Yves Achdou
email: achdou@ljll.univ-paris-diderot.fr

Mathieu Laurière

Mean field type control with congestion

published or not. The documents may come

Mean field type control with congestion

Introduction

In the recent years, an important research activity has been devoted to the study of stochastic differential games with a large number of players. In their pioneering articles [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF]Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF]Mean field games[END_REF], J-M. Lasry and P-L. Lions have introduced the notion of mean field games, which describe the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of players tends to infinity. In these models, it is assumed that the agents are all identical and that an individual agent can hardly influence the outcome of the game. Moreover, each individual strategy is influenced by some averages of functions of the states of the other agents. In the limit when N → +∞, a given agent feels the presence of the other agents through the statistical distribution of the states of the other players. Since perturbations of a single agent's strategy does not influence the statistical distribution of the states, the latter acts as a parameter in the control problem to be solved by each agent. Another kind of asymptotic regime is obtained by assuming that all the agents use the same distributed feedback strategy and by passing to the limit as N → ∞ before optimizing the common feedback. Given a common feedback strategy, the asymptotics are given by the McKean-Vlasov theory, [START_REF] Mckean | A class of Markov processes associated with nonlinear parabolic equations[END_REF][START_REF] Sznitman | Topics in propagation of chaos[END_REF] : the dynamics of a given agent is found by solving a stochastic differential equation with coefficients depending on a mean field, namely the statistical distribution of the states, which may also affect the objective function. Since the feedback strategy is common to all agents, perturbations of the latter affect the mean field. Then, having each player optimize its objective function amounts to solving a control problem driven by the McKean-Vlasov dynamics. The latter is named control of McKean-Vlasov dynamics by R. Carmona and F. Delarue [START_REF] Carmona | Control of McKean-Vlasov dynamics versus mean field games[END_REF][START_REF] Carmona | Mean field forward-backward stochastic differential equations[END_REF] and mean field type control by A. Bensoussan et al, [START_REF] Bensoussan | Control and Nash games with mean field effect[END_REF][START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF]. When the dynamics of the players are independent stochastic processes, both mean field games and control of McKean-Vlasov dynamics naturally lead to a coupled system of partial differential equations, a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. For mean field games, the coupled system of partial differential equations has been studied by Lasry and Lions in [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF]Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF]Mean field games[END_REF]. Besides, many important aspects of the mathematical theory developed by J-M. Lasry and P-L. Lions on MFG are not published in journals or books, but can be found in the videos of the lectures of P-L. Lions at Collège de France: see the web site of Collège de France, [START_REF] Lions | Cours du Collège de France[END_REF]. One can also see [START_REF] Gomes | Mean field games models-a brief survey[END_REF] for a brief survey, and we mention [START_REF] Porretta | Weak solutions to Fokker-Planck equations and mean field games[END_REF], a very nice article on weak solutions of Fokker-Planck equations and of MFG systems of partial differential equations. The analysis of the system of partial differential equations arising from mean field type control can be performed with rather similar arguments as for MFG, see [START_REF] Achdou | On the system of partial differential equations arising in mean field type control[END_REF] for a work devoted to classical solutions.

The class of MFG with congestion effects was introduced and studied in [START_REF] Lions | Cours du Collège de France[END_REF] in 2011, see also [START_REF] Achdou | Finite difference methods for mean field games, Hamilton-Jacobi equations: approximations, numerical analysis and applications[END_REF][START_REF] Achdou | On the system of partial differential equations arising in mean field type control[END_REF] for some numerical simulations, to model situations in which the cost of displacement of the agents increases in the regions where the density is large. A striking fact is that in general, MFG with congestion cannot be cast into an optimal control problem driven by a partial differential equation, in contrast with simpler cases. In the present paper, we aim at studying mean field type control with congestion, in a setting in which classical solutions of the system of partial differential equations seem difficult to obtain. But, in contrast with MFG, mean field type control can genuinely be seen as a problem of optimal control of a partial differential equation. This will allow us to use techniques from the calculus of variations. Inspired by the works of Cardaliaguet et al, see [START_REF] Cardaliaguet | Geodesics for a class of distances in the space of probability measures[END_REF][START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF], we will introduce a pair of primal and dual optimization problems, leading to a suitable weak formulation of the system of partial differential equations for which there exists a unique solution. Note that [START_REF] Cardaliaguet | Geodesics for a class of distances in the space of probability measures[END_REF] is devoted to some optimal transportation problems (i.e. finding the geodesics for a class of distances between probability measures), whereas [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF] deals with some special cases of MFG with possibly degenerate diffusions to which the above mentioned techniques from the calculus of variations can be applied.

Model and assumptions

This paper is devoted to the analysis of the second order system

∂u ∂t (t, x) + ν∆u(t, x) + H(x, m(t, x), Du(t, x)) + m(t, x) ∂H ∂m (x, m(t, x), Du(t, x)) = 0, (1.1) ∂m ∂t (t, x) -ν∆m(t, x) + div m(t, •) ∂H ∂p (•, m(t, •), Du(t, •)) (x) = 0, (1.2)
with the initial and terminal conditions

m(0, x) = m 0 (x) and u(T, x) = u T (x). (1.3)
Assumptions We now list the assumptions on the Hamiltonian H, the initial and terminal conditions m 0 and u T . These conditions are supposed to hold in all what follows.

H1

The Hamiltonian H :

T d × (0, +∞) × R d → R is of the form H(x, m, p) = - |p| β m α + (x, m), (1.4)
with 1 < β ≤ 2 and 0 ≤ α < 1, and where is continuous cost function that will be discussed below. It is clear that H is concave with respect to p. Calling β * the conjugate exponent of β, i.e.

β * = β/(β -1), it is useful to note that H(x, m, p) = inf ξ∈R d (ξ • p + L(x, m, ξ)) , (1.5)
L(x, m, ξ) = (β -1)β -β * m α β-1 |ξ| β * + (x, m), (1.6)
that L is convex with respect to ξ, and that

L(x, m, ξ) = sup p∈R d (-ξ • p + H(x, m, p)). (1.7)
H2 (conditions on the cost ) The function : T d × R + → R is continuous with respect to both variables and continuously differentiable with respect to m if m > 0. We also assume that m → m (x, m) is strictly convex, and that there exist q > 1 and two positive constants

C 1 and C 2 such that 1 C 1 m q-1 -C 1 ≤ (x, m) ≤ C 1 m q-1 + C 1 , (1.8) 1 C 2 m q-1 -C 2 ≤ m ∂ ∂m (x, m) ≤ C 2 m q-1 + C 2 .
(1.9)

Moreover, since we can always add a constant to L, we can assume that (x, m) ≥ 0, ∀x ∈ T d , ∀m ≥ 0.

(1.10)

The convexity assumption on m → m (x, m) implies that m → mH(x, m, p) is strictly convex with respect to m. Moreover, we assume that there exists a constant C 3 ≥ 0 such that

| (x, m) -(y, m)| ≤ C 3 (1 + m q-1 )|x -y|. (1.11)
H3 We assume that β ≥ q * . H4 (initial and terminal conditions) We assume that m 0 is of class

C 1 on T d , that u T is of class C 2 on
T d and that m 0 > 0 and

T d m 0 (x)dx = 1. H5 ν is a positive number.
Remark 1.1. Note that Assumption [H5] can be relaxed: all what follows can be generalized to degenerate diffusions, i.e. to the following system of PDEs:

∂u ∂t (t, x) + A i,j (x) ∂ 2 u ∂x i ∂x j (t, x) + H(x, m(t, x), Du(t, x)) + m(t, x) ∂H ∂m (x, m(t, x), Du(t, x)) = 0, (1.12) ∂m ∂t (t, x) -ν ∂ 2 ∂x i ∂x j (A i,j (•)m)(t, x) + div m(t, •) ∂H ∂p (•, m(t, •), Du(t, •)) (x) = 0, (1.13)
where

(A i,j ) 1≤i,j≤d (x) = 1 2 Σ(x)Σ T (x)
and Σ is a Lipschitz continous map from T d to R d×D with D possibly smaller than d. The necessary modifications can easily be found in [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF].

A heuristic justification of (1.1)-(1.3)

Consider a probability space (Ω, A, P) and a filtration F t generated by a d-dimensional standard Wiener process (W t ) and the stochastic process (X t ) t∈[0,T ] in R d adapted to F t which solves the stochastic differential equation

dX t = ξ t dt + √ 2ν dW t ∀t ∈ [0, T ], (1.14)
given the initial state X 0 which is a random variable F 0 -measurable whose probability density is m 0 . In (1.14), ξ t is the control, which we take to be

ξ t = v(t, X t ), (1.15)
where v(t, •) is a continuous function on T d . As explained in [START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF], page 13, if the feedback function v is smooth enough, then the probability distribution m t of X t has a density with respect to the Lebesgue measure, m v (t, •) ∈ P ∩L 1 (T d ) for all t, and m v is solution of the Fokker-Planck equation

∂m v ∂t (t, x) -ν∆m v (t, x) + div m v (t, •)v(t, •) (x) = 0, (1.16)
for t ∈ (0, T ] and x ∈ T d , with the initial condition

m v (0, x) = m 0 (x), x ∈ T d .
(1.17)

We define the objective function

J (v) = E T 0 L(X t , m v (t, X t ), ξ t )dt + u T (X T ) = [0,T ]×T d L(x, m v (t, x), v(t, x))m v (t, x)dxdt + T d u T (x)m v (T, x)dx.
(1.18)

The goal is to minimize J (v) subject to (1.16) and (1.17). Following A. Bensoussan, J. Frehse and P.

Yam in [START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF], it can be seen that if there exists a smooth feedback function v * achieving J (v * ) = min J (v) and such that m v * > 0 then

v * (t, x) = argmin v L(x, m v * (t, x), v) + ∇u(t, x) • v(t, x)
and (m v * , u) solve (1.1), (1.2) and (1.3). The issue with the latter argument is that we do not know how to guarantee a priori that m will not vanish in some region of (0, T ) × T d . Hereafter, we propose a theory of weak solutions of (1.1)- (1.3), in order to cope with the cases when m may vanish.

Remark 1.2. Note that the system of partial differential equations that arises in a mean field game is

∂u ∂t (t, x) + ν∆u(t, x) + H(x, m(t, x), Du(t, x)) = 0,
with (1.2) and (1.3). To the best of our knowledge, for such a system with the Hamiltonian given in (1.4), the existence of a solution is an open problem except in the stationary case with β = 2 and α = 1, see [START_REF] Gomes | Existence for stationary mean field games with quadratic hamiltonians with congestion[END_REF]; in the latter case, a very special trick can be used. Besides, the theory of weak solutions proposed below does not apply to MFG, because as explained above, MFG with congestion cannot be seen as an optimal control problem driven by a partial differential equation.

Two optimization problems

The first optimization is described as follows: consider the set K 0 :

K 0 = φ ∈ C 2 ([0, T ] × T d ) : φ(T, •) = u T
and the functional A on K 0 :

A(φ) = inf m ∈ L 1 ((0, T ) × T d ) m ≥ 0 A(φ, m) (2.1)
where

A(φ, m) = T 0 T d m(t,
+ T d m 0 (x)φ(0, x)dx (2.2)
with the convention that if m = 0 then mH(x, m, p) = 0. Then the first problem consists of maximizing

sup φ∈K0 A(φ). (2.3)
For the second optimization problem, we consider the set K 1 :

K 1 =          (m, z) ∈ L 1 ((0, T ) × T d ) × L 1 ((0, T ) × T d ; R d ) : m ≥ 0 a.e ∂m ∂t -ν∆m + divz = 0, m(0, •) = m 0          (2.4)
where the boundary value problem is satisfied in the sense of distributions. We also define

L(x, m, z) =    mL(x, m, z m ) if m > 0 0 if (m, z) = (0, 0) +∞ otherwise . (2.5) Note that (m, z) → L(x, m, z) is LSC on R × R d .
B(m, z), (2.7)
where if

T 0 T d L(x, m(t, x), z(t, x))dxdt < +∞, B(m, z) = T 0 T d L(x, m(t, x), z(t, x))dxdt + T d m(T, x)u T (x)dx, (2.8)
and if not, B(m, z) = +∞.

