2 dz + O(N -3/2-2α ).
Consequently as long as

A(η) = 0 -∞ h(z) -h(z + η) -h(η) (-2z) 3/2 dz
is of order 1 then t 0 -t δ 0 is of order N -1/2-α . Note that A(η) is small when η is, but it is always possible to find functions h s.t. A(η) is of order 1 at least for some η. One way to see this is by observing that

A ′ (η) = - 0 -∞ h ′ (y + η) + h ′ (η) (-2y) 3/2 dy
cannot vanish for all η and functions h. Taking h such that A ′ (η) ≥ 1 for η in some non-trivial interval, we can assume that A is of order 1 for η ∈ [η, η] for some 0 < η < η.

Coming back to the definition of η and

x δ 0 , η ∈ [η, η] is equivalent to δ 2 2 + 2vδ 2 ∈ φ([η/N, η/N ]). (3.1)
Using the formula for φ and the fact that η and η are independent of N or δ 2 , we find

δ 2 2 + 2vδ 2 + O(N -1-α ) ∈ [η/N, η/N ]. Chossing δ 2 = 1/N , we obtain v + O(N -α ) ∈ [η/2, η/2].
We denote by V the space of initial velocities v ∈ V 0 s.t. (3.1) is satisfied for N large enough. We may assume that V ⊂ {v > η/4}.

Since the initial position x is arbitrary, for all x in a given compact set Ω 1 , there exists c > 0 and N 0 ≥ 1 such that we can construct a set

V(x) ⊂ {v > η/4} satisfying Ω 1 |V(x)|dx > 0 and cN -1/2-α ≤ |t 0 (x, v) -t δ 0 (x, v)| ≤ c -1 N -1/2-α (3.2) for N ≥ N 0 for all x ∈ Ω 1 , v ∈ V(x).
We now consider the rest of the trajectories after times t 0 and t δ 0 . Since F ∈ W 1,∞ for all N , uniqueness holds for the flow (X, V ). Therefore, for all 0 ≤ t ≤ t 0 ,

X(t 0 + t, x, v) = X(t 0 -t, x, v), V (t 0 + t, x, v) = -V (t 0 -t, x, v)
and similarly for (X δ , V δ ). In particular,

X(2t 0 , x, v) = x, and X δ (2t δ 0 , x, v) = x.
If v ∈ V(x), we have v > η/4, and since F is bounded, V (t, x, v) stays of order 1 when t is at a distance o(1) from 2t 0 . Therefore, it follows from (3.2) that |X(2t δ 0 , x, v) -x| ≥ CN -1/2-α , for a sufficiently small constant C. Since |δ| = 1/N , taking T large enough, one then concludes that

Q δ,N (T ) = x,v log 1 + 1 |δ| 2 sup 0≤t≤T |X(t, x, v) -X δ (t, x, v)| 2 + T 0 |V (t, x, v) -V δ (t, x, v)| 2 dt dx dv
is bounded from below by

x∈Ω 1 v∈V(x) log 1 + |X(2t δ 0 , x, v) -X δ (2t δ 0 , x, v)| 2 |δ 2 | dx dv ≥ x∈Ω 1 v∈V(x) log 1 + C|δ| 2α-1 dx dv ≥ C log 1 + C|δ| 2α-1 .
If α < 1/2, a uniform control in N of Q δ,N as given by Proposition 2.1 (or Proposition 2.2) is impossible. This gives the required counter-example. The condition α ≥ 1/2 exactly corresponds to imposing

F ∈ W 1/2,1 . 4 Control of Q δ (T ) : Proof of Prop. 2.1
Recall the notation α for the Fourier transform of F . The assumption of Prop. 2.1 corresponds to the following bound:

R d |k| 4.1 Decomposition of Q δ (T ) Let A δ (t, x, v) =|δ| 2 + sup 0≤s≤t |X(s, x, v) -X δ (s, x, v)| 2 + t 0 |V (s, x, v) -V δ (s, x, v)| 2 ds.
From (1.1), we compute

d dt log 1 + 1 |δ| 2 sup 0≤s≤t |X(s, x, v) -X δ (s, x, v)| 2 + t 0 |V (s, x, v) -V δ (s, x, v)| 2 ds = 2 A δ (t, x, v) d dt sup 0≤s≤t |X(s, x, v) -X δ (s, x, v)| 2 + (V (t, x, v) -V δ (t, x, v)) t 0 (F (X(s, x, v)) -F (X δ (s, x, v))) ds .
Since, for any f ∈ BV ,

d dt max 0≤s≤t f (s) 2 ≤ 2|f (t)f ′ (t)| ≤ 4|f (t)| 2 + 1 2 |f ′ (t)| 2 ,
we deduce from the previous computation that

Q δ (T ) ≤ Ω T 0 4 |X -X δ | 2 + |V -V δ | 2 /2 A δ (t, x, v) dt dx dv + Qδ (T ) ≤ 4|Ω|(1 + T ) + Qδ (T ) + 1 2 Ω T 0 |V -V δ | 2 A δ (t, x, v) dt dx dv
where,

Qδ (T ) = -2 T 0 Ω V (t, x, v) -V δ (t, x, v) A δ (t, x, v) • t 0 R d α(k) e ik•X(s,x,v) -e ik•X δ (s,x,v) dk ds dx dv dt.
Remark that

Ω T 0 |V -V δ | 2 A δ (t, x, v) dt dx dv ≤ Ω T 0 ∂ t A δ (t, x, v) A δ (t, x, v) dt dx dv ≤ Ω log A δ (T, x, v) |δ| 2 dx dv ≤ Q δ (T ).
Therefore, we have

Q δ (T ) ≤ 8|Ω|(1 + T ) + 2 Qδ (T ),
and it is enough to bound Qδ (T ). We introduce a

C ∞ b function χ : R + → [0, 1] such that χ(x) = 0 if x ≤ 1 and χ(x) = 1 if x ≥ 2. Writing X t (resp. V t ) for X(t, x, v) (resp. V (t, x, v)) and X δ t (resp. V δ t ) for X δ (t, x, v) (resp. V δ (t, x, v)), and introducing α(k) = α(k) if |k| ≥ (log 1/|δ|) 2 0 otherwise, (4.1)
we may write

Qδ (T ) = Q(1) δ (T ) + Q(2) δ (T ) + Q(3) δ (T ) + Q(4) (T ),
where

Q(1) δ (T ) = -2 T 0 Ω t 0 χ |X s -X δ s | |δ| 4/3 V t -V δ t A δ (t, x, v) • R d α(k) e ik•Xs -e ik•X δ s dk ds dx dv dt, Q(2) δ (T ) = -2 T 0 Ω t 0 χ |X s -X δ s | |δ| 4/3 V t -V δ t A δ (t, x, v) • R d (α(k) -α(k)) e ik•Xs -e ik•X δ s dk ds dx dv dt, Q(3) δ (T ) = -2 T 0 Ω t 0 1 -χ |X s -X δ s | |δ| 4/3 V t -V δ t A δ (t, x, v) • {|k|≤|δ| -4/3 } α(k) e ik•Xs -e ik•X δ s dk ds dx dv dt, and
Q(4) δ (T ) = -2 T 0 Ω t 0 1 -χ |X s -X δ s | |δ| 4/3 V t -V δ t A δ (t, x, v) • {|k|>|δ| -4/3 } α(k) e ik•Xs -e ik•X δ s dk ds dx dv dt.
There are two ideas in the decomposition. The first is to classically separate the low frequency in Fourier from the high. As we wish to have a control on Qδ in terms of power of log 1/|δ|, this leads to the cuf-off in α. The corresponding term with only low k is Q(2) δ and will be bounded simply since it would lead to a force term having one full derivative of just the right order.

However it is also necessary in the proof to make sure that |X s -X δ s | is not too small as this can create problems in the singular integrals that we will introduce. There is therefore another cut-off with χ(|X s -X δ s |/δ 4/3 ), |δ 4/3 | being exactly the critical scale (this is of course directly connected with the H 3/4 regularity).

The corresponding term with very small |X s -X δ s | may be easily bounded if the frequency k is not too high (again by doing the usual proof and taking one full derivative of the regularized force term). Therefore we are doing a third cut-off in frequency for |k| ≤ |δ| -4/3 , obtaining Q(3) δ . The term Q(4) δ is the last remainder from this separation and can be controlled by the decay of α in |k|.

Finally the term Q

δ with all the cut-off will have to be bounded in a more subtle way and its control constitutes the heart of our proof in Subsection 4.4.

Control of Q(4) δ (T )

Let us first state and prove a result that is used repeatedly in the sequel.

Lemma 4.1 There exists a constant C such that, for |δ| small enough,

T s |V t -V δ t | A δ (t, x, v) dt ≤ C(log 1/|δ|) 1/2 .
Proof Using Cauchy-Schwartz inequality,

T s |V t -V δ t | A δ (t, x, v) dt ≤ T s |V t -V δ t | |δ| 2 + t 0 |V r -V δ r | 2 dr 1/2 dt ≤ √ T T s |V t -V δ t | 2 |δ| 2 + t 0 |V r -V δ r | 2 dr dt 1/2 = √ T log |δ| 2 + T 0 |V r -V δ r | 2 dr |δ| 2 + s 0 |V r -V δ r | 2 dr 1/2 ≤ C √ T (log 1/|δ|) 1/2
for |δ| small enough. 2

Let us define the function

F (x) = {|k|>|δ| -4/3 } α(k)e ik•x dx. Since A δ (t, x, v) ≥ |δ|, we have | Q(4) δ (T )| ≤ C T 0 Ω (| F (X s )| + | F (X δ s )|) × T s |V t -V δ t | |δ| A δ (t, x, v)
dt dx dv ds.