(2.9)

To give a meaning to the second integral in (2.8), we define w(t, x) = z(t,x) m(t,x) if m(t, x) > 0 and w(t, x) = 0 otherwise. From (1.6) and (1.8), we see that

T 0 T d L(x, m(t, x), z(t, x))dxdt < +∞ implies that m 1+ α β-1 |w| β * ∈ L 1 ((0, T ) × T d ), which implies that m|w| β β-1+α ∈ L 1 ((0, T ) × T d ).
In that case, the boundary value problem in (2.4) can be rewritten as follows:

∂m ∂t -ν∆m + div(mw) = 0, m(0, •) = m 0 , (2.10)
and we can use the following Lemma which can be found in [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF]:

Lemma 2.1. If (m, w) ∈ K 1 is such that T 0 T d L(
x, m(t, x), z(t, x))dxdt < +∞, then the map t → m(t) for t ∈ (0, T ) and t → m 0 for t < 0 is Hölder continuous a.e. for the weak * topology of P(T d ).

Remark 2.1. Following the proof of lemma 3.1 in [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF], we see that the Hölder exponent in Lemma 2.1 is greater than or equal to min 1 2 , 1-α β .

This lemma implies that the measure m(t) is defined for all t, so the second integral in (2.8) has a meaning.

Lemma 2.2. sup φ∈K0 A(φ) = min (m,z)∈K1 B(m, z). (2.11)
Moreover the latter minimum is achieved by a unique (m * , z * ) ∈ K 1 , and m * ∈ L q ((0, T ) × T d ).

Proof. Let us reformulate the optimization problem (2.3): take

E 0 = C 2 ([0, T ] × T d ) and E 1 = C 0 ([0, T ] × T d ) × C 0 ([0, T ] × T d ; R d ).
We define the functional F on E 0 :

F(φ) = χ T (φ) - T d m 0 (x)φ(0, x)dx
where χ T (φ) = 0 if φ| t=T = u T and χ T (φ) = +∞ otherwise. Let us also define the linear operator Λ :

E 0 → E 1 by Λ(φ) = ∂φ ∂t + ν∆φ, Dφ . For (a, b) ∈ E 1 , let G(a, b) be defined by G(a, b) = - inf m ∈ L 1 ((0, T ) × T d ) m ≥ 0 T 0 T d m(t, x) (a(t, x) + H(x, m(t, x), b(t, x))) dxdt.
(2.12)

Note that the infimum with respect to m is in fact a minimum: indeed, from (1.4) and (1.8), we see

that m → T 0 T d m(t, x) (a(t, x) + H(x, m(t, x), b(t, x))) dxdt is convex, coercive and continuous in the set m ∈ L q ((0, T ) × T d ) : m ≥ 0 .
Hence, this map is lower semi-continuous for the weak convergence in {m ∈ L q ((0, T ) × T d ), m ≥ 0}. On the other hand, since a minimizing sequence (m n ) n∈N for (2.12) is bounded in L q ((0, T ) × T d ), we can extract a subsequence which converges weakly in L q ((0, T ) × T d ) to a nonnegative function. The weak limit achieves the minimum in (2.12). We now aim at characterizing the optimal m. Let us first characterize

K(x, γ, p) = min µ≥0 (µγ + µH(x, µ, p)) , (2.13)
which is nonpositive and concave with respect to (γ, p); since µ → µH(x, µ, p) is C 1 , strictly convex on R + and tends to +∞ as µ → +∞, we see that for any x ∈ T d , if p = 0 and γ ∈ R, or if p = 0 and γ + (x, 0) ≤ 0, then there exists a unique µ = ψ(x, γ, p) ≥ 0 such that

γ + H(x, µ, p) + µH m (x, µ, p) = 0.
Note that if p = 0 and γ+ (x, 0) < 0, then µ = ψ(x, γ, 0) > 0 is characterized by γ+ (x, µ)+µ m (x, µ) = 0. We extend ψ by 0 in the set {(γ, 0) : γ + (x, 0) ≥ 0}. Therefore,

K(x, γ, p) = ψ(x, γ, p)γ + ψ(x, γ, p)H(x, ψ(x, γ, p), p), (2.14)
with the convention that mH(x, m, p) = 0 if m = 0. We claim that the map (γ, p)

→ ψ(x, γ, p) is continuous in R × R d . Indeed,
1. the continuity of (γ, m, p) → γ + H(x, m, p) + µH m (x, m, p) and the fact that this map is strictly increasing w.r.t. m in (0, +∞) implies that (γ, p)

→ ψ(x, γ, p) is continuous in R × R d \{0}.
2. Similarly, the continuity of γ → ψ(x, γ, 0) stems from the continuity of the map (γ, m) → γ + (x, m) + m m (x, m) and its strictly increasing character w.r.t. m in (0, +∞).

3. Let us prove that if (γ, p) tends to (γ, 0) with p = 0, then lim

(γ, p)→(γ,0) ψ(x, γ, p) = ψ(x, γ, 0). ( 2.15)
(a) If γ + (x, 0) < 0, then ψ(x, γ, 0) > 0 and we get (2.15) from the same argument as in point 1.

(b) Consider the case γ + (x, 0) > 0. Suppose that (γ, p) tends to (γ, 0) with p = 0, and set μ = ψ(x, γ, p) > 0. We see that

(1 -α)μ -α |p| β = γ + d dm (m → m (x, m))(μ) > γ + (x, 0) → γ + (x, 0).
This implies that μα |p| -β is bounded as (γ, p) → (γ, 0), hence (2.15).

(c) Finally, we consider the case when γ = -(x, 0) ≤ 0 and (γ, p) → (γ, 0) with p = 0; let us assume that for a subsequence, μ is bounded away from 0: passing to the limit in the identity

d dm (m → m (x, m))(μ) -(x, 0) -(1 -α)μ -α |p| β = γ -γ,
we obtain that lim

(γ, p)→(γ,0) d dm (m → m (x, m))(μ) = (x, 0),
which can happen only if μ → 0 and we reach a contradiction. Hence, (2.15) holds.

We have proved the continuity of ψ with respect to (γ, p). The continuity of ψ with respect to x follows from similar arguments, using the regularity assumptions on . Therefore, ψ is continuous in the set

T d × R × R d .
It is also useful to notice that

K(x, γ, p) = inf µ≥0 (µ(γ + H(x, µ, p)) = inf µ≥0 µγ + µ inf ξ (ξ • p + L(x, µ, ξ)) = inf (µ,z)∈R×R d µγ + z • p + L(x, µ, z)
and, from the Fenchel-Moreau theorem, see e.g. see [START_REF]Convex analysis[END_REF], that

L(x, µ, z) = sup (γ,p)∈R×R d (-µγ -z • p + K(x, γ, p)) .
(2.16)

Remark 2.2. Note that for all 0 = p ∈ R d and γ ∈ R, the map µ → µ (γ + H(x, µ, p)) is strictly decreasing in some interval [0, μ] where μ > 0 depends on p and γ, and that its derivative tends to -∞ as µ → 0 + . Hence, if µ * = 0 achieves the minimum of µ → µ (γ + H(x, µ, p)), then p must be 0. Similarly, γ must be such that γ + (x, 0) ≥ 0.

With (2.14), the optimality conditions for (2.12) yield that

G(a, b) = - T 0 T d K(x, a(t, x), b(t, x))dxdt. (2.17)
From (2.14) and the continuity of ψ, we see that G is continuous on E 1 . We observe that

sup φ∈K0 A(φ) = -inf φ∈E0 (F(φ) + G(Λ(φ))) . (2.18)
By choosing φ 0 (t, •) = u T , we see that F(φ 0 ) < +∞, G(Λ(φ 0 )) < +∞ and that G • Λ is continuous at φ 0 . We can thus apply Fenchel-Rockafellar duality theorem, see [START_REF]Convex analysis[END_REF]:

-inf φ∈E0 (F(φ) + G(Λ(φ))) = min (m,z)∈E * 1 (F * (Λ * (m, z)) + G * (-m, -z)) (2.19)
where E * 1 is the topological dual of E 1 i.e. the set of Radon measures (m, z) on (0, T )

× T d with values in R × R d . If E * 0 is the dual space of E 0 , the operator Λ * : E * 1 → E * 0 is the adjoint of Λ.
The maps F * and G * are the Legendre-Fenchel conjugates of F and G. Following [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF], we check that

F * (Λ * (m, z)) =      T d u T (x)dm(T, x) if ∂m ∂t -ν∆m + divz = 0, m(0, •) = m 0 +∞ otherwise.
where the boundary value problem is understood in the sense of distributions.

On the other hand, from Rockafellar, [START_REF] Rockafellar | Integrals which are convex functionals[END_REF] Theorem 5, and (2.16), see also [START_REF] Cardaliaguet | Geodesics for a class of distances in the space of probability measures[END_REF], we see that

G * (-m, -z) = T 0 T d L(x, m ac (t, x), z ac (t, x))dxdt + T 0 T d L ∞ x, dm sing dθ , dz sing dθ dθ,
where (m ac , z ac ) and (m sing , z sing ) respectively denote the absolutely continuous and singular parts of (m, z), θ is any measure with respect to which (m sing , z sing ) is absolutely continuous, (for instance m sing + |z sing |, and

L ∞ (x, •) is the recession function of L(x, •), i.e. L ∞ (x, m, z) = sup λ>0 1 λ L(x, λm, λz) = 0 if (m, z) = (0, 0), +∞ otherwise.
Therefore,

G * (-m, -z) =    T 0 T d L(x, m(t, x), z(t, x))dxdt if (m, z) ∈ L 1 ((0, T ) × T d ) × L 1 ((0, T ) × T d ; R d ) +∞ otherwise.
(2.20) Hence, min

(m,z)∈E * 1 (F * (Λ * (m, z)) + G * (-m, -z)) = min (m,z)∈K1 B(m, z).
and we obtain the desired result from (2.18) and (2.19).

Using (1.6), the strict convexity of m → m (x, m) assumed in (H2), the convexity of the map

(m > 0, z) → m 1+ α β-1 | z m | β * (see [2] paragraph 3.
2), and the convexity of K 1 , we obtain the uniqueness of m * such that (m * , z * ) ∈ K 1 achieves a minimum of B for some z * . Moreover, from the strict convexity of (m, z)

→ m 1+ α β-1 | z m | β * for m > 0, we deduce that z * m * is unique in {(t, x) : m * (t, x) > 0}. Since z * = 0 in {(t, x) : m * (t, x) = 0}, the uniqueness of z * follows. It is clear from (H2) that m * ∈ L q ((0, T ) × T d ).
3 A priori estimates for a maximizing sequence of (2.1)

Let M ∈ R be the optimal value in (2.1). Take a maximizing sequence (φ n ) n∈N for (2.1). For some > 0, it can be chosen in such a way that

M -< A(φ n ) ≤ M.
From the definition of A, we see that

A(φ n ) ≤ A(φ n , m 0 ). Hence, M - ≤ T 0 T d m 0 (x) ∂φ n ∂t (t, x) + ν∆φ n (t, x) + H(x, m 0 (x), Dφ n (t, x)) dxdt + T d m 0 (x)φ n (0, x)dx = T 0 T d (-νDm 0 (x) • Dφ n (t, x) + m 0 (x)H(x, m 0 (x), Dφ n (t, x))) dxdt + T d m 0 (x)u T (x)
0 =        T 0 T d m n (t, x) ∂φ n ∂t (t, x) + ν∆φ n (t, x) dxdt + T 0 T d m n (t, x) (H(x, m n (t, x), Dφ n (t, x)) + m n (t, x)H m (x, m n (t, x), Dφ n (t, x))) dxdt. (3.2)
From (3.2), we deduce that

A(φ n ) = - T 0 T d m 2 n (t, x)H m ((x, m n (t, x), Dφ n (t, x))dxdt + T d m 0 (x)φ n (0, x)dx.
From (H1) and (H2),

m 2 H m (x, m, p) ≥ 1 C 2 m q -C 2 m + m 1-α |p| β .
Hence,

T 0 T d 1 C 2 m q n (t, x) -C 2 m n (t, x) + m 1-α n (t, x)|Dφ n (t, x)| β dxdt - T d m 0 (x)φ n (0, x)dx ≤ -M + . (3.3)
On the other hand, (3.1), (H1) and (H2) imply that for some constant C 4 > 0,

∂φ n ∂t (t, x) + ν∆φ n (t, x) + C 4 m q-1 n (t, x) + C 4 ≥ 0, a.e. ( 3.4)
Multiplying (3.4) by (φ + n ) q * -1 e -λ(T -t) for λ large enough, integrating in (τ, T ) × T d , we obtain that

φ + n L ∞ (0,T ;L q * (T d )) ≤ C 1 + m n q-1 L q ((0,T )×T d ) + u + T L q * (T d ) , (3.5)
and that

D((φ + n ) q * /2 ) 2 L 2 ((0,T )×T d ) ≤ C 1 + m n q L q ((0,T )×T d ) + u + T q * L q * (T d ) .
Remark 3.1. Note that the latter estimate does not hold with a degenerate diffusion as in Remark 1.1, but it will not be used hereafter.

This implies that

T d m 0 (x)φ + n (0, x)dx ≤ C m n q-1 L q ((0,T )×T d ) + 1 . (3.6)
Combining (3.3) and (3.6), we obtain that

T 0 T d 1 C 2 m q n (t, x) -C 2 m n (t, x) + m 1-α n (t, x)|Dφ n (t, x)| β dxdt + T d m 0 (x)φ - n (0, x)dx ≤ -M + + C m n q-1 L q ((0,T )×T d ) + 1 .
The latter and (3.6) yield

T 0 T d m q n (t, x) + m 1-α n (t, x)|Dφ n (t, x)| β dxdt + T d |φ n (0, x)|dx ≤ C. (3.7)
Let m(t, x) = 1 τ <t : since A(φ n , m) ≥ A(φ n ), we obtain that

M - ≤ t 0 T d ∂φ n ∂t (τ, x) + ν∆φ n (τ, x) + H(x, 1, Dφ n (τ, x)) dxdt + T d m 0 (x)φ n (0, x)dx = t 0 T d H(x, 1, Dφ n (τ, x))dxdτ + T d φ n (t, x)dx + T d (m 0 (x) -1)φ n (0, x)dx.
This implies that t → T d φ n (t, x)dx is bounded from below uniformly w.r.t. n. Combining with (3.5), we get that φ - n is bounded in L ∞ (0, T ; L 1 (T d )), and finally that φ n is bounded in

L ∞ (0, T ; L 1 (T d )). Finally, setting γ 1 n = -(•, m n ) -m n ∂ ∂m (•, m n ) and 0 ≤ γ 2 n = (1 -α) |Dφn| β m α n 1 {mn>0} , we see that the sequence (γ 1 n ) n is bounded in L q * ((0, T ) × T d
), and that

∂φ n ∂t + ν∆φ n ≥ γ 1 n + γ 2 n .
Integrating the latter on (0, T ) × T d , we obtain that

T 0 T d γ 2 n (t, x)dxdt ≤ T d u T (x)dx - T d φ n (0, x)dx - T 0 T d γ 1 n (t, x)dxdt, which implies that the sequence of positive function (γ 2 n ) n is bounded in L 1 ((0, T ) × T d ).
To summarize, we have proven the following lemma: Lemma 3.1. The maximizing sequence (φ n ) n∈N introduced at the beginning of § 3.1 is uniformly bounded in L β (0, T ; W 1,β (T d )) and in L ∞ (0, T ; L 1 (T d )). Noting m n the nonnegative function achieving A(φ n , m n ) = A(φ n ), the sequence (m n ) n∈N is uniformly bounded in L q ((0, T ) × T d ). Calling γ n (t, x) = -H(x, m n (t, x), Dφ n (t, x)) -m n (t, x)H m (x, m n (t, x), Dφ n (t, x)), with the convention that H(x, m, p) + mH m (x, m, p) = (x, 0) if m = 0 and p = 0, we can split γ n as follows:

γ n = γ 1 n + γ 2 n , where γ 1 n = -(•, m n )-m n ∂ ∂m (•, m n ) and γ 2 n = (1-α) |Dφn| β m α n 1 {mn>0} . The sequence (γ 1 n ) n∈N is uniformly bounded in L q * ((0, T ) × T d ). The function γ 2
n is nonnegative and the sequence (γ 2 n ) n∈N is uniformly bounded in L 1 ((0, T ) × T d ).

A relaxed problem 4.1 Definition and first properties

Let K be the set of pairs (φ, γ) such that

• φ ∈ L β (0, T ; W 1,β (T d )) ∩ L ∞ (0, T ; M(T d )) • γ ∈ M([0, T ] × T d ) and γ ac = γ 1 + γ 2 , with γ 1 ∈ L q * ((0, T ) × T d ) is non positive, γ 2 is non negative and γ 2 ∈ L 1 ((0, T ) × T d ) , γ sing ∈ M + ([0, T ] × T d ) • ∂φ ∂t + ν∆φ ≥ γ, and φ| t=T ≤ u T . (4.1)
It is clear that K is convex. The following lemma implies that φ has a trace in a very weak sense:

Lemma 4.1. Consider (φ, γ) ∈ K. For any Lipschitz continuous map ξ :

T d → R, the map t → T d ξ(x)φ(t, x)dx has a BV representative on [0, T ].
Moreover, if we note T d ξ(x)φ(t + , x)dx its right limit at t ∈ [0, T ), then the map ξ → T d ξ(x)φ(t + , x)dx can be extended to a bounded linear form on C 0 (T d ).

Proof. Consider first a nonnegative and Lipschitz continuous function ξ : T d → R + ; the following identity holds in the sense of distributions:

- d dt t → T d ξ(x)φ(t, x)dx -ν T d ∇φ(t, x) • ∇ξ(x)dx ≥ T d ξ(x)γ ac (t, x)dx,
The second and third integral in the latter inequality belong to L 1 ((0, T )). From this, we deduce that t → T d ξ(x)φ(t, x)dx has a BV representative. If now ξ is a Lipschitz continuous function that may change sign, ξ : T d → R, then we write ξ = ξ + -ξ - and use the above argument separately for ξ + and ξ -: we still obtain that t →

T d ξ(x)φ(t, x)dx = T d ξ + (x)φ(t, x)dx - T d ξ -(x)φ(t,
x)dx has a BV representative. The continuity of ξ → T d ξ(x)φ(t + , x)dx comes from the fact that φ ∈ L ∞ (0, T ; M(T d )). Thanks to Lemma 4.1, we may define the concave functional J on K by

J(φ, γ) = T 0 T d K(x, γ ac (t, x), Dφ(t, x))dxdt + T d m 0 (x)φ(0 + , x)dx, (4.2)
and the relaxed optimization problem: sup

(φ,γ)∈K J(φ, γ). (4.3)
In (4.2), note that (x, t) → K(x, γ ac (t, x), Dφ(t, x)) is a measurable nonpositive function, so the first integral is meaningful and has a value in [-∞, 0]. Note also that, from (2.14), it is possible to restrict ourselves to the pairs (φ, γ) ∈ K such that γ ac (t, x) ≤ -(x, 0) for almost every (t, x) such that Dφ(t, x) = 0. Noting K the set

K = {(φ, γ) ∈ K : γ ac (t, x) ≤ -(x,
T 0 T d K(x, γ ac (t, x), Dφ(t, x))dxdt > -∞,
for any (m, z) ∈ K 1 such that m ∈ L q ((0, T ) × T d ) and

T 0 T d L(x, m(t, x), z(t, x))dxdt < +∞,
the following holds: for almost any t ∈ (0, T ),

- T d m(T, x)u T (x)dx + T d m(t, x)φ(t, x)dx + T s=t T d m(s, x) (γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) dxds ≤ T s=t T d L(x, m(s, x), z(s, x))dxds, (4.5)
-

T d m(t, x)φ(t, x)dx + T d m 0 (x)φ(0 + , x)dx + t s=0 T d m(s, x) (γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) dxds ≤ t s=0 T d L(x, m(s, x), z(s, x))dxds, (4.6)
and the meaning of T d m(t, x)φ(t, x)dx will be explained in the proof. Moreover, if

- T d m(T, x)u T (x)dx + T d m 0 (x)φ(0 + , x)dx + T s=0 T d m(s, x) (γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) dxds = T s=0 T d L(x, m(s, x), z(s, x))dxds, (4.7)
then z(t, x) = m(t, x)H p (x, m(t, x), Dφ(t, x)) holds almost everywhere. Remark 4.1. Before proving Lemma 4.2, note that the integrals of (s, x) → m(s, x) (γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) in (4.5)-(4.7) have a meaning in (-∞, +∞], because m(s, x) (γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) ≥ K(x, γ ac (s, x), Dφ(s, x)) a.e. and from the hypothesis of Lemma 4.2, the latter is integrable in (0, T ) × T d with a nonpositive integral.