≤ C(log 1/|δ|) 1/2 |δ| -1 T 0 Ω ′ | F (x)| dx dv ds ≤ C(log 1/|δ|) 1/2 |δ| -1 Ω ′ 1 | F (x)| 2 dx 1/2
, where the second inequality follows from Lemma 4.1 and from Property 1 applied to the change of variables (x, v) = (X s , V s ) and (x, v) = (X δ s , V δ s ). Then, it follows from Plancherel's identity that

| Q(4) δ (T )| ≤ C(log 1/|δ|) 1/2 |δ| -1 {|k|>|δ| -4/3 } |α(k)| 2 dk 1/2 ≤ C(log 1/|δ|) 1/2 |δ| 4a/3 R d |k| 3 2 +2a |α(k)| 2 dk 1/2 . 4.3 Control of Q(2) δ (T ) and Q(3) δ (T )
We recall that the maximal function

M f of f ∈ L p (R d ), 1 ≤ p ≤ +∞, is defined by M f (x) = sup r>0 C d r d B(x,r) f (z) dz, ∀x ∈ R d .
We are going to use the following classical results. First (see [START_REF] Crippa | Estimates and regularity results for the DiPerna-Lions flow[END_REF] in the appendix), there exists a constant C such that, for all x, y ∈ R d and f ∈

L p (R d ), |f (x) -f (y)| ≤ C |x -y|(M |∇f |(x) + M |∇f |(y)). (4.2)
Second (see [29, Ch. 1, Thm. 1]), for all 1 < p < ∞, the operator M is a continuous application from L p (R d ) to itself and is sublinear, i.e. M (f +g) ≤ M f + M g. We begin with the control of

Q(3) δ (T ). Let F (x) = {|k|≤|δ| -4/3 } α(k)e ik•x dx.
It follows from the previous inequality that

{|k|≤|δ| -4/3 } α(k)(e ik•Xs -e ik•X δ s ) dk = | F (X s ) -F (X δ s )| ≤ |X s -X δ s | M |∇ F |(X s ) + M |∇ F |(X δ s ) .
Therefore, since 1 -χ(x) = 0 if |x| ≥ 2, following the same steps as for the control of Q( 4)

δ (T ), | Q(3) δ (T )| ≤ C T 0 Ω T s |V t -V δ t | |δ| A δ (t, x, v) |δ| 4/3 M |∇ F |(X s ) + M |∇ F |(X δ s ) dt dx dv ds. ≤ C(log 1/|δ|) 1/2 |δ| 1/3 Ω ′ 1 (M |∇ F |(x)) 2 dx 1/2 ≤ C(log 1/|δ|) 1/2 |δ| 1/3 Ω ′ 1 |∇ F | 2 (x) 1/2 . Then | Q(3) δ (T )| ≤ C(log 1/|δ|) 1/2 |δ| 1/3 {|k|≤|δ| -4/3 } |k| 2 |α(k)| 2 dk 1/2 ≤ C(log 1/|δ|) 1/2 |δ| 4a/3 R d |k| 3 2 +2a |α(k)| 2 dk 1/2
.

The control of Q( 2) δ (T ) follows from a similar computation: introducing

F 0 (x) = {k<(log 1/|δ|) 2 } α(k)e ik•x dx, we obtain | Q(2) δ (T )| ≤ C T 0 Ω T s |V t -V δ t | A δ (t, x, v) |X s -X δ s | A δ (t, x, v) M |∇F 0 |(X s ) + M |∇F 0 |(X δ s ) dt dx dv dt. Since |X s -X δ s | ≤ A δ (t, x, v) for all s ≤ t | Q(2) δ (T )| ≤ C(log 1/|δ|) 1/2 T 0 Ω ′ M |∇F 0 |(x) 2 dx dv 1/2 ds ≤ C(log 1/|δ|) 1/2 {|k|<(log 1/|δ|) 2 } |k| 2 |α(k)| 2 dk 1/2 ≤ C(log 1/|δ|) 1-2a R d |k| 3 2 +2a |α(k)| 2 dk 1/2 .

Control of Q(1) δ (T )

The inequality (4.2) is insufficient to control Q(1) δ (T ). Our estimate relies on a more precise version of this inequality, Proposition 4.2 For any α(k), one has

α(k) e ik•Xs -e ik•X δ s dk = R d α(k) R d k |z| d-1 • ψ z |z| , X s -X δ s |X s -X δ s | , |z| |X s -X δ s | e ik•(X δ s +z) dz dk - R d α(k) R d k |z| d-1 • ψ z |z| , X δ s -X s |X δ s -X s | , |z| |X δ s -X s | e ik•(Xs+z) dz dk, (4.3) X X δ h {X θ,h , 0 ≤ θ ≤ 1} Figure 2: The graph of θ → X θ,h
where the function

ψ belongs to C 0,∞,∞ b (S d-1 , S d-1 , R \{0}) and has support in {(u, v) ∈ (S d-1 ) 2 : cos(u, v) ≥ 17 -1/2 } × [0, 3/4], (4.4)
where (u, v) denotes the angle between the vectors u and v.

Note that, since |z| ≤ |X δ s -X s | when ψ = 0 in both terms of the righthand side of (4.3), this proposition has as consequence the control by the maximal function of the derivatives (4.2). Our case requires the stronger result of Prop. 4.2 since we need to keep all the cancellations and not simply take the absolute values. For any θ ∈ [0, 1] and h ∈ R d , we define

X θ,h (t, x, v) = θX(t, x, v) + (1 -θ)X δ (t, x, v) + (1 -(2θ -1) 2 )h,
and we write for simplicity X θ,h t for X θ,h (t, x, v). For any fixed h ∈ R d , by differentiation in θ

R d α(k) e ik•Xs -e ik•X δ s dk = R d α(k) 1 0 e ik•X θ,h s k • (X s -X δ s + 4(1 -2θ)h) dθ dk. (4.5)
For any x, y ∈ R d , we introduce the hyperplane orthogonal to x -y

H(x, y) = {h ∈ R d : h • (x -y) = 0}.
If x = y, we define for example H(x, y) = H(0, e 1 ), where e 1 = (1, 0, . . . , 0). Fix a C ∞ b function ψ : R + → R + such that ψ(x) = 0 for x > 1 and H(0,e 1 ) ψ(|h|) dh = 1. By invariance of |h| with respect to rotations, we also have

H(x,y) ψ(|h|) dh = 1 for all x, y ∈ R d .
Since the left-hand side of (4.5) does not depend on h, we have

R d α(k) e ik•Xs -e ik•X δ s dk = 1 |X -X δ | d-1 H(Xs,X δ s ) ψ |h| |X -X δ | R d α(k) 1 0 e ik•X θ,h s k • (X s -X δ s + 4(1 -2θ)h) dθ dk dh in the case where X s = X δ s . If X s = X δ s , the previous quantity is 0. Let ρ : [0, 1] → R + be a C ∞ b function such that ρ(x) = 1 for 0 ≤ x ≤ 1/4, ρ(x) = 0 for 3/4 ≤ x ≤ 1 and ρ(x) + ρ(1 -x) = 1 for 0 ≤ x ≤ 1. Then, one has R d α(k) e ik•Xs -e ik•X δ s dk = B δ (s, x, v) + C δ (s, x, v),
where

B δ (s, x, v) = 1 |X s -X δ s | d-1 H(Xs,X δ s ) ψ |h| |X s -X δ s | R d α(k) 1 0 ρ(θ)e ik•X θ,h s k • (X s -X δ s + 4(1 -2θ)h) dθ dk dh (4.6)
and

C δ (s, x, v) = 1 |X s -X δ s | d-1 H(Xs,X δ s ) ψ |h| |X s -X δ s | R d α(k) 1 0 ρ(1 -θ)e ik•X θ,h s k • (X s -X δ s + 4(1 -2θ)h) dθ dk dh. (4.7)
We focus on B δ (s, x, v) as by symmetry between X and X δ , C δ is dealt with in exactly the same manner.

K(x) = {y ∈ R d : ∃θ ∈ [0, 1], h ∈ H(x, 0) s.t. |h| ≤ |x| and y = θ(x + 4(1 -θ)h)}. (4.8) Observing that θ = y |x| • x |x| ,
this set may also be defined as

K(x) = y ∈ R d : y |x| • x |x| ∈ [0, 1] and y |x| - y |x| • x |x| x |x| ≤ 4 y |x| • x |x| 1 - y |x| • x |x| .
Note that, for any y ∈ K(x), taking θ and h as in (4.8), we have

|y| 2 = θ 2 (|x| 2 + 16(1 -θ 2 )|h| 2 ) ≤ 17θ 2 |x| 2 . Therefore, cos(x, y) = x |x| • y |y| = θ|x| |y| ≥ 17 -1/2 . (4.9)
For fixed x, y ∈ R d , we now introduce the application

F x,y : [0, 1] × {h ∈ H(x, y) : |h| ≤ |y -x|} → K(x -y) (θ, h) → θ(x -y + 4(1 -θ)h).
It is elementary to check that F x,y is a bijection when x = y, with inverse

F -1 x,y (z) =   z |x -y| • x -y |x -y| , z - z |x-y| • x-y |x-y| (x -y) 4 z |x-y| • x-y |x-y| 1 -z |x-y| • x-y |x-y|  
for z ∈ K(x -y). Moreover, F x,y is differentiable and its differential, written in an orthonormal basis of R d with first vector (x -y)/|x -y|, is

∇F x,y (θ, h) = |x -y| 4(1 -2θ)h 0 4θ(1 -θ)Id .
Therefore, the Jacobian of

F x,y at (θ, h) is (4θ(1 -θ)) d-1 |x -y|.
Making the change of variable z = F Xs,X δ s (θ, h) in (4.6), we can now compute

B δ (s, x, v) = R d α(k) 1 0 H(Xs,X δ s ) ρ(θ) ψ |h| |Xs-X δ s | |X s -X δ s | d (4θ(1 -θ)) d-1 e ik•X θ,h s k • (X s -X δ s + 4(1 -θ)h -4θh) (4θ(1 -θ)) d-1 |X s -X δ s | dh dθ dk = B ′ δ (s, x, v) -B ′′ δ (s, x, v), (4.10)
with

B ′ δ (s, x, v) = R d α(k) R d k • z |z| d ψ (1) z |z| , X s -X δ s |X s -X δ s | , |z| |X s -X δ s | e ik•(X δ s +z) dz dk, and
B ′′ δ (s, x, v) = - R d α(k) R d k |z| d-1 • ψ (2) z |z| , X s -X δ s |X s -X δ s | , |z| |X s -X δ s | e ik•(X δ s +z) dz dk.
We defined, for (a, b, c)

∈ S d-1 × S d-1 × (R \ {0}), ψ (1) (a, b, c) = ρ((a • b)c) ψ |a-(a•b)b| 4(a•b)(1-(a•b)c) 4 d-1 (a • b) d (1 -(a • b)c) d-1 and ψ (2) (a, b, c) = ρ((a • b)c) ψ |a-(a•b)b| 4(a•b)(1-(a•b)c) 4 d-1 (a • b) d-1 (1 -(a • b)c) d c(a -(a • b)b),
where ρ(x) = ρ(x) if x ∈ [0, 1], and ρ(x) = 0 otherwise.

It follows from (4.9) and from the definition of ρ that these two functions have support in (4.4). Moreover, they belong to C 0,∞,∞ b (S d-1 , S d-1 , R\{0}). Indeed, since ρ(x) = 0 for x ≥ 3/4, the terms (1-(a•b)c) in the denominators do not cause any regularity problem. Moreover, since ψ(x) = 0 for x > 1 and |a -

(a • b)b| |a • b| ≥ 1 |a • b| -1 for all a, b ∈ S d-1
, the terms a•b in the denominators do not cause any worry either. Finally, since ρ ∈ C ∞ b (R \ {0}), the discontinuity of ρ at 0 can only cause a problem in the neighborhood of points such that a • b = 0 (c being nonzero). Therefore, the previous observation also solves this difficulty.

This concludes the proof of Prop. 4.2 by putting ψ = z |z| ψ (1) -ψ (2) .

Averaging along the trajectory : integration by parts in time

With the obvious symmetries in the problem we focus on only one term coming from Prop. 4.2 and in particular, with the previous notation,

B δ (s, x, v) = R d R d α(k) e ik•(X δ s +z) k • ψ s |z| d dk dz, writing ψ t for ψ z |z| , X t -X δ t |X t -X δ t | , |z| |X t -X δ t | . (4.11)
The aim now is to prove Lemma 4.3 One has for some numerical constant

C T 0 Ω V t -V δ t A δ (t, x, v) • B δ (s, x, v) dx dv dt ≤ C (log 1/|δ|) 1-2a R d |k| 3/2+2a |α(k)| 2 dk 1/2 .
For this it is necessary to gain some regularity by integrating along the trajectories and accordingly, we decompose B δ (s, x, v)

B δ (s, x, v) = R d R d α(k) e ik•(X δ s +z) k • ψ s |z| d-1 i k |k| • V δ s |k| -1/2 + i k |k| • V δ s dk dz + R d R d α(k) e ik•(X δ s +z) k • ψ s |z| d-1 |k| -1/2 |k| -1/2 + i k |k| • V δ s dk dz =: B 1 δ (s, x, v) + B 2 δ (s, x, v). Now, let us write χ s for χ |X s -X δ s | |δ| 4/3 , (4.12)
and let us define similarly as in (4.11) and (4.12) the notation ∇ 2 ψ s (which is a Jacobian matrix), ∇ 3 ψ s (which is a gradient vector) and χ ′ s . The whole point in this somewhat artificial decomposition is that the term i k |k| • V δ s e ik•(X δ s +z) is exactly the time derivative of 1 |k| e ik•(X δ s +z) . So integrating by parts in time, we obtain

t 0 χ s B 1 δ (s, x, v)ds = I(t, x, v)-II(t, x, v)-III(t, x, v)-IV(t, x, v)-V(t, x, v), with I(t, x, v) = R d R d α(k) k • ψ t |k| |z| d-1 χ t e ik•(X δ t +z) |k| -1/2 + i k |k| • V δ t dk dz, II(t, x, v) = R d R d α(k) k • ψ 0 |k| |z| d-1 χ 0 e ik•(x+δ 1 +z) |k| -1/2 + i k |k| • (v + δ 2 )
dk dz,

III(t, x, v) = t 0 R d R d α(k) k • ψ s |k| |z| d-1 χ ′ s e ik•(X δ s +z) |k| -1/2 + i k |k| • V δ s (X s -X δ s ) • (V s -V δ s ) |δ| 4/3 |X s -X δ s | dk dz ds, correspondingly IV(t, x, v) = t 0 R d R d α(k) k |k| |z| d-1 and V(t, x, v) = t 0 R 2d α(k) k • ψ s |k| |z| d-1 χ s e ik•(X δ s +z) |k| -1/2 + i k |k| • V δ s 2 i k |k| •F (X δ s )dkdzds.
Let us define

I(T ) = T 0 Ω V t -V δ t A δ (t, x, v) • I(t, x, v) dx dv dt,
and II(T ), III(T ), IV(T ) and V(T ) similarly.

We are going to bound each of these terms, the last one being the more worrying as it has the larger power in the denominator (|k| First, we make the change of variables z ′ = z +X δ s , followed by the change of variable (x ′ , v ′ ) = (X δ s , V δ s ) in the integral defining V(T ). When (x, v) ∈ Ω, the variable (x ′ , v ′ ) belongs to the set Ω s = {(X δ (s, x, v), V δ (s, x, v)), (x, v) ∈ Ω}. Recall that, by assumption, either (X δ , V δ ) satisfy (1.4), or

-1/2 + i k |k| • V δ s ) 2 . Hence Lemma 4.
(X ′ (t, x, v), V ′ (t, x, v)) = (X δ (t, x -δ 1 , v -δ 2 ), V δ (t, x -δ 1 , v -δ 2 )) (4.13)
satisfy (1.4). In the second case, one has X ′ (-s, x ′ , v ′ ) = x + δ 1 and

V ′ (-s, x ′ , v ′ ) = v + δ 2 . So for example X δ t ((X δ s , V δ s ) -1 (x ′ , v ′ )) = X ′ t ((X δ s , V δ s ) -1 (x ′ , v ′ ) + δ) = X ′ t ((X ′ s , V ′ s ) -1 (x ′ , v ′ ) -δ + δ) = X ′ t-s (x ′ , v ′
). Writing for convenience x, v, z instead of x ′ , v ′ , z ′ , it follows from these changes of variables and from Property 1 that In the case where (X δ , V δ ) satisfy (1.4), the same formula holds but setting δ = 0 in the equations (4.13) and (4.14).

V(T ) = T 0 t 0 Ωs R R d χs Ṽ δ t,s -V ′ t-s |δ| 2 + sup 0≤r≤t | Xδ r,s -X ′ r-s | 2 + t 0 | Ṽ δ r,s -V ′ r-s | 2 dr • α(k) k • ψs |k| |z -x| d-1 e ik•z i k |k| • F (x) |k| -1/2 + i k |k| • v 2 dk dz dx dv ds dt, where Xδ t,s = X(t, X ′ (-s, x, v) -δ 1 , V ′ (-s, x, v) -δ 2 ), Ṽ δ t,s = V (t, X ′ (-s, x, v) -δ 1 , V ′ (-s, x, v) -δ 2 ), (4.14
Writing the tensor (remember that α(k

) ∈ R d ) G V (v, z) = R d k ⊗ k |k| 2 ⊗ α(k) e ik•z |k| -1/2 + i k |k| • v 2 dk,
and reminding that Ω s ⊂ Ω ′ for all s ∈ [0, T ], we have

|V(T )| ≤ C T 0 Ω ′ t 0 R d χs | ψs | |F (x)| G V (v, z) |z -x| d-1 |δ| + | Xδ s,s -x| | Ṽ δ t,s -V t-s | |δ| 2 + t 0 | Ṽ δ r,s -V r-s | 2 dr 1/2 dz ds dx dv dt,
where a 2 = d i,j,k=1 a 2 ijk for any tensor with three entries a = (a ijk ) with

1 ≤ i, j, k ≤ d. So |V(T )| ≤ C T 0 Ω ′ R d | ψs | G V (v, z) |z -x| d-1 |δ| + | Xδ s,s -x| T s | Ṽ δ t,s -V t-s | |δ| 2 + t 0 | Ṽ δ r,s -V r-s | 2 dr
1/2 dt dz dx dv ds.

Now, on the one hand, following the same computation as in Lemma 4.1, the integral with respect to t can be upper bounded by C(log 1/|δ|) 1/2 for |δ| small enough. On the other hand,

Ω ′ R d | ψs | G V (v, z) |z -x| d-1 |δ| + | Xδ s,s -x| dz dx dv ≤   Ω ′ R d | ψs | |z -x| d-1 |δ| + | Xδ s,s -x| dz dx dv   1/2   Ω ′ 2 R d Ω 1 | ψs | G V (v, z) 2 |z -x| d-1 |δ| + | Xδ s,s -x| dx dz dv   1/2
, and, because of the properties of the support of ψ, this last term is bounded by

C Ω ′ 1 | Xδ s,s -x| x+K( Xδ s,s -x) dz |z -x| d-1 dx dv 1/2 Ω ′ 2 R d G V (v, z) 2 Ω ′ 1 dx |z -x| d-1 (|δ| + |z -x|) dz dv 1/2 ≤ C(log 1/|δ|) 1/2 Ω ′ 2 R d G V (v, z) 2 dz dv 1/2 , (4.15)
where we have used that, for any z ∈ x+K( Xδ s,s -x), |z -x| ≤ | Xδ s,s -x|, and where the last inequality can be obtained by a spherical coordinate change of variable centered at x in the variable z in the first term, and centered at z in the variable x in the second term. Now,

Ω ′ 2 R d G V (v, z) 2 dz dv = Ω ′ 2 R d d i,j,n=1 R d R d k i l i k j l j |k| 2 |l| 2 αn (k)α n (l)e iz•(k-l) |k| -1/2 + i k |k| • v 2 |l| -1/2 -i l |l| • v -2
dl dk dz dv, and integrating first in z and l, this is equal to

Ω ′ 2 d i,j R d k 2 i k 2 j |k| 4 |α(k)| 2 |k| -1/2 + i k |k| • v 4 dk dv.
Therefore

Ω ′ 2 R d G V (v, z) 2 dz dv ≤ C R d Ω ′ 2 |α(k)| 2 1 |k| + k•v |k| 2 2 dv dk ≤ C R d |k| 2 |α(k)| 2 +∞ -∞ dv 1 (1 + |k|v 2 1 ) 2 dk,
where we write the vector v as (v 1 , . . . , v d ) in an orthonormal basis of R d with first vector k/|k|. In conclusion, using the definition (4.1) of α,