Proof. We first extend m to [-1, T + 1] by setting m(t) = m 0 for t ≤ 0 and m(t) = m(T ) for t > T . Note that m(T ) is well defined from Lemma 2.1. Similarly, we extend z to [-1,

T + 1] by setting z(t) = 0 if t / ∈ (0, T ). Consider a regularizing kernel η (t, x) = -d-δ h 1 ( t δ )h 2 ( x ) where h 1 is a smooth even and nonnegative function supported in [-1/2, 1/2] such that R h 1 (t)dt = 1, h 2 is a smooth symmetric nonnegative function supported in [-1/2, 1/2] d such that R d h 2 (x)dx = 1
, and δ will be chosen later. We define m = η m and z = η z in (-1/2, T + 1/2) × T d . We can see that in (0,

T ) × T d , ∂m ∂t -ν∆ m 1 t∈(0,T ) + divz = divR (4.8)
where R = η Dm 1 t∈(0,T ) -1 t∈(0,T ) D(η m). From [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], we know that R → 0 in L q ((0, T ) × T d ) as → 0, because m ∈ L q ((0, T ) × T d ). We know that φ(T ) ≤ u T . Thus T d φ(T, x)m (T, x)dx ≤ T d u T (x)m (T, x)dx and the latter integral converges to T d u T (x)m(T, x)dx in view of Lemma 2.1. We know that m → m in L q ((0, T ) × T d ) and that φ ∈ L β ((0, T ) × T d ). From Assumption (H3), this implies that φm → φm in L 1 ((0, T ) × T d ). From Lemma 4.1, we know that out of a countable set, φ(t -) = φ(t + ) = φ(t) and that T d φ(t, x)m (t, x)dx is well defined. From the latter two observations, up to the extraction of a subsequence, we may assume that

T d φ(t, x)m (t, x)dx → T d φ(t, x)m(t, x)dx as → 0, with φ(t -) = φ(t + ) = φ(t)
, for almost all t ∈ (0, T ). Let t ∈ (0, T ) be such that the latter is true. Since (φ, γ) ∈ K,

T t T d φ(s, x) ∂m ∂t (s, x) + νDφ • D(1 s∈(0,T ) m (s, x)) dxds - T d u T (x)m (T, x)dx + T d φ(t, x)m (t, x)dx ≤ - T t T d γ ac (s, x)m (s, x)dxds (4.9)
On the other hand, from (4.8),

T t T d φ(s, x) ∂m ∂t (s, x) + νDφ • D(1 s∈(0,T ) m (s, x)) dxds = - T t T d Dφ(s, x) • R (s, x)dxds + T t T d Dφ(s, x) • z (s, x)dxds (4.10)
We deduce from (4.9) and (4.10) that

- T t T d Dφ(s, x) • R (s, x)dxds + T t T d Dφ(s, x) • z (s, x)dxds - T t T d m (s, x)H(x, m (s, x), Dφ(s, x))dsds - T d u T (x)m (T, x)dx + T d φ(t, x)m (t, x)dx ≤ - T t T d γ ac (s, x)m (s, x)dxds - T t T d m (s, x)H(x, m (s, x), Dφ(s, x))dsds. (4.11)
Finally, we use the fact that

T t T d (Dφ(s, x) • z (s, x) -m (s, x)H(x, m (s, x), Dφ(s, x))) dxds ≥ - T t T d L(x, m (s, x), z (s, x))dxds.
(4.12)

Combining this with (4.11), we find that 1. We have seen that the first line of (4.13) tends to - (γ ac (s, x) + H(x, m (s, x), Dφ(s, x))) m (s, x)dxds.

- T d u T (x)m (T, x)dx + T d φ(t, x)m (t, x)dx ≤ - T t T d m (s, x) (γ ac (s, x) + H(x, m (s, x), Dφ(s, x))) dsds + T t T d L(x, m (s, x), z (s, x))dxds + T t T d Dφ(s, x) • R (s, x)dxds.
T d u T (x)m(T, x)dx + T d φ(t,

3.

T t

T d Dφ(s, x) • R (s, x)dxds tends to 0 because R tends to 0 in L q ((0, T ) × T d ) and Dφ ∈ L β ((0, T ) × T d )) with β ≥ q * .
4. Finally, from the convexity of L with respect to (m, z) and from (1.11), we see that lim sup

→0 T t T d L(x, m (s, x), z (s, x))dxds ≤ T t T d L(x, m(s, x), z(s, x))dxds.
From (4.13) and all the points above, we deduce (4.5).

Similarly as for (4.13), we obtain that

- T d φ(t, x)m (t, x)dx + T d φ(0, x)m (0, x)dx ≤ - t 0 T d γ ac ( s, x)m (s, x)dxds - t 0 T d m (s, x)H(x, m (s, x), Dφ(s, x))dxds
+ t 0 T d L(x, m (s, x), z (s, x))dxds +
t 0 T d Dφ(s, x) • R (s, x)dxds, (4.14)
in which T d φ(0, x)m (0, x)dx has a meaning from Lemma 4.1. We claim that T d φ(0, x)m (0, x)dx → T d φ(0, x)m 0 (x)dx as → 0: indeed, let ζ ∈ (0, 1) be the Hölder exponent in Lemma 2.1: in view of Remark 2.1, we know that ζ ≥ min(1/2, (1 -α)/β); calling h 2, (x) = -d h 2 ( x ) and h 1, (t) = -δ h 1 ( t δ ), we get that for all x ∈ T d ,

|m (0, x) -(h 2, m 0 )(x)| = (m(s, y) -m 0 (y))h 1, (-s)h 2, (x -y)dsdy ≤ C s δζ-d-1 h 1, (-s)ds ≤ C δζ-d-1 .
Choosing δ large enough, (i.e. such that δζ -d -1 > 0) and using the fact that

m 0 is C 1 , lim →0 h 2, m 0 -m 0 L ∞ (T d ) = 0. Therefore, lim →0 m (0, •) -m 0 L ∞ (T d ) = 0.
The claim follows from the continuity stated in Lemma 4.1.

The fact that T d φ(0, x)m (0, x)dx → T d φ(0, x)m 0 (x)dx and the arguments above imply (4.6). Let us now suppose that (4.7) holds: then the inequalities in (4.5) and (4.6) are equalities, for almost all t. For σ > 0 let us introduce the set

E σ (t) = (s, y) ∈ [t, T ] × T d : m(s, y)H(x, m(s, y), Dφ(s, y)) -L(y, m(s, y), z(s, y)) ≤ z(s, y) • Dφ(s, y) -σ . If |E σ (t)| > 0, then for > 0 small enough, |E ,σ (t)| > |E σ (t)|/2
, where

E ,σ (t) = (s, y) ∈ [t, T ] × T d : m (s, y)H(x, m (s, y), Dφ(s, y)) -L(y, m (s, y), z (s, y)) ≤ z (s, y) • Dφ(s, y) -σ 2 .
Then (4.12) becomes

T t T d (Dφ(s, x) • z (s, x) -m (s, x)H(x, m (s, x), Dφ(s, x))) dxds ≥ - T t T d L(x, m (s, x), z (s, x))dxds + σ 4 |E σ (t)|, (4.15)
which implies that

- T d m(T, x)u T (x)dx + T d m(t, x)φ(t, x)dx + T s=t T d m(s, x) (γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) dxds ≤ T s=t T d L(x, m(s, x), z(s, x))dxds - σ 4 |E σ (t)|,
in contradiction with the fact that there is an equality in (4.5).

Hence, m(s, y)H(x, m(s, y), Dφ(s, y)) -L(y, m(s, y), z(s, y)) = z(s, y) • Dφ(s, y) holds almost everywhere.

In view of (2.6), this shows that z(s, y) = m(s, y)H p (x, m(s, y), Dφ(s, y)) a.e. in {m > 0}. Furthermore, from (2.5) and the fact that

T 0 T d L(
x, m(t, x), z(t, x))dxdt < +∞, we see that z(s, y) = 0 a.e. in {m = 0}. Hence, z(s, y) = m(s, y)H p (x, m(s, y), Dφ(s, y)) a.e. in (0,

T ) × T d . Proposition 4.1. sup (φ,γ)∈K J(φ, γ) = sup φ∈K0 A(φ). Proof. It is clear that ∀φ ∈ K 0 , -∞ < A(φ) ≤ sup (φ,γ)∈K J(φ, γ): indeed consider m achieving A(φ, m) = A(φ), i.e. m = ψ ∂φ ∂t + ν∆φ, Dφ . We know that T 0 T d m 1-α |Dφ| β < +∞ and that T 0 T d m q < ∞. Let us take γ = γ 1 + γ 2 where γ 1 = -(•, m) -m ∂ ∂m (•, m) and γ 2 = (1-α) m α |Dφ| β 1 m>0 .
It is easy to check that (φ, γ) ∈ K and that A(φ) = J(φ, γ). Hence sup (φ,γ)∈K J(φ, γ) ≥ sup φ∈K0 A(φ). For the reverse inequality, consider (φ, γ) ∈ K such that J(φ, γ) > -∞. This implies that

T 0 T d K(x, γ ac (s, x), Dφ(s, x))dxds > -∞.
Let (m * , z * ) be the pair of functions achieving (2.7), see Lemma 2.2. From Lemma 4.2,

J(φ, γ) ≤ T 0 T d m * (t, x) (γ ac (t, x) + H(x, m * (t, x), Dφ(t, x))) dxdt + T d m 0 (x)φ(0 + , x)dx ≤ T s=0 T d L(x, m * (s, x), z * (s, x))dxds + T d m * (T, x)u T (x)dx = B(m * , z * ). Hence, J(φ, γ) ≤ min (m,z)∈K1 B(m, z) = sup φ∈K0 A(φ).
and we conclude using (4.4). Proof.

Existence of a solution of the relaxed problem

Step 1 Consider the maximizing sequence φ n for problem (2.3) described in Lemma 3.1 and call m n the function such that A(φ n ) = A(φ n , m n ). Also, let the functions γ 1 n and γ 2 n be defined as in Lemma 3.1. We know that ∂φ n ∂t

+ ν∆φ n = γ n ≥ γ 1 n + γ 2 n ,
and (φ n , γ n ) ∈ K. The definition of γ 1 n ensures that γ 1 n ≤ -(•, 0). Up to the extraction of a subsequence, we may assume that φ n φ in L β (0, T, W 1,β (T d )), m n m in L q ((0, T ) × T d ), γ 1 n γ 1 in L q * ((0, T ) × T d ), with γ 1 ≤ -(•, 0) a.e., γ 2 n γ2 in M((0, T ) × T d ) * . It is clear that γ2 is a positive Radon measure that we write γ2 = γ 2 + γ sing . We see that

∂φ ∂t + ν∆φ ≥ γ = γ 1 + γ 2 .
Thus for any function ξ in Lip(T d ), the function t → T d ξ(x)φ(t, x)dx has a BV representative. Moreover, since

t → ζ n (t) = T d ξ(x)φ n (t, x)dx is bounded in L ∞ ((0, T )) by C ξ L ∞ (T d )
, where C is independent of ξ, we can assume (up to the extraction of a subsequence) that ζ n ζ in L ∞ ((0, T )) weak *, and ζ L ∞ ((0,T )) ≤ C ξ L ∞ (T d ) . Since for all smooth function ψ : [0, T ] → R, we know that T 0

T d ξ(x)φ n (t, x)ψ(t)dxdt tends to T 0 T d ξ(x)φ(t, x)ψ(t)dxdt, we see that ζ = T d ξ(x)φ(•, x)dx. There- fore T d ξ(x)φ(t, x)dx ≤ C ξ L ∞ (T d )
. This shows that (φ, γ) ∈ K.

Step 2 It can be proved that the map (φ, γ) ∈ K → -

T 0 T d K(x, -γ ac (t, x), -Dφ(t, x))dxdt is the restriction to K of the convex conjugate of the map (µ, z) ∈ E 1 → T 0 T d L(x, µ(t, x), z(t, x))dxdt.
Therefore, the map (φ, γ) ∈ K → T 0

T d K(x, γ ac (t,
dζ n dt (t) -ν T d Dm 0 (x)Dφ n (t, x)dx - T d γ 1 n (t, x)m 0 (x)dx ≥ 0.
Thanks to the a priori bounds on φ n and γ 1 n , this implies that

ζ n (t) ≥ ζ n (0) -C(t 1 β * + t 1 q ), ∀t ∈ [0, T ].
Letting n tend to +∞,

ζ(t) ≥ lim sup n→∞ ζ n (0) -C(t 1 β * + t 1 q ), a.e. t ∈ [0, T ]. Hence ζ(0) ≥ lim sup n→∞ ζ n (0), i.e. T d m 0 (x)φ(t, x)dx ≥ lim sup n→∞ T d m 0 (x)φ n (t, x)dx.
Step 4 By combining the results of steps 2 and 3, we see that

J(φ, γ) ≥ lim sup n→∞ A(φ n ) = sup φ∈K0 A( φ) = sup ( φ,γ)∈K J( φ, γ),
Using the fact that γ 1 ≤ -(•, 0) a.e. and (2.14), we can always decrease γ 2 where Dφ = 0 in such a way that (φ, γ) ∈ K and the value of J(φ, γ) is preserved.

5 Weak solutions to the system of PDEs Definition 5.1. A pair (φ, m) ∈ L β (0, T ; W 1,β (T d )) × L q ((0, T ) × T d ) with m ≥ 0 almost everywhere, is a weak solution of (1.1)-

(1.3) if 1. m -α |Dφ| β 1 m>0 ∈ L 1 ((0, T ) × T d ),
φ(T, x) ≤ u T (x), (5.1)
holds in the sense of distributions, with the convention that H(x, m, p) + mH m (x, m, p) = (x, 0) if m = 0 and p = 0.

(5.2)

3. The following:

∂m ∂t (t, x) -ν∆m(t, x) + div(m(t, •)H p (•, m(t, •), Dφ(t, •)))(x) = 0, m(0, x) = m 0 (x) (5.3)
holds in the sense of distributions 4. There exists a constant C such that for each function ξ ∈ Lip(T d ),

T d ξ(x)φ(t, x)dx L ∞ (0,T ) ≤ C ξ L ∞ (T d ) .
5. The following identity holds

0 = T 0 T d m(t,
γ(t, x) = -H(x, m(t, x), D φ(t, x)) -m(t, x)H m (x, m(t, x), D φ(t, x)),
with the convention (5.2).

Proof. Let (m * , z * ) ∈ K 1 be the minimizer of (2.7), (which implies that m * ∈ L q ((0, T ) × T d ) and that T 0

T d L(x, m * (t, x), z * (t, x
))dxdt < +∞) and (φ * , γ * ) ∈ K be a maximizer of (4.3). We know that T 0

T d K(x, γ * ,ac (t, x), Dφ * (t, x))dxdt > -∞. Hence, T 0 T d L(x, m * (t, x), z * (t, x))dxdt + T d m * (T, x)u T (x)dx = T 0 T d K(x, γ * ,ac (t, x), Dφ * (t, x))dxdt + T d m 0 (x)φ * (0, x)dx ≤ T 0 T d m * (t, x) (γ
) -α |Dφ * | β 1 m * >0 ∈ L 1 ((0, T ) × T d ).
Using also (5.5), we see that

(m * ) 1-α |Dφ * | β ∈ L 1 ((0, T ) × T d ).
From the inequalities satisfied by γ * ,ac , we see that (5.1) holds. Finally, (5.4) is obtained by combining all the points above. We have proved that (φ * , m * ) is a weak solution in the sense of Definition 5.1.

Suppose now that ( φ, m) is a weak solution in the sense of Definition 5.1. Let us choose γ(t, x) = -H(x, m(t, x), D φ(t, x)) -m(t, x)H m (x, m(t, x), D φ(t, x)) always with the convention (5.2), (we split γ as follows: γ = γ1 + γ2 , where γ1

= -(•, m) -m ∂ ∂m (•, m), γ2 = (1 -α) |D φ| β mα 1 { m>0}
) and z(t, x) = m(t, x)H p (x, m(t, x), D φ(t, x)). It is clear that ( φ, γ) ∈ K and that ( m, z) ∈ K 1 .

From the definition of γ, we see that a.e., m(t, x) = argmin µ≥0 µγ(t, x) + µH(x, µ, D φ(t, x)) . Moreover, since a.e., H p (x, m(t, x), D φ(t, x)) = argmin ξ ξ • D φ(t, x) + L(x, m(t, x), ξ) , we see that a.e.

( m(t, x), z(t, x)) = argmin (µ,z) µγ(t, x) + z • D φ(t, x) + L(x, µ, z) , i.e. K(x, γ(t, x), D φ(t, x)) = m(t, x)γ(t, x) + z(t, x) • D φ(t, x) + L(x, m(t, x), z(t, x)).

(5.6)

Note also that 0 ≤ = m(t, x)γ(t, x) + z(t, x) • D φ(t, x) + L(x, m(t, x), z(t, x))

=K(x, γ(t, x), D φ(t, x))

where the second line is obtained using the definition of z and the third line is obtained using (5.6).

Combining the latter two observations, we see that max 3). We know that mH p (x, m, D φ1 ) = mH p (x, m, D φ2 ): this implies that |D φ1 (t, x)| β-2 D φ1 (t, x) = |D φ2 (t, x)| β-2 D φ2 (t, x) at almost every (t, x) such that m(t, x) > 0: hence at almost every (t, x) such that m(t, x) > 0, |D φ1 (t, x)| = |D φ2 (t, x)| and finally D φ1 (t, x) = D φ2 (t, x). Moreover, D φ1 (t, x) = D φ2 (t, x) = 0 at almost every (t, x) such that m(t, x) = 0. Therefore, D φ1 (t, x) = D φ2 (t, x) at almost every (t, x). This means that φ1 -φ2 only depends on t, and that γ1 = γ2 a.e., where γi (t, x) = -H(x, m(t, x), D φi (t, x)) -m(t, x)H m (x, m(t, x), D φi (t, x)), using convention (5.2). Using (5.4), we deduce from the previous points that φ1 (t = 0 + , •) = φ2 (t = 0 + , •) a.e. in T d . We set γ = γ1 = γ2 ∈ L 1 ((0, T ) × T d ) and z = mH p (x, m, D φ1 ) = mH p (x, m, D φ2 ). Going back to the proof of Lemma 4.2, we see that for almost all t, both ( φ1 , γ) and ( φ2 , γ) achieve the equality in (4.6) with (m, z) = ( m, z). From the previous points, this implies that

T d m(t, x) φ1 (t, x)dx = T d m(t, x) φ2 (t, x)dx
for almost every t. Since φ1 (t, x) -φ2 (t, x) only depends on t and m(t) is a probability measure, the latter implies that φ 1 = φ 2 holds almost everywhere.

Proposition 4 . 2 .

 42 The relaxed problem (4.3) has at least a solution (φ, γ) ∈ K.

Step 3

 3 x), Dφ(t, x))dxdt is upper semi-continuous for the weak * topology of L β (0, T ; W 1,β (T d )) × M, and therefore,T 0 T d K(x, γ ac (t, x), Dφ(t, x))dxdt ≥ lim sup n→∞ T 0 T d K(x, γ n (t, x), Dφ(t, x))dxdt. (4.16) Let ζ n = T d m 0 (x)φ n (t, x)dx and ζ = T d m 0 (x)φ(t, x)dx. We have seen that ζ n ζ in L ∞ ((0, T )) weak *; from the inequation satisfied by φ n , dζ n dt (t) -ν T d Dm 0 (x)Dφ n (t, x)dx -T d (γ 1 n (t, x) + γ 2 n (t, x))m 0 (x)dx ≥ 0, which implies that

T 0 T

 0 d m(t, x)γ 2 (t, x)dxdt < +∞ and thatT 0 T d |z(t, x) • D φ(t, x)|dxdt ≤ C T 0 T d m1-α (t, x)|D φ(t, x)| β dxdt < +∞.Let (m * , z * ) be the solution of (2.7). From Lemma 4.2,B(m * , z * ) = T 0 T d L(x, m * (t, x), z * (t, x))dxdt + T d m * (T, x)u T (x)dx ≥ T 0 T d m * (t, x) γ(t, x) + H(x, m * (t, x), D φ(t, x)) dxdt + T d m 0 (x) φ(0, x)dx.Hence,B(m * , z * ) ≥ T 0 T d K(x, γ(t, x), D φ(t, x))dxdt + T d m 0 (x) φ(0, x)dx = T 0 T d m(t, x) γ(t, x) + H(x, m(t, x), D φ(t, x)) dxdt + T d m 0 (x) φ(0, x)dx,where the first line comes from (5.6) and the definition of K, and the last line comes from the definition of γ. Finally, from the latter inequality and (5.4), we deduce thatB(m * , z * ) ≥ T 0 T d L(x, m(t, x), z(t, x))dxdt + T d m(T, x)u T (x)dx =B( m, z).Therefore ( m, z) achieves the minimum in (2.7) and m = m * , z = z * . It remains to prove that ( φ, γ) achieves the maximum in (4.3). We deduce from Lemma 2.2, Proposition 4.1 and the latter point that max(φ,γ)∈K J(φ, γ) = B( m, z) = T 0 T d L(x, m(t, x), z(t, x))dxdt + T d m(T, x)u T (x)dx = -x)H m (x, m(t, x), D φ(t, x))dxdt + T d m0 (x) φ(0 + , x)dx,where the last line comes from (5.4) and the definition of z. But, using the definition of γ, -m2 (t, x)H m (x, m(t, x), D φ(t, x)) = m(t, x)γ(t, x) + m(t, x)H(x, m(t, x), D φ(t, x))

Theorem 5 . 2 .

 52 , γ(t, x), D φ(t, x))dxdt + T d m0 (x) φ(0 + , x)dx = J( φ, γ),which concludes the proof. There exists a unique weak solution of (1.1)-(1.3).Proof. Existence is a direct consequence of Theorem 5.1, Proposition 4.2 and Lemma 2.2. From Theorem 5.1, we also see that any weak solution ( φ, m) of (1.1)-(1.3) is such that the pair ( m, mH p (x, m, D φ)) is the minimizer of (2.7). Thus ( m, mH p (x, m, D φ)) is unique. Consider now two weak solutions ( φ1 , m) and ( φ2 , m) of (1.1)-(1.

  x)m(t, x)dx.2. From Remark 4.1, or more precisely the facts that T 0 T d K(x, γ ac (t, x), Dφ(t, x))dxdt > -∞ and that γ ac (s, x) + H(x, m (s, x), Dφ(s, x))m (s, x) ≥ K(x, γ ac (s, x), Dφ(s, x)),

	we can use Fatou lemma and get that
	T
			(γ ac (s, x) + H(x, m(s, x), Dφ(s, x))) m(s, x)dxds
	t	T d
			T
	≤ lim inf →0	t	T d

  Dφ(t, x) = 0 a.e. in the region {(t, x) : m(t, x) = 0}, m 1-α |Dφ| β ∈ L 1 ((0, T ) × T d ) and

		T
			m(t, x)L (x, m(t, x), H p (x, m(t, x), Dφ(t, x))) dxdt < +∞.
		0	T d
	2. The following:
	∂φ ∂t	(t, x) + ν∆φ(t, x) + H(x, m(t, x), Dφ(t, x)) + m(t, x)H m (x, m(t, x), Dφ(t, x)) ≥ 0,

  But from Lemma 4.2 and especially (4.5)-(4.6), we see that the latter inequality is in fact an equality; then, from the last part of Lemma 4.2, this yields that z * (t, x) = m * (t, x)H p (x, m * (t, x), Dφ * (t, x)) and that L(x, m * (t, x), z * (t, x)) = L (x, m * (t, x), m * (t, x)H p (x, m * (t, x), Dφ * (t, x)) almost everywhere. We have proved that m * H p (•, m * , Dφ * ) ∈ L 1 ((0, T ) × T d ) and that (5.3) holds. Moreover * ,ac (t, x), Dφ * (t, x)) = m * (t, x) (γ * ,ac (t, x) + H(x, m * (t, x), Dφ * (t, x))) (5.5) almost everywhere. In view of Remark 2.2, this implies that Dφ * (t, x) = 0 almost everywhere in the region where m * (t, x) = 0, and that γ * ,ac (t, x) = -H(x, m * (t, x), Dφ * (t, x)) -m * (t, x)H m (x, m * (t, x), Dφ * (t, x)), a.e. where m

	T	T
	0 m This implies that T d K(x, γ * ,ac (t, x), Dφ * (t, x))dxdt = 0 T d
	K(x, γ

* ,ac (t, x) + H(x, m * (t, x), Dφ * (t, x))) dxdt +

T d m 0 (x)φ * (0, x)dx * (t, x) (γ * ,ac (t, x) + H(x, m * (t, x), Dφ * (t, x))) dxdt. * (t, x) > 0

and with the convention (5.2), we see that

γ * ,ac (t, x) ≥ -H(x, m * (t, x), Dφ * (t, x)) -m * (t, x)H m (x, m * (t, x), Dφ * (t, x)),

a.e..