Ω ′ 2 R d G V (v, z) 2 dz dv ≤ C {|k|>(log 1/|δ|) 2 } |k| 3/2 |α(k)| 2 dk ≤ C(log 1/|δ|) -4a R d |k| 3 2 +2a |α(k)| 2 dk, (4.16)
Combining this inequality with (4.15), we finally get Applying to IV(T ) the same change of variable as we did for V(T ), we have

|V(T )| ≤ C(log 1/|δ|) 1-2a R d
|IV(T )| ≤ C T 0 Ω ′ t 0 R d | Ṽ δ s,s -v| | Ṽ δ t,s -V t-s | |δ| 2 + t 0 | Ṽ δ r,s -V r-s | 2 dr χs | ψs | G IV (v, z) |z -x| d-1 | Xδ s,s -x| dz ds dx dv dt where a 2 = d i,j=1 a 2 ij for any matrix a = (a ij ) 1≤i,j≤d , G IV (v, z) = R d k |k| ⊗ α(k) e ik•z |k| -1/2 + i k |k| • v dk and ψs = -∇ 3 ψ z -x |z -x| , Xδ s,s -x | Xδ s,s -x| , |z -x| | Xδ s,s -x| ⊗ ( Xδ s,s -x) |z| | Xδ s,s -x| 2 -∇ 2 ψ z -x |z -x| , Xδ s,s -x | Xδ s,s -x| , |z -x| | Xδ s,s -x| Xδ s,s -x | Xδ s,s -x| ⊗ Xδ s,s -x | Xδ s,s -x| +∇ 2 ψ z -x |z -x| , Xδ s,s -x | Xδ s,s -x| , |z -x| | Xδ s,s -x| .
Note that, because of the properties of ψ in Prop. 4.2,

| ψs | ≤ CI {z-x∈K( Xδ s,s -x)}
for some constant C. Then, following a similar computation as the one leading to (4.15), Now, making the change of variable (x ′ , v ′ ) = (X δ (-s, x.v), V δ (-s, x, v)) and denoting (x ′ , v ′ ) as (x, v) for convenience, we have Next, similarly as in the computation leading to (4.16), we have Next, the computation is very similar to (4.17): As will appear below, this term is very similar to V(T ). We apply the same method as before, without integrating by parts in time:

|IV(T )| ≤ C Ω ′ 2 R d G IV (v, z) 2 Ω ′ 1 dx dz dv |z -x| d-1 |δ| 4/3 + |z -x| 1/2    Ω ′ T 0 t 0 | Ṽ δ s,s -v| 2 | Ṽ δ t,s -V t-s | 2 |δ| 4 + t 0 | Ṽ δ r,s -V r-s | 2 dr 2 1 | Xδ s,s -x| x+K( Xδ s,s -x) dz |z -x| d-1 ds dt dx dv 1/2 . Hence |IV(T )| ≤ C(log 1/|δ|) 1/2 Ω ′ 2 R d G IV (v, z) 2 1/2    Ω ′ T 0 t 0 | Ṽ δ s,s -v| 2 | Ṽ δ t,s -V t-s | 2 |δ| 4 + t 0 | Ṽ δ r,s -V r-s | 2 dr 2    1/2 (
Ω ′ T 0 t 0 | Ṽ δ s,s -v| 2 | Ṽ δ t,s -V t-s | 2 |δ| 4 + t 0 | Ṽ δ r,s -V r-s | 2 dr 2 ≤ Ω ′′ T 0 |V t -V δ t | 2 t 0 |V s -V δ s | 2 ds |δ| 4 + t 0 |V s -V δ s | 2 ds
Ω ′ 2 R d G IV (v, z) 2 dz dv ≤ C R d Ω ′
|I(T )| ≤ C Ω ′ 2 R d G IV (v, z)
|B 2 δ (T )| ≤ C T 0 Ω ′ t 0 R d | Ṽ δ t,s -V t-s | |δ| 2 + t 0 | Ṽ δ r,s -V r-s | 2 dr

Figure 1 :

 1 Figure 1: The potential φ and the construction of x 0 and x δ 0

4. 4 . 1

 41 First part of the proof of Prop. 4.2.

Figure 3 :

 3 Figure 3: The set K(x)

2 .χ s B 2 δ 2 . 4 . 4 . 4

 222444 3 is implied by the following two Lemma 4.4 One has|I(T )| + |II(T )| + |III(T )| + |IV (T )| + |V (T )| ≤ C (log 1/|δ|) 1-2a R d |k| 3/2+2a |α(k)| 2 dk 1/(s, x, v) ds dx dv dt ≤ C (log 1/|δ|) 1-2a R d |k| 3/2+2a |α(k)| 2 dk1/Proof of Lemma 4.4: Upper bound for |V(T )|

|k| 3 2 2 . 4 . 4 . 5

 32445 +2a |α(k)| 2 dk 1/Proof of Lemma 4.4: Upper bound for |IV(T )|

≤

  C log(1/|δ|).

2 |αd |k| 3 2d |k| 3 2

 233 2 |α(k)| 2 dk ≤ C(log 1/|δ|) -2-4a R +2a |α(k)| 2 dk.The combination of these inequalities finally yields|IV(T )| ≤ C R +2a |α(k)| 2 dk 1/2 if |δ| < 1/e.

4. 4 . 6 2 I 2 , 3 2

 46223 Proof of Lemma 4.4: Upper bound for |III(T )| As before, we compute|III(T )| ≤ C -v| | Ṽ δ t,s -V t-s | |δ| 2 + t 0 | Ṽ δ r,s -V r-s | 2 dr | ψs | | χ′ s | G IV (v, z) |δ| 4/3 |z -x| d-1 dz ds dx dv dt.Then, proceeding as in (4.17),|III(T )| ≤ C |δ| 4/3 Ω ′ 2 R d G IV (v, z) 2 B(z,2|δ| 4/3 ) dx |z -x| d-1 dz dv s,s -v| 2 | Ṽ δ t,s -V t-s | 2 |δ| 4 + t 0 | Ṽ δ r,s -V r-s | 2 dr {| Xδ s,s -x|≤2|δ| 4/3 } x+K( Xδ s,s -x) dz |z -x| d-1 ds dt dx dv1/so that |III(T )| ≤ C(log 1/|δ|) used that |z -x| ≤ | Xδ s,s -x| ≤ 2|δ| 4/3 when | ψs | | χ′ s | = 0. Finally, |III(T )| ≤ C R d |k| +2a |α(k)| 2 dk 1/2 if |δ| < 1/e.

4. 4 . 7

 47 End of proof of Lemma 4.4: Upper bound for |I(T )| and |II(T )| We only detail the computation of a bound for |I(T )|. The case of |II(T )| is very similar and is left to the reader.We compute as before|I(T )| ≤ C δ r,t -V r-t | 2 dr 1/2 χs | ψs | G IV (v, z) |z -x| d-1 |δ| + | Xδ t,t -x|dz dx dv dt.

1 / 2 . 2 , 2 . 4 . 4 . 8 χ s B 2 δ

 12224482 Proceeding again as before |I(T )| ≤ C(log 1/|δ|) so that eventually |I(T )| ≤ C log(1/|δ|) Proof of Lemma 4.5: Upper bound for |B 12 δ ((s, x, v) ds dx dv dt.

1 / 2 | 2 ≤d |k| 3 2χ s B 1 δd |k| 3 2

 122313 ψs | G 12 (v, z) |z -x| d-1 (|δ| + | Xδ s,s -x|)dz ds dx dv dt whereG 12 (v, z) = R d k |k| 1/2 ⊗ α(k) e ik•z |k| -1/2 + i k |k| • v dk.Again,|B 12 δ (T )| ≤ C log(1/|δ|) Ω ′ 2 R d G 12 (v, z) 2 dz dv 1/C log(1/|δ|)It follows from Lemma 4.4 that|B 1 δ (T )| ≤ C 1 + (log |1/|δ|) 1-2a R +2a |α(k)| 2 dk (s, x, v) ds dx dv dt.Combining this with Lemma 4.5, we obtain that|B δ (T )| ≤ C 1 + (log 1/|δ|) 1-2a R +2a |α(k)| 2 dk 1/2

  4.17) where we have used that | Xδ s,s -x| ≥ |δ| 4/3 and | Xδ s,s -x| ≥ |z -x| when χs | ψs | = 0.

+2a |α(k)| 2 dk = F 2 H

3/4+a (Ω ′′ ) < +∞.


Nicolas Champagnat
email: nicolas.champagnat@sophia.inria.fr

Pierre-Emmanuel Jabin
email: jabin@unice.fr

The evolutionary limit for models of populations interacting competitively via several resources

Keywords: MSC 2000 subject classifications: 35B25, 35K55, 92D15 adaptive dynamics, Hamilton-Jacobi equation with constraints, Dirac concentration, metastable equilibrium

We consider a integro-differential nonlinear model that describes the evolution of a population structured by a quantitative trait. The interactions between individuals occur by way of competition for resources whose concentrations depend on the current state of the population. Following the formalism of [16], we study a concentration phenomenon arising in the limit of strong selection and small mutations. We prove that the population density converges to a sum of Dirac masses characterized by the solution ϕ of a Hamilton-Jacobi equation which depends on resource concentrations that we fully characterize in terms of the function ϕ.

Introduction

We are interested in the dynamics of a population subject to mutation and selection driven by competition for resources. Each individual in the population is characterized by a quantitative phenotypic trait x ∈ R (for example the size of individuals, their age at maturity, or their rate of intake of nutrients). We study the following equation

∂ t u ε (t, x) = 1 ε k i=1 I ε i (t) η i (x) -1 u ε (t, x) + M ε (u ε )(t, x), (1.1)
where M ε is the mutation kernel

M ε (f )(x) = 1 ε R K(z) (f (x + εz) -f (x)) dz, (1.2)
for a K ∈ C ∞ c (R) such that R zK(z) dz = 0. Among many other ecological situations [START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory[END_REF], this model is relevant for the evolution of bacteria in a chemostat [START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF][START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF]. With this interpretation, u ε (t, x) represents the concentration of bacteria with trait x at time t, the function η i (x) represents the growth rate of the population of trait x due to the consumption of a resource whose concentration is I ε i , and the term -1 corresponds to the decrease of the bacteria concentration due to the constant flow out of the chemostat. Note that the growth and mutation terms in (1.1) are of different nature, as the total mutation rate R K(z) of the population of trait x, which could either be produced by mutations during individuals' lives or occuring at births, does not depend on the resources concentrations I ε i . Note also that the resources consumption rates depend linearly on the resources concentrations. From a biological point of view, one might prefer a saturating functional response. Our method actually extends to all the models of [9, Thm. 1], including competitive Lotka-Volterra PDEs, but we chose here to keep the model simple to ease the presentation of the mathematical arguments. This model extends the one proposed in [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF] to an arbitrary number of resources. This equation has to be coupled with equations for the resources I i , namely

I i (t) = c i 1 + R η i (x) u ε (x) dx , (1.3)
where c i > 0. This corresponds to an assumption of fast resources dynamics with respect to the evolutionary dynamics. The resources concentrations are assumed to be at a (quasi-)equilibrium at each time t, which depends on the current concentrations u ε . The limit ε → 0 corresponds to a simultaneous scaling of fast selection and small mutations. It was already considered in [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF]. The following argument explains what limit behaviour for u ε can be expected when ε → 0. Defining ϕ ε as u ε = e ϕε/ε , or ϕ ε = ε log u ε , (1.4) one gets the equation

∂ t ϕ ε = k i=1 I ε i (t) η i (x) -1 + H ε (ϕ ε ), (1.5)
where

H ε (f ) = R K(z) e (f (x+ε z)-f (x))/ε -1 dz.
(1.6)