These observation imply that (m *

Acknowledgements

The first author was partially funded by the ANR projects ANR-12-MONU-0013 and ANR-12-BS01-0008-01.


Fabienne Comte
email: fabienne.comte@parisdescartes.fr

Claire Lacour
email: claire.lacour@math.u-psud.fr

Pointwise deconvolution with unknown error distribution

This note presents rates of convergence for the pointwise mean squared error in the deconvolution problem with estimated characteristic function of the errors.

Résumé

Déconvolution ponctuelle avec distribution de l'erreur inconnue. Cette note présente les vitesses de convergence pour le risque quadratique ponctuel dans le problème de déconvolution avec fonction caractéristique des erreurs estimée.

Introduction

Let us consider the following model:

Y j = X j + ε j j = 1, . . . , n (1)
where (X j ) 1≤j≤n and (ε j ) 1≤j≤n are independent sequences of i.i.d. variables. We denote by f the density of X j and by f ε the density of ε j . The aim is to estimate f when only Y 1 , . . . , Y n are observed. Contrary to the classical convolution model, we do not assume that the density of the error is known, but that we additionally observe ε -1 , . . . , ε -M , a noise sample with distribution f ε , independent of (Y 1 , . . . , Y n ). Note that the availability of two distinct samples makes the problem identifiable.

Although there exists a huge literature concerning the estimation of f when f ε is known, this problem without the knowledge of f ε has been less studied. One can cite [START_REF] Efromovich | Density estimation for the case of supersmooth measurement error[END_REF] in a context of circular data and [START_REF] Diggle | A Fourier approach to nonparametric deconvolution of a density estimate[END_REF] who examine the case M ≥ n. [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] gives an upper bound and a lower bound for the integrated risk in the case where both f and f ε are ordinary smooth, and [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF] gives upper bounds for the integrated risk in a larger context of regularities. An other practical issue to the considered problem is the study of the model of repeated observations, see [START_REF] Delaigle | On deconvolution with repeated measurements[END_REF].

The contribution of this note is to provide a class of estimators and compute upper bounds for their pointwise rates of convergence depending on M and n in a general setting.

Notations. For z a complex number, z denotes its conjugate and |z| its modulus. For a function t : R → R belonging to L 1 ∩ L 2 (R), we denote by t the L 2 norm of t and by t 1 the L 1 norm of t. The Fourier transform t * of t is defined by t * (u) = e -ixu t(x)dx.

Estimation procedure

It easily follows from Model (1) and independence assumptions that, if f Y denotes the common density of the Y j 's, then

f Y = f * f ε and thus f * Y = f * f * ε .
Therefore, under the classical assumption:

(A1) ∀x ∈ R, f * ε (x) = 0, the equality f * = f * Y /f * ε yields an estimator of f * by considering the following estimate of f * Y : f * Y (u) = n -1 n j=1 e -iuYj . Indeed, if f * ε is known, we can use the estimate of f * : f * Y /f * ε .
Then, we should use inverse Fourier transform to get an estimate of f . As 1/f * ε is in general not integrable (think of a Gaussian density for instance), this inverse Fourier transform does not exist, and a cutoff is used. The final estimator for known f ε can thus be written:

(2π) -1 |u|≤πm e iux f * Y (u)/f * ε (u)du.
Here m is a real positive bandwidth parameter. This estimator is classical in the sense that it corresponds both to a kernel estimator built with the sinc kernel (see [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF]) or to a projection type estimator as in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]. Now, f * ε is unknown and we have to estimate it. Therefore, we use the preliminary sample and we define the natural estimator of

f * ε : f * ε (x) = 1 M M j=1 e -ixε-j
. Next, we introduce as in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] the truncated estimator:

1 f * ε (x) = 1 {| f * ε (x)|≥M -1/2 } f * ε (x) = 1 f * ε (x) if | f * ε (x)| ≥ M -1/2 and 1 f * ε (x) = 0 otherwise. Then our estimator is fm (x) = 1 2π πm -πm e ixu f * Y (u) f * ε (u)
du.

(2)

Study of the pointwise mean squared error

We introduce the notations

∆(m) = 1 2π πm -πm |f * ε (u)| -2 du, ∆ 0 (m) = 1 2π πm -πm |f * ε (u)| -1 du 2 , ∆ 0 f (m) = 1 2π πm -πm |f * (u)| |f * ε (u)| du 2 .
Proposition 3.1. Consider Model (1) under (A1), then there exist constants C, C > 0 such that for all positive real m and all positive integers n, M ,

E[( fm (x) -f (x)) 2 ] ≤ 2 1 2π |t|≥πm |f * (t)|dt 2 + C n min( f * Y 1 ∆(m), ∆ 0 (m)) + C ∆ 0 f (m) M .
Note that the result of Proposition 3.1 holds for any fixed and independent integers M and n. Assumption (A1) is generally strengthened by the following description of the rate of decrease of f * ε :

(A2) There exist s ≥ 0, b > 0, γ ∈ R (γ > 0 if s = 0) and k 0 , k 1 > 0 such that ∀x ∈ R k 0 (x 2 + 1) -γ/2 exp(-b|x| s ) ≤ |f * ε (x)| ≤ k 1 (x 2 + 1) -γ/2 exp(-b|x| s ).
Moreover, the density function f to estimate generally belongs to the following type of smoothness spaces:

A δ,r,a (l) = {f density on R and |f * (x)| 2 (x 2 + 1) δ exp(2a|x| r )dx ≤ l} (3) with r ≥ 0, a > 0, δ ∈ R and δ > 1/2 if r = 0, l > 0.
When r > 0 (respectively s > 0), the function f (respectively f ε ) is known as supersmooth, and as ordinary smooth otherwise. The spaces of ordinary smooth functions correspond to classic Sobolev classes, while supersmooth functions are infinitely differentiable. For example normal (r = 2) and Cauchy (r = 1) densities are supersmooth.

Corollary 3.2. If f * ε satisfies (A2) and if f ∈ A δ,r,a (l), the rates of convergence for the Mean Squared Error E[( fm0 (x) -f (x)) 2 ] are given in Table 1 (which also contains the chosen m 0 ). Indeed, if f ∈ A δ,r,a (l), the bias term can be bounded in the following way

2 1 2π |t|≥πm |f * (t)|dt 2 ≤ K 1 (πm) -2δ+1-r exp(-2a(πm) r ).
and straightforward computation gives ∆(m) ≤ K 2 (πm) 2γ+1-s exp(2b(πm) s ) and ∆ 0 (m) ≤ K 3 (πm) 2γ+2-2s exp(2b(πm) s ); lastly, denoting by v = 2γ + 1 -s, we have

∆ 0 f (m)K -1 4 ≤ (πm) (2γ+1-2δ)+ (log(m)) 1 δ=γ+1/2 1 {r=s=0} + (πm) v-max(2δ,s-1) exp(2b(πm) s )1 {s>r} +(πm) v-2δ exp(2(b -a)(πm) s )1 {r=s,b≥a} + 1 {r>s}∪{r=s,b<a}
where K 1 , K 2 , K 3 , K 4 are positive constants. Then the rates of Table 1 are obtained by choosing adequate m 0 depending on n, M and the smoothness indices. For the case (r > 0, s > 0), the rules for the compromise between supersmooth terms in both squared bias and variance are given in [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF] in the case of a known noise. The computations are similar for the present study. As this case is very tedious to write and contains several sub-cases, we omit the precise rates: it is sufficient to know that they decrease faster than any logarithmic functions, both in M and n.

s = 0 s > 0 r = 0 n -2δ-1 2δ+2γ + M -[min(1, 2δ-1 2γ )] (log M ) 1 δ=γ+1/2 (log n) -(2δ-1)/s + (log M ) -(2δ-1)/s for m 0 = min(n 1/(2δ+2γ) , M 1/ max(2γ,2δ-1) ) for m 0 = π -1 (log(min(n, M ))/(2b + 1)) 1/s r > 0 (log n) (2γ+1)/r n + 1 M for See comment below. m 0 = π -1 [(log(n) -(1 + 2(δ + γ)/r) log log(n))/(2a)] 1/r ,
The rates in term of n are known to be the optimal one for the deconvolution with known error (see [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] and [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF]). They are recovered as soon as M ≥ n. Extending the proof of [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], we can prove the optimality of the rate M -1 in the cases where f is smoother than f ε and r ≤ 1. Note that even for M ≥ n, automatic selection of m should be performed in the spirit of [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF], but none of the quoted works proves theoretical results about it.

Notice that Corollary 3.2 has not only a theoretical importance but also provides an answer to practical problems of noised observations by studying in detail the effect of preliminary measurements.

Proof of Proposition

3.1 First, let us denote f m (x) = (2π) -1 πm -πm e ixu f * (u)du and R(x) = ( f * ε (x)) -1 -(f * ε (x)) -1 . Then E[( fm (x) -f (x)) 2 ] ≤ 2(f m (x) -f (x)) 2 + 2E[( fm (x) -f m (x)) 2 ] ≤ 2(f m (x) -f (x)) 2 + 4Var 1 2π πm -πm e ixu f * Y (u) f * ε (-u) du + 4E 1 2π πm -πm e ixu f * Y (u)R(u)du 2 (4) Since (f -f m )(x) = (1/2π)(f * -f * m ) * (-x)
, we can bound the bias term in the following way

(f m (x) -f (x)) 2 ≤ 1 2π |t|≥πm |f * (t)|dt 2 . ( 5
)
The second term of the right-hand-side of ( 4) is the variance term when f * ε is known and has already been studied: it follows from [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF] that

Var 1 2π πm -πm e ixu f * Y (u) f * ε (-u) du ≤ 1 2πn min( f * Y 1 ∆(m), ∆ 0 (m)). (6)
For the last remaining term in the right-hand-side of ( 4), we bound it by [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] proved that there exists a positive constant C 1 such that

2E 1 2π πm -πm e ixu ( f * Y (u) -f * Y (u))R(u)du 2 + 2E 1 2π πm -πm e ixu f * Y (u)R(u)du 2 := 2T 1 + 2T 2 .
E|[R(u)| 2 ] = E 1 f * ε (u) - 1 f * ε (u) 2 ≤ C 1 min 1 |f * ε (u)| 2 , 1 M |f * ε (u)| 4 .
Then we find

T 1 = 1 4π 2 e ix(u-v) Cov( f * Y (u), f * Y (v))E(R(u) R(v))dudv ≤ 1 4π 2 n |f * Y (u -v)| E(|R(u)| 2 )E(|R(v)| 2 )dudv ≤ C 1 4π 2 n |f * Y (u -v)| |f * ε (u)f * ε (v)| dudv.
This term is clearly bounded by C 1 (2πn) -1 ∆ 0 (m). Moreover writing it as

C 1 4π 2 n |f * Y (u -v)| |f * ε (u)| |f * Y (u -v)| |f * ε (v)|
and thus it has the same order as the usual variance term. Lastly,

T 2 ≤ 1 4π 2 |u|,|v|≤πm |f * Y (u)f * Y (v)| E(|R(u)| 2 )E(|R(v)| 2 )dudv ≤ 1 4π 2 πm -πm |f * Y (u)| E(|R(u)| 2 )du 2 ≤ C 1 4π 2 M πm -πm |f * Y (u)| |f * ε (u)| 2 du 2 = C 1 ∆ 0 f (m) 2πM . (8)
Inserting the bounds (5) to (8) in Inequality (4), we obtain the result of Proposition 3.1.

  dudv and using the Schwarz Inequality, and the Fubini Theorem yields the bound C 1 (2πn) -1 f * Y 1

Table 1 :

 1 Rates of convergence for the MSE if f

* ε satisfies (A2) and f ∈ A δ,r,a (l).