At the limit ε → 0 the Hamiltonian H ε simply becomes

H(p) = R K(z) (e p z -1) dz, (1.7)
where p stands for ∂ x f (x). So one expects Eq. (1.5) to lead to

∂ t ϕ = k i=1 I i (t) η i (x) -1 + H(∂ x ϕ), (1.8)
for some I i which are unfortunately unknown since one cannot pass to the limit directly in (1.3). Therefore one needs to find a relation between the ϕ and the I i at the limit. Under quite general assumptions on the parameters (see Lemma 3.1 below), the total population mass R u ε is uniformly bounded over time. This suggests that max x∈R ϕ(t, x) = 0 should hold true for all t ≥ 0. Together with (1.8), this gives a candidate for the limit dynamics as a solution to a Hamilton-Jacobi equation with Hamiltonian H and with unknown Lagrange multipliers I i , subject to a maximum constraint. The limit population is then composed at time t of Dirac masses located at the maxima of ϕ(t, •). This heuristics was justified in [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF] in the case of a single resource (and when the resources evolve on the same time scale as the population), and the case of two resources was only partly solved. The mathematical study of the convergence to the Hamilton-Jacobi equation with maximum constraint and the study of the Hamilton-Jacobi equation itself have only be done in very specific cases [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF][START_REF] Carrillo | Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model[END_REF][START_REF] Perthame | Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit[END_REF]. In fact the main problem in this proposed model is that the number of unknowns (the resources) may easily be larger than the dimension of the constraint (formally equal to the number of points where ϕ = 0). Our goal in this paper is to prove the convergence of ϕ ε to a solution of (1.8), where we give a full characterization of the functions I i . Those are no more considered as Lagrange multipliers for a set of constraints but are given by the solution ϕ itself. The new resulting model describes the evolution of a population as Dirac masses and is formally well posed. The general problem of characterizing evolutionary dynamics as sums of Dirac masses under biologically relevant parameter scalings is a key tool in adaptive dynamics-a branch of biology studying the interplay between ecology and evolution [START_REF] Hofbauer | Adaptive dynamics and evolutionary stability[END_REF][START_REF] Metz | How should we define 'fitness' for general ecological scenarios?[END_REF][START_REF] Metz | Adaptive Dynamics, a geometrical study of the consequences of nearly faithful reproduction[END_REF][START_REF] Dieckmann | The dynamical theory of coevolution: a derivation from stochastic ecological processes[END_REF][START_REF] Champagnat | The canonical equation of adaptive dynamics: a mathematical view[END_REF]. The phenomenon of evolutionary branching, where evolution drives an (essentially) monotype population to subdivide into two (or more) distinct coexisting subpopulations, is particularly relevant in this framework [START_REF] Metz | Adaptive Dynamics, a geometrical study of the consequences of nearly faithful reproduction[END_REF][START_REF] Geritz | Dynamics of adaptation and evolutionary branching[END_REF][START_REF] Geritz | Evolutionayr singular strategies and the adaptive growth and branching of the evolutionary tree[END_REF]. When the population state can be approximated by Dirac masses, this simply corresponds to the transition from a population composed of a single Dirac mass to a population composed of two Dirac masses. Several mathematical approaches have been explored to study this phenomenon. One approach consists in studying the stationary behaviour of an evolutionary model involving a scaling parameter for mutations, and then letting this parameter converge to 0. The stationary state has been proved to be composed of one or several Dirac masses for various models (for deterministic PDE models, see [START_REF] Calsina | Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics[END_REF][START_REF] Carrillo | Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Jabin | Selection dynamics with competition[END_REF][START_REF] Genieys | Mathematical model of evolutionary branching[END_REF], for Fokker-Planck PDEs corresponding to stochastic population genetics models, see [START_REF] Bürger | Stationary distributions under mutationselection balance: structure and properties[END_REF], for stochastic models, see [START_REF] Yu | Stationary distributions of a model of sympatric speciation[END_REF], for game-theoretic models, see [START_REF] Cressman | Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics[END_REF]). Closely related to these works are the notions of ESS (evolutionarily stable strategies) and CSS (convergence stable strategies) [START_REF] Metz | Adaptive Dynamics, a geometrical study of the consequences of nearly faithful reproduction[END_REF][START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF], which allow one in some cases to characterize stable stationary states [START_REF] Calsina | Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Jabin | Selection dynamics with competition[END_REF][START_REF] Cressman | Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics[END_REF]. The other main approach consists in studying a simultaneous scaling of mutation and selection, in order to obtain a limit dynamics where transitions from a single Dirac mass to two Dirac masses could occur. Here again, deterministic and stochastic approaches have been explored. The deterministic approach consists in applying the scaling of (1.1). The first formal results have been obtained in [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF]. This was followed by several works on other models and on the corresponding Hamilton-Jacobi PDE [START_REF] Carrillo | Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model[END_REF][START_REF] Perthame | Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit[END_REF]. For models of the type we consider here, rigorous results (especially for the well posedness of the Hamilton-Jacobi eq. at the limit) mainly only exist in the case with just one resource, see [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF] and [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF] (one resource but multidimensional traits).

The stochastic approach is based on individual-based models, which are related to evolutionary PDE models as those in [START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Jabin | Selection dynamics with competition[END_REF] through a scaling of large population [START_REF] Champagnat | From individual stochastic processes to macroscopic models in adaptive evolution[END_REF]. Using a simultaneous scaling of large population and rare mutations, a stochastic limit process was obtained in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF] in the case of a monotype population (i.e. when the limit process can only be composed of a single Dirac mass), and in [START_REF] Champagnat | Polymorphic evolution sequence and evolutionary branching[END_REF] when the limit population can be composed of finitely many Dirac masses. Finally note that the total population of individuals is typically very high, for bacteria for example. This is why even stochastic models will usually take some limit with infinite populations. Of course, this has some drawbacks. In particular the population of individuals around a precise trait may turn out to be low (even though the total population is large). As in the scaling under consideration, one has growth or decay of order exp(C/ε), it is in fact quite common that the population density of large sets of traits becomes much smaller than the density corresponding to a single individual, making the notion of density meaningless. One of the most important open problem would be to derive models that are both able of dealing with very large populations and still treat correctly the small subpopulations (keeping the stochastic effects or at least truncating the population with less than 1 individual). There are already some attempts in this direction, mainly proposing models with truncation, see [START_REF] Perthame | Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations[END_REF] and very recently [START_REF] Mirrahimi | Singular Hamilton-Jacobi equation for the tail problem[END_REF]. In these works however the truncation is only made at some level exp(-C/ε).

In order to state our main result, we need some regularity and decay assumptions on the η i , namely (1.9) where C 0 (R) is the set of continuous function, tending to 0 as x → ±∞. Our method of characterization of the environmental variables I i in (1.8) is based on the general principle that there exists a unique environment I 1 , . . . , I k corresponding to any given population state. Here, the population state at time t is given by the set {x ∈ R : ϕ(t, x) = 0}. This principle was the basis of the work of Barles and Perthame [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF] in the case of a single resource, where the previous correspondance is automatically satisfied under very general assumptions. In our case, we characterize I i from ω := {ϕ(t, •) = 0} by first constructing a population measure corresponding to the set ω, which has support in this set and which is a sort of metastable population equilibrium when there is no mutations (more specifically, we require it to be an ESS in the set of population measures with support in ω, see Prop. 1.1 below and [START_REF] Jabin | Selection dynamics with competition[END_REF]). Next, we construct the I i from this population measure. The following assumptions characterize the situations where our approach can be applied, that is where uniqueness holds in the two steps previously described. The first assumption deals with the number of possible roots of the reproduction rate:

η i > 0, ∃η ∈ C 0 (R), ∀x, k i=1 |η i (x)| + |η i (x)| + |η i (x)| ≤ η(x),
∃1 ≤ k ≤ k, ∀ I 1 ∈ [0, c 1 ], . . . , I k ∈ [0, c k ], the function x → k i=1 I i η i (x) -1 has at most k roots. (1.10)
We also require an invertibility condition on the matrix η i (x j )

∀ x 1 . . . xk distinct, the k vectors of R k (η i (x 1 )) i=1...k , . . . , (η i (xk)) i=1...k are linearly independent (1.11)
Assumption (1.11) holds for generical choices of the functions η i . Assumption (1.10) is more restrictive, specifically if k = 1, since it basically means that η 1 is monotone. Note however that sometimes, one may have uniqueness of the environmental variables whereas the population measure is not unique (and (1.11) is violated). Most of our method would remain valid in such a case. This is the situation of a single resource, where almost nothing is required [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF]. Nevertheless in more general situations, the conditions for which this kind of property holds are not currently identified. Even though (1.10) and (1.11) are likely more restrictive than would be strictly necessary, they have the advantage of being explicit and of enabling us to reconstruct the limit of the original u ε . Moreover from the point of view of the biological interpretation, it is interesting to have a population equilibrium. It may be hard to check (1.10) in specific models, but it is at least satisfied in large classes of parameters, for example if the derivatives η (k) i are positive (or negative) for all i = 1, . . . , k, in which case k = k. For instance in the case k = 2, (1.10) and (1.11) are ensured by the convexity (or concavity) of η 1 and η 2 . This issue of uniqueness of the environmental variables is of course easier if the environment is larger. In particular, it would be considerably simplified by considering a model with (implicitly) infinitely many resources, like the Lotka-Volterra model

∂ t u ε (t, x) = 1 ε r(x) - R b(x -y)u ε (t, y)dy u ε (t, x) + M ε (u ε )(t, x),
with an even function b. In this case, assumptions (1.10) and (1.11) should be replaced by the much simpler assumption that b as an operator is positive, i.e. that the Fourier transform of b is positive (see [START_REF] Champagnat | Convergence to equilibrium in competitive LotkaVolterra and chemostat systems[END_REF]). However, we chose to study here a model with explicit resources, as these are easier to interpret in terms of biological modeling. Now, we may uniquely define the metastable measure associated with a set ω by Proposition 1.1 For any closed ω ⊂ R, there exists a unique finite nonnegative measure µ(ω

) satisfying i) supp µ ⊂ ω ii) denoting Īi (µ) = c i /(1 + η i (x) dµ(x)), k i=1 Īi (µ) η i (x) -1 ≤ 0 in ω, k i=1 Īi (µ) η i (x) -1 = 0 on supp µ.
Now the limiting I i in (1.8) are directly obtained by

I i (t) = Īi (µ({x ∈ R : ϕ(t, x) = 0})), (1.12)
i.e. ϕ is solution to the (closed) Hamilton-Jacobi equation