Matthieu J Mercier

Louis Gostiaux

Karl Helfrich

Joel Sommeria

Samuel Viboud

Henri Didelle

Sasan J Ghaemsaidi

Thierry Dauxois

Thomas Peacock

Large-scale, realistic laboratory modeling of M 2 internal tide generation at the Luzon Strait

The complex double-ridge system in the Luzon Strait in the South China Sea (SCS) is one of the strongest sources of internal tides in the oceans, associated with which are some of the largest amplitude internal solitary waves on record. An issue of debate, however, has been the specific nature of their generation mechanism. To provide insight, we present the results of a large-scale laboratory experiment performed at the Coriolis platform. The experiment was carefully designed so that the relevant dimensionless parameters, which include the excursion parameter, criticality, Rossby and Froude numbers, closely matched the ocean scenario. The results advocate that a broad and coherent weakly-nonlinear, three-dimensional, M2 internal tide that is shaped by the overall geometry of the double-ridge system is radiated into the South China Sea and subsequently steepens, as opposed to being generated by a particular feature or localized region within the ridge system.

Introduction

The Luzon Strait (LS), a complex double-ridge system between Taiwan and the Philippines, generates some of the strongest internal tides found anywhere in the oceans [START_REF] Alford | Energy flux and dissipation in Luzon Strait: two tales of two ridges[END_REF]. Consequently, a striking characteristic of the South China Sea (SCS) to the west of the LS is the regular observation of large amplitude Internal Solitary Waves (ISWs) with amplitudes up to 170 m and speeds of 3 m/s, taking about 33 hours to traverse the SCS basin [START_REF] Alford | Speed and evolution of nonlinear internal waves transiting the South China Sea[END_REF]. A summary of observations over a three-year time period reveals that these ISWs are primarily directed slightly north of west, with surface signatures of the ISWs are rarely observed east of 120.3 • [START_REF] Jackson | An empirical model for estimating the geographic location of nonlinear internal solitary waves[END_REF].

In recent years, there has been a concerted effort to more clearly establish the early stages of the formation of ISWs in the SCS. The lee wave mechanism [START_REF] Maxworthy | A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge[END_REF], initially alluded to as a candidate process [START_REF] Ebbesmeyer | New observations on internal waves (solitons) in the South China Sea using an acoustic doppler current profiler[END_REF], is now considered inappropriate [START_REF] Alford | Speed and evolution of nonlinear internal waves transiting the South China Sea[END_REF] due to the lack of reported detection between the ridges, or observations in satellite imagery, of ISWs east of 120.3 o [START_REF] Jackson | An empirical model for estimating the geographic location of nonlinear internal solitary waves[END_REF]. Instead, it is now widely accepted that a westward propagating, weakly-nonlinear, mode-1 internal tide generated at the LS steepens to form ISWs [START_REF] Lien | Energy of nonlinear internal waves in the South China Sea[END_REF][START_REF] Li | The generation and evolution of nonlinear internal waves in the deep basin of the South China Sea[END_REF].

A prevailing theme has been to identify generation hot spots within the LS. Two-dimensional models [START_REF] Zhao | Source and propagation of internal solitary waves in the northeastern South China Sea[END_REF][START_REF] Alford | Speed and evolution of nonlinear internal waves transiting the South China Sea[END_REF] and numerical simulations employing idealized or actual transects containing purported generation sites [START_REF] Buijsman | On the generation and evolution of nonlinear internal waves in the South China Sea[END_REF][START_REF] Warn-Varnas | Solitary waves generation dynamics at Luzon Strait[END_REF], have been used in this regard, producing discordant results. The inconsistency of the outcome of twodimensional studies is most likely due to the inherently three-dimensional nature of the internal tide generation process, with extended sections of the two ridges playing a key role [START_REF] Zhang | Threedimensional, nonhydrostatic numerical simulation of nonlinear internal wave generation and propagation in the South China Sea[END_REF][START_REF] Guo | Numerical investigation of internal solitary waves from the Luzon Strait: generation process, mechanism and three-dimensional effects[END_REF].

Here, we present the results of a large-scale, realistic, laboratory project of internal tide generation in the LS. The experiments, performed at the Coriolis Platform of LEGI (Grenoble, France), model key constituents of the generation process: three-dimensional bathymetry, nonlinear stratification, semi-diurnal (M2) tidal forcing, and background rotation. The goal of the experiments is to elucidate the geometry and characterize the nonlinearity of the radiated three-dimensional M2 internal tide, thereby providing insight into the consequent generation of ISWs in the SCS. Due to the difference in the dissipation processes, laboratory experiments are a valuable complement to realistic threedimensional numerical simulations. Any result reproduced by both approaches can be considered robust. We describe the experimental arrangement in §2 and discuss the scalings chosen to preserve dynamic similarity with the ocean in §3. Results for the M2 barotropic and baroclinic (internal) tides are presented in §4 and §5 respectively. The nonlinearity of the internal tide is characterized in §6 before discussion and conclusions in §7.

Experimental configuration

A schematic of the experiment is presented in Fig. 1(a). It was performed at the Coriolis Platform, the world's largest turntable facility for geophysical fluid dynamics experiments, in order to incorporate the appropriate influence of background rotation and to curtail viscous effects. A 5 m long, 2 m wide and 0.6 m tall scale model of the LS constructed based on topographic data provided by the Ocean Data Bank, National Science Council, Taiwan, was positioned near the center of the tank. Partition barriers were introduced north and south of the model, representing Taiwan and the Philippines, respectively. Damping material was affixed to the inner walls of the tank to damp reflections effectively, as discussed in §5. One of a pair of vertically oscillating prismatic plungers was used to generate the equivalent of a M2 barotropic tide [START_REF] Biesel | Les appareils générateurs de houle en laboratoire[END_REF]. Located in the Pacific Ocean (PO) basin behind vertical partitions blocking internal waves emanating from the plunger itself, each plunger was tested individually. No differences were found in the resulting tidal flows.

A three-dimensional rendering of the experimental topography is presented in Fig. 1(b). The horizontal and vertical scalings of the topography compared to the ocean were 1:100,000 and 1:5,000, respectively. To ensure these choices did not affect the dynamical processes being modeled, other physical quantities such as the stratification, tidal forcing and background rotation frequencies were appropriately rescaled, as described in §3. Only topographic features above 3000 m deep were reproduced, since the weak deepocean stratification has negligible effect on the wavelengths and phase speeds of the vertical modes. Several metrics were calculated to ascertain the quality of reproduction of the original ocean data set; one measure, the cross-correlation coefficient, was 0.90 ± 0.03, indicating a good transcription from the ocean to the experiment. Over the same domain, the distribution of slopes, ∇h, presented in Fig. 1(c), confirms that the laboratory topography was indeed 20 times steeper than that of the ocean and with the same distribution.

A nonlinear density stratification was established using salt water introduced via a computer-controlled doublebucket system (see Fig. 1(d)). Velocity field data was obtained using Particle Image Velocimetry (PIV). For horizontal velocity field measurements, small particles of diameter 350 µm with a density in a narrow range (±1 kg/m 3 ) corresponding to the base of the pycnocline were used to seed the fluid, ensuring they would clearly reveal horizontal motions associated with mode-1 and mode-2 internal tides; the particles were illuminated from below by submerged white lights. For vertical velocity field measurements, a moveable laser sheet in a vertical plane cutting across the ridge illuminated similar particles distributed uniformly in density to cover the entire fluid depth. Local density oscillations in the radiated wave field were measured using conductivity and temperature (CT) probes, and acoustic probes on either side of the ridge measured the surface deflection due to the barotropic tide with an accuracy of 0.05 mm.

Governing parameters

There are ten dimensional parameters that characterize internal tide generation at the LS: the barotropic tidal frequency (ω) and velocity (U0), the Coriolis frequency (f ), the maximum and bottom value of the ocean stratifications (Nm, N b ), the depth of the pycnocline (δ) and of the ocean (H), the vertical (h0) and across-ridge horizontal (L) scales of the topography, and the fluid viscosity (ν), be it kinematic or turbulent. With two dimensions, there are thus eight governing dimensionless parameters. The practical choice of these parameters is established [START_REF] Garrett | Internal tide generation in the deep ocean[END_REF], but prudence is necessary when estimating their values. In particular, one must differentiate between dynamics associated with global and local scale processes. We therefore consider both a global scale (Luzon) for the entire LS and a local scale (Batanes) characterized by the channel between Batan and Itbayat islands, which has been touted as a hot spot for internal tide generation.

The eight governing dimensionless parameters are: (1,2) h * = h0/H and δ * = δ/H, the ratio of the topographic height, h0, and the pycnocline depth, δ, to the depth of the surrounding ocean, H, respectively; (3) the aspect ratio of the topographic features h0/L; (4) the ratio of the upper and deep ocean stratifications,

N * = Nm/N b ; (5) the criti- cality, ε * = ∇h/ (ω 2 -f 2 )/(N 2 -ω 2 ),
which is the ratio of the local topographic slope to the internal wave ray slope; (6,7) the Reynolds and Rossby numbers, Re * = U0L/ν and Ro * = U0/f L, respectively, these being the ratios of inertia to viscous and Coriolis forces. Three alternative definitions of the wave nonlinearity parameter are (8): the tidal excursion parameter, A * = U0/ωL, which is the ratio of the barotropic tidal excursion to the horizontal scale of the topography; the Long number, Lo * = U0/Nmh0, quantifying the tendency for upstream blocking and flow separation due to the stratification; or the Froude number, F r * n = U0/c n φ , the ratio of the flow speed to the propagation speed c n φ of mode-n, with n = 1 being the lowest baroclinic mode. A comparison of the dimensional and dimensionless parameters for the ocean and the lab is given in Tables 1 and2, respectively.

Table 1. Characteristic values of key dimensional parameters for internal tide generation on the scale of the Luzon Strait (Luzon) and on the local scale (Batanes). The corresponding laboratory values are Lab L and Lab B , respectively. The scaling factor relating laboratory values to the ocean ones is indicated at the last line.