∂ t ϕ = k i=1 Īi (µ({ϕ(t, •) = 0})) η i (x) -1 + H(∂ x ϕ). (1.13)
We prove

Theorem 1.1 Assume K ∈ C ∞ c (R), R zK(z) dz = 0, (1.9), (1.10), (1.11), that the initial data u ε (t = 0) > 0 or ϕ ε (t = 0) are C 2 , satisfy sup ε R u ε (t = 0, x) dx < ∞, sup ε ∂ x ϕ ε (t = 0, •) L ∞ (R) < ∞, (1.14) inf ε inf x∈R ∂ xx ϕ(t = 0, x) > -∞, (1.15)
and that ϕ ε (t = 0, •) converges to a function ϕ 0 for the norm

• W 1,∞ (R) .
Then up to the extraction of a subsequence in ε, ϕ ε converges to some ϕ uniformly on any compact subset of [0, T ] × R and in W 1,p ([0, T ] × K) for any T > 0, p < ∞ and any compact K. In particular, ϕ is continuous. The function I ε i converges to I i in L p ([0, T ]) for any T > 0, p < ∞, where I i is defined from ϕ as in (1.12), and I i is approximately right-continuous for all t ≥ 0. The function ϕ is a solution to (1.8) almost everywhere in t, x with initial condition ϕ(t = 0,

•) = ϕ 0 . Moreover if one defines ψ = ϕ -k i=1 t 0 I i (s) ds η i (x), then ψ is a viscosity solution to ∂ t ψ(t, x) = H ∂ x ψ + k i=1 t 0 I i (s) ds η i (x) .
(1.16)

We recall that a function

f on [0, +∞) is approximately right-continuous at t ≥ 0 if t is a point of Lebesgue right-continuity of f , i.e. lim s→0 1 s t+s t |f (θ) -f (t)|dθ = 0.
Notice that, under the assumptions of Theorem 1.1, ϕ ε (t = 0, x) → -∞ when x → ±∞ since R u ε < ∞ and ϕ ε is uniformly Lipschitz. Be also careful that we assume ϕ ε (t = 0, •) -ϕ 0 W 1,∞ (R) → 0 even though ϕ ε (t = 0) (and thus ϕ 0 ) is not bounded. From a practical point of view, computing the solution u ε of Eq. (1.1) is often too costly for small ε. This result allows to approximate the population density u ε for small ε by the simpler µ({ϕ(t, •) = 0}), where ϕ may be obtained by a discretization of (1.13), in the fashion of those done in [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF]. Rigorous numerical analysis of this kind of Hamilton-Jacobi equations is however still very preliminary. On a more theoretical level, this theorem justifies in simple cases the classical view of evolution (population at local equilibrium at every time). It also indicates that for small ε, the behaviour of the population is determined by a closed equation which means that evolution remains deterministic and stable in the parameter ε.

In the proofs below, C denotes a numerical constant which may change from line to line.

2 Proof of Prop. 1.1

Uniqueness

Assume that two measures µ 1 and µ 2 satisfy both points of Prop. 1.1. We first prove that they induce the same resources Īi and then conclude that they are equal.

1st step: Uniqueness of the Īi . The argument here is essentially an adaptation of [START_REF] Jabin | Selection dynamics with competition[END_REF]. First note that

R k i=1 Īi (µ 1 ) η i (x) -1 dµ 2 + R k i=1 Īi (µ 2 ) η i (x) -1 dµ 1 ≤ 0, (2.1)
since µ 1 and µ 2 are non negative and by the point ii, k i=1 Īi (µ j ) η i (x) -1 is non positive on ω for j = 1, 2. On the other hand since k i=1 Īi (µ j ) η i -1 vanishes on the support of µ j , one has for instance

R k i=1 Īi (µ 1 ) η i -1 dµ 2 = R k i=1 ( Īi (µ 1 ) -Īi (µ 2 )) η i dµ 2 = k i=1 ( Īi (µ 1 ) -Īi (µ 2 )) R η i dµ 2 = k i=1 ( Īi (µ 1 ) -Īi (µ 2 )) (c i / Īi (µ 2 ) -1),
by the definition of Īi (µ 2 ). Since one has

k i=1 c i ( Īi (µ 1 ) -Īi (µ 2 )) (1/ Īi (µ 2 ) -1/ Īi (µ 1 )) = k i=1 c i ( Īi (µ 1 ) -Īi (µ 2 )) 2 Īi (µ 1 ) Īi (µ 2 ) ≥ 0. one deduces from (2.1) that Īi (µ 1 ) = Īi (µ 2 ), i = 1 . . . k. (2.2)
2nd step: Uniqueness of µ. It is not possible to deduce that µ 1 = µ 2 directly from (2.2). This degeneracy (the possibility of having several equilibrium measures, all corresponding to the same environment) is the reason why we require additional assumptions on the η i . First of all by Assumption (1.10), point i and thanks to (2.2), we know that µ 1 and µ 2 are both supported on a set consisting of at most k points {x 1 , . . . , xk}, which are the roots of i Īi (µ 1 )η i (x) -1 = i Īi (µ 2 )η i (x) -1 (possibly completed by arbitrary distinct points if this function has less than k roots). Therefore one may write

µ j = k l=1 α j l δ x l . Now (2.2) tells that η i dµ 1 = η i dµ 2 which means that k l=1 α 1 l η i (x l ) = k l=1 α 2 l η i (x l ), ∀i = 1 . . . k.
To conclude it remains to use condition (1.11) and get that α1 l = α 2 l .

Existence

The basic idea to get existence is quite simple: Solve the equation 1

∂ t ν = k i=1 Īi (ν) η i (•) -1 ν, (2.3)
and obtain the equilibrium measure µ as the limit of ν(t) as t → +∞. This is done by considering the entropy

L(ν) = k i=1 c i log Īi (ν) + dν = k i=1 c i log c i - k i=1 c i log 1 + η i dν + dν. (2.4)
As -log is convex and η i ≥ 0, then L itself is a convex function of ν. Moreover if ν(t) solves (2.3), one has

d dt L(ν(t)) = - k i=1 Īi (ν) η i (x) -1 2 dν.
(2.5)

Therefore one expects the limit of ν and the equilibrium measure we are looking for to be the minimum of L.

Since the η i are bounded, one finds

L(ν) ≥ -C + c dν,
for two numerical constants C and c. Consequently L is bounded from below on M 1 + (ω) the set of nonnegative Radon measures on ω. In addition, one may restrict to a bounded subset of M 1 + (ω) to compute the infimum of L. As any ball of M 1 + (ω) is compact for the weak-* topology (dual of continuous functions with compact support), L attains its infimum, or

M 0 = {ν ∈ M 1 + (ω), L(ν) ≤ L(ν ) ∀ν ∈ M 1 + (ω)} = ∅.
Now take any µ ∈ M 0 then take ν the solution to (2.3) with ν(t = 0) = µ. L(ν) is non increasing and since it is already at a minimum initially, it is necessarily constant. By (2.5), this means that

k i=1 Īi (µ) η i -1 = 0 on supp µ.
Hence µ is in fact a stationary solution to (2.3) and it satisfies point i and the second part of point ii of Prop. 1.1. Note by the way that the uniqueness argument in fact tells that there is a unique element in M 0 . It only remains to check the first part of point ii. By contradiction assume that there exists a point x 0 ∈ ω s.t.

k i=1 Īi (µ) η i (x 0 ) -1 > 0.
Let α > 0 and define

ν α = µ + αδ x 0 ∈ M 1 + (ω). Now compute L(ν α ) = dµ + α - k i=1 c i log 1 + η i dµ + αη i (x 0 ) + k i=1 c i log c i = dµ + α - k i=1 c i log 1 + η i dµ + α η i (x 0 ) 1 + η i dµ + O(α 2 ) + k i=1 c i log c i = L(µ) -α k i=1 Īi (µ) η i (x 0 ) -1 + O(α 2 ).
Thus L(ν α ) < L(µ) for α small enough which is impossible as µ is an absolute minimum of L.

Consequently the first part of ii is satisfied and the proof of Prop. 1.1 complete.

3 Proof of Theorem 1.1

3.1 A priori estimates for Eq. (1.5)

We denote by BV loc (R) the set of functions on R with bounded variation on any compact subset of R, by M 1 (ω) the set of signed Radon measures on the subset ω of R equipped with the total variation norm.

We show the following estimates on the solution to (1.5)

Lemma 3.1 Let ϕ ε be a solution to (1.5) with initial data ϕ 0 ε such that R e ϕ 0 ε (x)/ε dx < ∞, ∂ x ϕ 0 ε ∈ L ∞ (R)
and ∂ xx ϕ 0 ε uniformly bounded from below. Then for any T > 0

∂ t ϕ ε L ∞ ([0,T ]×R) + ∂ x ϕ ε L ∞ ([0,T ],BV loc (R)∩L ∞ (R)) + ∂ tx ϕ ε L ∞ ([0,T ],M 1 ) ≤ C T , ∀ t ≤ T, x ∈ R, ∂ xx ϕ ε (t, x) ≥ -C T , H ε (ϕ ε ) ≥ -C T ε, ∀ t ≤ T, R u ε (t, x) dx ≤ C T , ϕ ε (t, x) ≤ C T ε log 1/ε.
where C T only depends on the time T , R e ϕ 0 ε (x)/ε dx, ∂ x ϕ 0 ε L ∞ (R) and the infimum of ∂ xx ϕ 0 ε (x).

Proof. We start with the easy bound on the total mass.

Step 0: Bound on the total mass. First notice that because of (1.9), there exists R > 0 s.t.

∀|x| > R, k i=1 η i (x) ≤ 1/2.
Let ψ be a regular test function with support in |x| > R, taking values in [0, 1] and equal to 1 on |x| > R+1. Using the fact that

I ε i (t) ≤ 1, we compute d dt R ψ(x) u ε (t, x) dx ≤ - 1 2ε R ψ(x) u ε (t, x) dx + 1 ε R 2 K(z)(ψ(x -εz) -ψ(x)) u ε (t, x) dz dx ≤ - 1 2ε R ψ(x) u ε (t, x) dx + C R u ε (t, x) dx.
On the other hand as each η i > 0, one has for some constant C

I ε i (t) = c i 1 + η i u ε dx ≤ C 1 + (1 -ψ)u ε dx .
Therefore with the same kind of estimate

d dt R (1 -ψ(x)) u ε (t, x) dx ≤C R u ε (t, x) dx + 1 ε C 1 + (1 -ψ)u ε dx -1 R (1 -ψ(x)) u ε (t, x) dx.
Summing the two

d dt R u ε (t, x) dx ≤ 1 ε C 1 + (1 -ψ)u ε dx -1 R (1 -ψ(x)) u ε (t, x) dx - 1 2ε R ψ(x) u ε (t, x) dx + C R u ε (t, x) dx.
Since the sum of the first two terms of the r.h.s. is negative if u ε is larger than a constant independent of ε, this shows that u ε (t, x) dx remains uniformly bounded on any finite time interval.