ω (rad/s)

f (rad/s) Nm (rad/s) N b (rad/s) U 0 (m/s) δ (m) H (m) L (m) h 0 (m) ν (m 2 /s) Luzon
1.40×10 -4 5.00×10 -5 1.57×10 -2 3.65×10 -4 0.1 100 3.0×10 3 10 5 1.5×10 3 10 -4 Batanes 1.40×10 -4 5.00×10 -5 1.57×10 -2 1.28×10

* δ * h 0 /L N * ε * Re * Ro * A * Lo * F r * 1 F r * 2 Luzon 0.50 3.3×10 -2 1.5×10 -2 43.0 [0.004 -14] 1.0×10 8 2.0×10 -2 7.14×10 -3 2.5×10 -2 3.6×10 -2 7.3×10 -2 Batan 0.53 6.6×10 -2 8.0×10 -2 12.3 [0.004 -14] 1.0×10 8 2.0 7.14×10 -1 2.7×10 -1 4.2×10 -1 8.3×10 -1 Lab L 0.50 2.7×10 -2 0.3 42.0 [0.01 -14] 2.76×10 3 2.0×10 -2 7.15×10 -3 1.2×10 -2 2.1×10 -2 4.3×10 -2 Lab B 0.53 5.5×10 -2 1.6 1.82 [0.01 -14] 2.76×10 3 2.0 7.15×10 -1 3.5×10 -1 8.9×10 -1 1.8 Lab L Luzon 1.0 0.82 20 0.98 O(1) 2.76×10 -5 1.0 1.00 0.5 0.6 0.6
The values in Table 2 reveals that the LS can be characterized as a weakly nonlinear system on the large scale, with a relevant influence of background rotation and topographic blocking; and a more nonlinear one on the local scale, with a large excursion parameter promoting lee waves generation and internal wave energy trapping by supercritical flow. The increase by a factor 20 in the aspect ratio and the ray slope (based on the rescaled stratification in Fig. 1(d)), preserving criticality as can be seen when comparing the ocean and laboratory distributions of ε * in Fig. 1(c), is irrelevant in the long wave limit (equivalent to the hydrostatic approximation), which is expected to be valid in the generation process; this is of importance for the subsequent wave steepening leading to ISWs, however, as discussed in §6, which therefore is not reproduced in the experiment. Overall the values of the dimensionless parameters for the LS and the laboratory model are consistent, except for Re * . As such, the nature of the oceanic internal tide generation process will be faithfully reproduced in the experiments.

The M 2 barotropic tide

Experiments were first performed using unstratified water to investigate solely the barotropic tide. Horizontal views of the flow at z = -0.12m (corresponding to -600m in the ocean) are compared with TPXO 7.1 predictions [START_REF] Egbert | Efficient inverse modeling of barotropic ocean tides[END_REF] in Fig. 2(a) and (b); the ellipses and colorbars are scaled according to the parameters listed in Table 1 so that, in principle, they should appear the same. The orientations of the tidal ellipses are in very good agreement; in both cases, the ellipses are almost flat, indicating a weak influence of background rotation for most of the strait, except in very shallow regions where ellipses are quite large, which stress the importance of Coriolis effects in setting the amplitude of the tidal velocity near shallow features. The spatial distributions of the tidal amplitude are also very similar, the intensification of the flow with decreasing depth being well reproduced. Intense magnification above z = -0.12m could not be visualized due to the shadows cast by the laser light (coming from the west) striking the topography.

Figure 2(c) presents a vertical transect of the barotropic tide taken along the blue dashed line in Figure 2(a). Away from topography a depth-uniform, nearly horizontal velocity is observed, as expected for barotropic tidal flow. Amplification of the flow near seamounts is again visible over the west ridge. The acoustic surface probes revealed a phase difference of ∼ 155 o upon passing across the strait, which is in good agreement with the ∼ 180 o phase shift found in TPXO models and numerical simulations [START_REF] Niwa | Three-dimensional numerical simulation of M 2 internal tides in the East China Sea[END_REF]. Finally, the introduction of a density stratification induced no discernible change in the depth-averaged velocities taken along the blue transect in Figure 2(a). Overall, we conclude that the barotropic tidal flow responsible for internal tide generation in the laboratory experiments successfully reproduced that in the ocean.

The M 2 internal tide

Figure 3(a) presents a snapshot of the east-west velocity in an isopycnal plane around z = -0.04m, with instantaneous velocity vectors inside the M2 filtered tidal ellipses. This view essentially contains the baroclinic velocity component by virtue of considering the time instant of flow reversal of the barotropic tide (i.e. the onset of flow from the SCS to the PO). Large-scale coherent waves radiate from the strait in a west-northwest direction. The maps of the amplitude and phase of the M2-filtered baroclinic and barotropic tides, which cannot be readily discriminated, are presented in Fig. 3(b) and (c), respectively. There is a clear modulation of the amplitude oriented in the ∼ 285 o N direction, which is not due to interference with internal waves reflected at the wall of the tank, as no signal was observed in the SCS when filtering solely eastward propagating waves [START_REF] Mercier | Reflection and diffraction of internal waves analyzed with the Hilbert transform[END_REF]. The apparent wavelength of the signal, 1.3±0.15 m, is consistent with an internal tide dominated by the mode-1 wavelength of 1.4 m for this stratification. A notable feature in Fig. 3(c) is that a wave front, evident as a sharp line of constant phase, connects coherently from the central and southern sections of the eastern ridge to the far field west of both ridges, on the way smoothly adjoining the phase of the wave field generated by the northern and central sections of the western ridge; the combined wavefront coming from these sections of the two ridges propagates in a ∼ 285 o N direction. In the south, another distinct wavefront is oriented in a southwestward direction. In the northern section of the strait, between the two ridges, no clear phase propagation exists, a behavior characteristic of a standing wave pattern. In the absence of background rotation, although the wavelength of the radiated internal tide is noticeably diminished, we found little discernible change in the direction of radiation, revealing the dominant role of the topography in shaping the radiated M2 internal tide. Figure 3(d) presents the same data as in Fig. 3(a), but for a vertical transect indicated by the blue dashed line. The structure of the velocity field takes the form of a classic mode-1 signal, with a vertical velocity maxima of alternating signs along the length of the transect and in the vicinity of the pycnocline, as expected from the observations in Fig. 3(a). Modal decomposition confirms these results, determining that ∼ 95% of the energy flux is in mode-1, and the remaining 5% in modes 2 and 3 mainly.

Nonlinearity

Although the ISWs in the SCS are best described by fully nonlinear models [START_REF] Li | The generation and evolution of nonlinear internal waves in the deep basin of the South China Sea[END_REF], in order to reasonably quantify nonlinearity we assume the waves to be governed by a weakly nonlinear, KdV-like equation. The nonlinearity of the wave field is assessed by the ratio ζ/H (it is appropriate to scale with H since the waves are dominantly mode-1), where ζ is a measure of the wave amplitude given by η(x, z, t) = ζ(x, t)φ(z), η being the vertical displacement of an isopycnal initially located at (x, z), and φ the long wave vertical structure function of mode-1 evaluated at the initial depth of the isopycnal [START_REF] Grimshaw | Long nonlinear surface and internal gravity waves in a rotating ocean[END_REF]. Estimations of ζ/H are based on time series presented in Fig. 4, extracted at z = -3.6 cm and coordinates (20.5 • N,120.5 • E). Given the stratification profile, η is obtained from the CT probe recordings of the density perturbation, ρ in Fig. 4 agreement with oceanic observations and a recent analysis of [START_REF] Li | The generation and evolution of nonlinear internal waves in the deep basin of the South China Sea[END_REF] based upon the rotationally-modified KdV equation. The steepening of the radiated internal tide, producing sharp ISWs, is not observed in the experiments. This is not due to a stronger dissipation, but rather the consequence of the modified aspect ratio in the experiments. The characteristic wavelength of a soliton with amplitude ζ0 is LS ∼ (H 3 /ζ0) 1/2 , while the wavelength of the linear mode-1 is λ1 H/ tan θ, with tan θ = (ω 2 -f 2 )/(N 2 -ω 2 ). For our experiments (ζ0/H = 0.05), the ratio LS/λ1 = (H/ζ0) 1/2 tan θ ∼ 1, meaning that ISWs have the same horizontal scale as mode-1, while in the ocean the horizontal scale of ISWs is roughly 1/25 that of the mode-1 tide. The experiments do not preserve the scale separation between ISWs and the internal tide required to observe the subsequent steepening.

Conclusions and Discussion

These experiments provide a clear demonstration that despite having complex three-dimensional geometry, the LS radiates a spatially coherent, mode-1 dominated, weakly nonlinear M2 internal tide oriented in a west-northwest direction. The spacing of the ridges being on the order of the mode-1 internal wavelength for M2, extended sections of both ridges shape the westward propagating M2 internal tide. We observed modulation of the amplitude of the radiated wave field due to interference between the barotropic and baroclinic tide, an effect observed in numerical models [START_REF] Buijsman | On the generation and evolution of nonlinear internal waves in the South China Sea[END_REF]. The commensurate spacing of the ridges in the northern section of the LS results in a standing wave pattern. Being weakly nonlinear, the radiated internal tide is prone to steepening, and this is the origin of the ISWs in the SCS, although the steepening process itself is not captured in these experiments. These results do not preclude the ability of local topographic features to produce ISWs, observed on a smaller scale [START_REF] Ramp | Observations of a mode-2 nonlinear internal wave on the northern Heng-Chun ridge south of Taiwan[END_REF]. We expect that further experimental work incorporating both the diurnal and semi-diurnal tides will provide insight on ISWs with differing characteristics, such as the reported A and B waves [START_REF] Ramp | Internal solitons in the northeastern South China Sea part I: Sources and deep water propagation[END_REF]. Other potentially important considerations, such as the Kuroshio, could not reasonably be incorporated into the experiments.

Figure 1 .

 1 Figure 1. (a) Schematic of the experiment. Vertical partitions (1) extend from Taiwan and The Philippines to the side of the tank . After filling, the PO and SCS are separated using inserted barriers (2) and the barotropic tide is generated using prismatic plungers (3) located behind vertical walls (4). Dashed green lines and the dotted square indicate the vertical PIV planes and overhead field of view, respectively. Red stars and circles indicate the location of the CT and acoustic probes, respectively. (b) The experimental topography, with Taiwan (T), The Philippines (P), Itbayat Island (I), Sabtang & Batan Islands (S) and Babuyan Island (B) indicated. (c) Distributions of the topographic slope (black) and the criticality parameter ε * (red) for the experiments (solid) and the ocean (dashed). The ocean slopes have been multiplied by 20 to match the laboratory range. (d) Typical winter and summer stratifications (red) in the LS, compared to a laboratory stratification (black).

Figure 2 .

 2 Figure 2. M2 tidal ellipses and horizontal velocity magnitude (a) in the experiment and (b) from TPXO 7.1. The ellipses and colorbars are scaled using the relevant scaling parameters listed in Table 1. (c) Vertical view of the velocity (arrows) and its time-averaged horizontal component (colormap) extracted along the W-NW transect (blue dashed line) in (a), when the tidal flow from the SCS to the PO is maximum.

Figure 3 .

 3 Figure 3. (a) Colormap of the east-west velocity in the isopycnal plane at z = -0.04m at an instant of barotropic tide flow reversal. The local velocity direction is indicated by the black arrows inside the tidal ellipses. (b) Amplitude of the total velocity and (c) phase of the east-west velocity of the combined M2 baroclinic and barotropic tides, filtered at the forcing frequency. (d) Same data as in (a) for the vertical transect indicated by the dashed blue line in (a), arrows indicate the in-plane velocity field.

Figure 4 .

 4 Figure 4. Time series of (a) density perturbations ρ from a conductivity probe located at z = -3.6cm and coordinates (20.5 • N,120.5 • E); (b) the east-west and vertical velocity (U and W respectively) at the same location, along with the barotropic east-west velocity U bt (experiments with no stratification) and the vertical velocity dη/dt estimated from the vertical displacements η; and (c) the nonlinear first internal mode amplitude ζ compared to the water depth H. Gray (resp. white) regions correspond to flow from PO to SCS (resp. SCS to PO).

Table 2 .

 2 Characteristic values of key dimensionless parameters for the ocean and the measured experimental values. The ratio of laboratory to ocean values actually achieved is given at the end of the table.h

	-3	1.0	100	1.5×10 3	10 4	8.0×10 2	10 -4

Acknowledgments. We thank the IWISE collaboration for many productive interactions. This work is funded by ONR grants N00014-09-1-0282 & N00014-09-1-0227, CNRS-PICS grant 5860, ANR grant 08-BLAN-0113-01, and the MIT-France Program.