Step 1: Bound on ∂ x ϕ ε . This is a classical bound for solutions to Hamilton-Jacobi equations. Here we still have to check that it remains true uniformly at the ε level. Compute

∂ t ∂ x ϕ ε = k i=1 I ε i (t) η i (x) + K(z) e ϕε(t,x+εz)-ϕε(t,x) ε ∂ x ϕ ε (t, x + εz) -∂ x ϕ ε (t, x) ε dz. (3.1)
We first observe that, as

I ε i ∈ [0, max i c i ] and i |η i (x)| ≤ η(x) |∂ t ∂ x ϕ ε | ≤ max i c i η(x) + 2 ε R K(z)e |z| ∂xϕε(t,•) L ∞ (R) ∂ x ϕ ε (t, •) L ∞ (R) dz ≤ C ε e C ∂xϕε(t,•) L ∞ (ρ) ,
since K has compact support. This entails

∂ x ϕ ε (t, •) L ∞ (R) ≤ ∂ x ϕ 0 ε L ∞ (R) + C ε t 0 e C ∂xϕε(s,•) L ∞ (R) ds, from which easily follows that ∂ x ϕ ε ∈ L ∞ ([0, t ε ], R)
for some t ε > 0, which may (for the moment) depend on ε. Now we use the classical maximum principle. Fix t ∈ [0, T ] such that C ε,t :=

∂ x ϕ ε (t, •) L ∞ (R) < ∞. For any x ∈ R such that ∂ x ϕ ε (t, x) > sup y ∂ x ϕ ε (t, y)- α,
where the constant α > 0 will be specified later, we have

∂ t ∂ x ϕ ε (t, x) ≤ max i c i η(x) + R K(z)e |z| Ct,ε α ε dz ≤ C 1 + α ε e C Ct,ε .
Therefore, choosing α = εe -C Ct,ε , we obtain

d dt sup x ∂ x ϕ ε (t, x) ≤ C,
for a constant C independent of t < t ε and of ε. Using a similar argument for the minimum, we deduce that t ε > T and that ∂ x ϕ ε is bounded on [0, T ] × R by a constant depending only on T and

∂ x ϕ 0 ε L ∞ (R) .
Step 2: First bound on H ε (ϕ ε ) and bounds on ∂ t ϕ ε and ϕ ε . Simply note that

- R K(z) dz ≤ H ε (ϕ ε (t))(x) = R K(z) e ϕε(t,x+εz)-ϕε(t,x) ε dz - R K(z) dz ≤ K(z) e |z| ∂xϕε L ∞ ([0,T ],R) dz ≤ C.
Consequently, directly from Eq. (1.5),

|∂ t ϕ ε | ≤ max i c i η(x) + C,
hence ending the proof of the full Lipschitz bound on ϕ ε .

To get the upper bound on ϕ ε , simply note that because of the uniform Lipschitz bound on ϕ ε

ϕ ε (t, y) ≥ ϕ ε (t, x) -C T |y -x|, so R u ε (t, y) dy ≥ R e ϕε(t,x)/ε e -C T |y-x|/ε dy ≥ 2C -1 T ε e ϕε(t,x)/ε .
Hence the bound on the total mass yields that ϕ ε ≤ C T ε log 1/ε.

Step 3: BV bound on ∂ x ϕ ε . As for ∂ x ϕ ε , we begin with a maximum (actually, minimum) principle. First from (1.5)

∂ t ∂ xx ϕ ε ≥ -η(x) + R K(z) e ϕε(t,x+εz)-ϕε(t,x) ε ∂ xx ϕ ε (t, x + εz) -∂ xx ϕ ε (t, x) ε dz + R K(z) e ϕε(t,x+εz)-ϕε(t,x) ε (∂ x ϕ ε (t, x + εz) -∂ x ϕ ε (t, x)) 2 ε dz.
The last term is of course non negative and so with the same argument as before, we get

d dt inf x ∂ xx ϕ ε (t, x) ≥ -C,
where C does not depend on ε. This proves the uniform lower bound on ∂ xx ϕ ε . On the other hand, for any measurable subset A of [x 1 , x 2 ],

x 2

x 1

(I x∈A -I x ∈A )∂ xx ϕ ε (t, x) dx = x 2 x 1 ∂ xx ϕ ε (t, x) dx -2 x 2 x 1 ∂ xx ϕ ε (t, x) I x ∈A dx ≤ ∂ x ϕ ε (t, x 2 ) -∂ x ϕ ε (t, x 1 ) + C |x 2 -x 1 | ≤ 2 ∂ x ϕ ε L ∞ ([0,T ],R) + C |x 2 -x 1 |.
This indeed shows that

∂ xx ϕ ε (t, •) belongs to M 1 ([x 1 , x 2 ]) with total variation norm less than 2 ∂ x ϕ ε L ∞ ([0,T ],R) + C |x 2 -x 1 |. Thus, ∂ xx ϕ ε belongs to the space L ∞ ([0, T ], M 1 (R)), which entails ∂ x ϕ ε ∈ L ∞ ([0, T ], BV loc (R)).
Finally, it follows from (3.1) that

|∂ tx ϕ ε (t, x)| ≤ η(x) max i c i + K(z)e |z| ∂xϕε L ∞ |∂ x ϕ ε (t, x+εz)-∂ x ϕ ε (t, x)| ε dz ≤ c η(x) + K(z) z 0 |∂ xx ϕ ε (t, x+εθ)| dθdz .
Integrating, by Fubini

x 2 x 1 |∂ tx ϕ ε | dx ≤ C x 2 x 1 η(x) dx + C x 2 +ερ x 1 -ερ |∂ xx ϕ ε | dx,
where ρ is such that the support of K is included in the ball centered at 0 of radius ρ. This ends the proof of all the bounds on the derivatives of ϕ ε .

Conclusion. It only remains to show the sharp lower bound on

H ε (ϕ ε ). Let us write H ε (ϕ ε ) ≥ R K(z) exp 1 0 z ∂ x ϕ ε (t, x + θz ε) dθ dz - R K(z) dz.
The BV bound on ∂ x ϕ ε shows that this function admits right and left limits at all x ∈ R. Let us denote ∂ x ϕ ε (t, x + ) the limit on the right and ∂ x ϕ ε (t, x -) the limit on the left. As ∂ xx ϕ ε is bounded from below, we know in addition that ∀x,

∂ x ϕ ε (t, x + ) ≥ ∂ x ϕ ε (t, x -).
By differentiating once more

1 0 z ∂ x ϕ ε (t, x + θz ε) dθ ≥ z ∂ x ϕ ε (t, x + ) + 1 0 z 1 0 θ z ε ∂ xx ϕ ε (t, x + θ θzε) dθ dθ ≥ z ∂ x ϕ ε (t, x + ) -C ε z 2 , again as ∂ xx ϕ ε is bounded from below. Finally H ε (ϕ ε ) ≥ R K(z) exp(z ∂ x ϕ ε (t, x + ) -C ε z 2 ) dz - R K(z) dz ≥ H(∂ x ϕ ε (t, x + )) -C ε,
where H is defined as in (1.7) and since K is compactly supported. Because we assumed that R zK(z) dz = 0, we have H(p) ≥ 0 for any p, which ends the proof of Lemma 3.1.

Passing to the limit in ϕ ε

From the assumptions in Theorem 1.1, Lemma 3.1 gives uniform bounds on ϕ ε . Therefore up to an extraction in ε (still denoted with ε), there exists a func-

tion ϕ on [0, T ] × R such that ∂ t ϕ ∈ L ∞ ([0, T ] × R), ∂ x ϕ ∈ L ∞ ([0, T ], BV loc ∩ L ∞ (R)), ∂ tx ϕ ∈ L ∞ ([0, T ], M 1 loc (R)) and ∂ xx ϕ uniformly lower bounded on [0, T ] × R, satisfying ϕ ε -→ ϕ uniformly in C(K) for any compact K of [0, T ] × R, ∂ x ϕ ε -→ ∂ x ϕ in any L p loc ([0, T ], R), p < ∞. (3.2)
The first convergence follows from Arzéla-Ascoli theorem. For the second convergence, observe that

∂ x ϕ ε L ∞ ([0,T ],BV loc (R)) + ∂ tx ϕ ε L ∞ ([0,T ],M 1 (R)) ≤ C T implies that ∂ x ϕ ε is uniformly bounded in L ∞ ([0, T ] × R) ∩ BV loc ([0, T ] × R).
The convergence in L p loc follows by compact embedding. We also have ϕ ≤ 0 since otherwise the uniform bound on R u ε (t, x) dx would be contradicted. As the I ε i are bounded, it is possible to extract weak-* converging subsequences (still denoted with ε) to some I i (t). Now, we write again

H ε (ϕ ε ) = R K(z) exp 1 0 z ∂ x ϕ ε (t, x + ε z θ) dθ -1 dz.
From the L ∞ bound on ∂ x ϕ ε and its strong convergence, one deduces that

H ε (ϕ ε ) -→ H(∂ x ϕ) in L 1 loc . (3.3)
Therefore one may pass to the limit in (1.5) and obtain (1.8) (for the moment in the sense of distribution; the equality a.e. will follow from the convergence of I ε i in L p ([0, T ]), proved below). In addition by following [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF] or [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF], one may easily show that ψ(t, x) = ϕ(t, x) -k i=1 t 0 I i (s) ds η i (x) is a viscosity solution to (1.16). We refer the reader to these references for this technical part. It remains to obtain (1.12), the approximate right-continuity of I i for all time t and the convergence of I i to I i in L p ([0, T ]) for p < ∞. This requires some sort of uniform continuity on the I ε i which is the object of the rest of the proof.

Continuity in time for the I ε i

First of all note that, as suggested by the simulations of [START_REF] Diekmann | The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach[END_REF], there are examples where the I i have jumps in time at the limit. So we will only be able to prove their right-continuity. This regularity in time comes from the stability of the equilibrium defined through (1.12) and Prop. 1.1. Therefore let us define Īi (t) = Īi (µ({ϕ(t, .) = 0})),

where Īi and µ are given by Prop. 1.1 and ϕ is the uniform limit of ϕ ε as taken in the previous subsection. Our first goal is the following result. Lemma 3.2 For any fixed s, there exist functions σ s , σ ∈ C(R + ) with σ s (0) = σ(0) = 0 s.t.

t s |I ε i (r) -Īi (s)| 2 dr ≤ (t -s) σ s (t -s) + σ(ε).
Remark. Of course the whole point is that σ s is uniform in ε. It is also crucial for the following that σ does not depend on s.

Proof of Lemma 3.2

Step 0: ϕ has compact level sets.

Observe that ϕ ε (t = 0, x) → -∞ when x → ±∞ since R u ε (t = 0, x) dx < ∞ and ∂ x ϕ(t = 0) is bounded. Because of the uniform convergence of ϕ ε (t = 0) to ϕ 0 on R, one deduces that ϕ 0 (x) → -∞ when x → ±∞. Since ∂ x ϕ ∈ L ∞ ([0, T ], R) and I i (t) ∈ [0, max i c i ], it follows from (1.8) that ∂ t ϕ ∈ L ∞ ([0, T ], R) and thus ϕ(t, x) → -∞ when x → ±∞ for all t ≥ 0. Therefore, the set Since all the y n belong to the compact set Ω of Step 0, we can extract a converging subsequence y n → y. As ϕ is continuous, ϕ(s, y) = 0 or y ∈ {ϕ(s, .) = 0}. On the other hand one would also have d(y, {ϕ(s, .) = 0}) ≥ τ 0 which is contradictory.

Ω := {(t, x) ∈ [0, T ] × R : ϕ(t, x) ≥ -1} is compact.
Step 2: The functional. Denote µ s = µ({ϕ(s, .) = 0}), as given by Prop. 1.1. We look at the evolution of

F ε (t) = R log u ε (t, x) dµ s (x) = 1 ε R ϕ ε (t, x) dµ s (x), for t ≥ s. Compute d dt F ε (t) = 1 ε R k i=1 I ε i (t) η i (x) -1 dµ s (x) + 1 ε R H ε (ϕ ε (t)) dµ s . Now write 1 ε R k i=1 I ε i (t) η i (x) -1 dµ s (x) = d dt R u ε (t, x) dx - 1 ε R k i=1 I ε i (t) η i (x) -1 (u ε (t, x) dx -dµ s (x)). As k i=1 Īi (s) η i (x) -1 vanishes on the support of µ s , 1 ε R k i=1 I ε i (t) η i (x) -1 dµ s (x) = d dt R u ε (t, x) - A(t) ε - 1 ε R k i=1 (I ε i (t) -Īi (s)) η i (x) (u ε (t, x) dx -dµ s (x)), with
A(t) = R k i=1 Īi (s) η i (x) -1 u ε (t, x) dx. Notice that R k i=1 (I ε i (t) -Īi (s)) η i (x) (u ε (t, x) dx-dµ s (x)) = - k i=1 c i (I ε i (t) -Īi (s)) 2 I ε i (t) Īi (s)
.

So we deduce

1 ε t s k i=1 c i (I ε i (r) -Īi (s)) 2 I ε i (r) Īi (s) dr = R log u ε (t, x) u ε (s, x) dµ s - R (u ε (t, x)-u ε (s, x)) dx + t s A(r) ε dr - 1 ε t s R H ε (ϕ ε (r)) dµ s . (3.5)
Step 3: Easy bounds. Lemma 3.1 tells that -H ε (ϕ ε ) ≤ C T ε.

The total mass stays bounded in time so

- R (u ε (t, x)-u ε (s, x)) dx ≤ R (u ε (t, x)+u ε (s, x)) dx ≤ C. And furthermore R log u ε (t, x) u ε (s, x) dµ s = 1 ε R (ϕ ε (t, x) -ϕ ε (s, x)) dµ s ≤ 1 ε R (ϕ(t, x) -ϕ(s, x)) dµ s + 2 ε ϕ ε -ϕ L ∞ (Ω) ,
where the last bound comes from the fact that, by Prop. 1.1, µ s is supported on {ϕ(s, .) = 0} ⊂ Ω, where Ω is defined in Step 0. Since in addition we know that ϕ ≤ 0,

R log u ε (t, x) u ε (s, x) dµ s ≤ 2 ε ϕ ε -ϕ L ∞ (Ω) .
Consequently we deduce from (3.5) the bound

1 ε t s k i=1 c i (I ε i (r) -Īi (s)) 2 I ε i (r) Īi (s) dr ≤ C + 2 ε ϕ ε -ϕ L ∞ (Ω) + t s A(r) ε dr. (3.6)
Step 4: Control on A and the measure of {x, ϕ ε ∼ 0}.

For some α ε to be chosen later, decompose

t s A(r) dr = t s R k i=1 Īi (s) η i (x) -1 u ε (r, x)I ϕε(r,x)≤-αε dx dr + t s R k i=1 Īi (s) η i (x) -1 u ε (r, x)I ϕε(r,x)≥-αε dx dr.
For the first part, note again that by (1.9), there exists R s.t.

∀|x| > R, k i=1 Īi (s) η i (x) ≤ 1/2.
Therefore we may simply dominate Concerning the second part, we constrain 1/2 ≥ α ε ≥ ϕ -ϕ ε L ∞ (Ω) and may therefore bound

t s k i=1 Īi (s) η i (x) -1 u ε (r, x)I ϕε(r,x)≥-αε dr ≤ t s k i=1 Īi (s) η i (x) -1 + u ε (r, x)I ϕ(r,x)≥-2αε dr,
where (a) + denotes the positive part of a ∈ R. Now k i=1 Īi (s) η i (x) -1 is nonpositive on {ϕ(s, .) = 0} and so Of course σ s is continuous and, as τ s is continuous and τ s (0) = 0, then trivially σ s (0) = 0. Since {ϕ(r, .) ≥ -2α ε } and {ϕ(r, .) = 0} are subsets of Ω, σ(ε) is bounded for ε ≤ 1, and thus, in order to complete the proof of Lemma 3.2, we only have to check that σ(ε) → 0 when ε → 0 for a convenient choice of α ε . If we take α ε ≥ ϕ ε -ϕ L ∞ (Ω) converging to 0 slowly enough to have α ε /ε → +∞, we only have to prove that C T 0 δ({ϕ(r, .) ≥ -2α ε }, {ϕ(r, .) = 0}) dr -→ 0 as ε → 0.

k i=1 Īi (s) η i -1 + I ϕ(r,.)=0 ≤ C sup x∈{ϕ(r,.)=0}
(I ε i (r) -Īi (s)) 2 I ε i (r) Īi (s) dr ≤ C ε + 2 ϕ ε -ϕ L ∞ (Ω) + C (t -s) e -αε/ε + C t s τ s (r -s) dr + C t s δ({ϕ(r, .) ≥ -2α ε }, {ϕ(r, .) = 0}).
By dominated convergence it is enough that for any r δ({ϕ(r, .) ≥ -2α ε }, {ϕ(r, .) = 0}) -→ 0.

Just as in

Step 1 this is a direct consequence of the continuity of ϕ.

Compactness of the I ε

i and the proof of (1.12)

First notice that, simply passing to the limit in Lemma 3. This means that at any point of Lebesgue continuity of I i , one has I i = Īi . We recall that a.e. point is a Lebesgue point for I i . As the I i were defined only almost everywhere anyhow (they are weak-* limits), we may identify I i and Īi . This proves (1.12) and that Īi is approximately continuous on the right for any time t (and not only a.e. t). Now let us prove the compactness in L 1 loc of each I ε i . We apply the usual criterion and hence wish to control The second term can be handled the same way after swapping the order of integration This is enough to get compactness of the I ε i in L 1 loc and then in any L p loc with p < ∞, which concludes the proof of Theorem 1.1.

Step 1 :

 1 One basic property of {ϕ = 0}. Let us start with the following crucial observation ∀s, ∃τ s ∈ C(R + ) with τ s (0) = 0, s.t. ∀t ≥ s, ∀x ∈ {ϕ(t, .) = 0}, ∃y ∈ {ϕ(s, .) = 0} with |y -x| ≤ τ s (t -s). (3.4) This is a sort of semi-continuity for {ϕ = 0}. It is proved very simply by contradiction. If it were not true, then ∃s, ∃τ 0 > 0, ∃t n → s, t n ≥ s, ∃y n ∈ {ϕ(t n , .) = 0}, d(y n , {ϕ(s, .) = 0}) ≥ τ 0 , where d(y, ω) = inf x∈ω |x -y| is the usual distance.

  ) η i (x) -1 u ε (r, x)I ϕε(r,x)≤-αε dx dr ≤ C (t -s) e -αε/ε .

(3. 7 )

 7 Conclusion.Eq. (3.5) indeed gives Lemma 3.2 if one definesσ s (t -s) = 1 t -s t s τ s (r -s) dr, σ(ε) = Cε + 2 ϕ ε -ϕ L ∞ (Ω) + C T e -αε/ε + C T 0 δ({ϕ(r, .) ≥ -2α ε }, {ϕ(r, .) = 0}) dr.

2 , 3 . 3

 233 Lemma There existsσ s ∈ C(R + ) with σ s (0) = 0 s.t. ∀i t s |I i (r) -Īi (s)| 2 dr ≤ (t -s) σ s (t -s).

  t) -I i (s)| dt ds.The first and third terms are bounded directly from Lemmas 3.2 and 3.3: for example, (h) + σ(ε)/h) 1/2 ds.

3 T +h 0 σ

 30 ) -I i (t)| dt ds = (t) -I ε i (s)| dt ds ≤ 3 T +h 0 (σ s (h) + σ(ε)/h) 1/2 ds ≤ s (h) ds + 3(T + h) σ(ε)/h.Since of course the functions σ s (•) can be chosen uniformly bounded in Lemma 3.2, again by dominated convergence, this shows that ∀τ > 0, ∃h, ∃ε 0 (h) s.t. ∀ε < ε 0 (h) ) -I ε i (s)| dt ds ≤ τ.

  by Step 1 as the η i are uniformly Lipschitz. For two sets O 1 and O 2 , define in general δ(O 1 , O 2 ) = sup

			x∈O 1	inf y∈O 2	|x -y|.
	By the same argument, one gets
	k	+
		Īi (s) η i -1	I ϕ(r,.)≥-2αε ≤C τ s (t -s)
	i=1
			+ C δ({ϕ(r, .) ≥ -2α ε }, {ϕ(r, .) = 0}).
	Inequality (3.5) now becomes
	t	k
		c i
	s	i=1
				inf y∈{ϕ(s,.)=0}	|y -x| ≤ C τ s (t -s),

Existence and uniqueness are trivial for (2.3), for example by Cauchy-Lipschitz theorem in the set of finite positive measures equipped with the total variation norm.