R Émi Carles
email: remi.carles@math.cnrs.fr

Wolfgang Pauli

WKB ANALYSIS FOR THE NONLINEAR SCHR ÖDINGER EQUATION AND INSTABILITY RESULTS

For the semi-classical limit of the cubic, defocusing nonlinear Schrödinger equation with an external potential, we explain the notion of criticality before a caustic is formed. In the sub-critical and critical cases, we justify the WKB approximation. In the super-critical case, the WKB analysis provides a new phenomenon for the (classical) cubic, defocusing nonlinear Schrödinger equation, which can be compared to the loss of regularity established for the nonlinear wave equation by G. Lebeau. We also show some instabilities at the semi-classical level.

Introduction

We consider the solution to the nonlinear Schrödinger equation (NLS) with a cubic, defocusing nonlinearity, and an external potential:

iε∂ t u ε + ε 2 2 ∆u ε = V u ε + ε κ |u ε | 2 u ε , (t, x) ∈ [0, T ] × R n , (1.1)
u ε (0, x) = a ε 0 (x)e iφ0(x)/ε . (1.2)

We are interested in the behavior of the solution u ε as the positive parameter ε goes to zero. According to the cultural background, this field goes under the name of semi-classical limit, WKB analysis (which is a little bit more specific, see below), or geometrical optics. The general idea is to describe the asymptotic behavior of u ε with a simplified model, which involves geometric quantities. In the case of (1.1), these quantities are called either classical trajectories (in view of classical mechanics), or rays (in view of geometric optics).

There are at least two motivations for such a study. We outline them here, and refer to the survey [START_REF] Rauch | Lectures on geometric optics, Hyperbolic equations and frequency interactions[END_REF] for a broader discussion on this subject. The first one comes from the applied mathematics, and may find its origins in physics. In the case of (1.1), suppose that ε represents the (rescaled) Planck constant. It may be small compared to the other parameters at stake. In this case, it is sensible to consider that the asymptotic behavior of u ε as ε → 0 provides a reliable approximation of the exact solution. Hopefully, the asymptotic model is easier to describe than the initial one (1.1)-(1.2). Another motivation stems from the propagation of singularities for equation where the small parameter ε is not necessarily present initially. Most of the studies in this direction concern hyperbolic equations. However, the belief according to which Schrödinger equations share properties with hyperbolic equations in the semi-classical limit is a first hint that this field is applicable to Schrödinger equations as well (see e.g. [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF][START_REF] Szeftel | Propagation et réflexion des singularités pour l'équation de Schrödinger non linéaire[END_REF]). To illustrate this statement, we give a result, whose proof will be straightforward after the analysis of (1.1)-(1.2).

R. CARLES

Theorem 1.1 ([6], Cor. 1.7). Let n 3. Consider the cubic, defocusing NLS:

(1.3) i∂ t u + 1 2 ∆u = |u| 2 u ; u |t=0 = u 0 .
Denote s c = n 2 -1. Let 0 < s < s c . We can find a family (u ε 0 ) 0<ε 1 in S(R n ) with u ε 0 H s (R n ) → 0 as ε → 0 , and 0 < t ε → 0 such that the solution u ε to (1.3) associated to u ε 0 satisfies:

u ε (t ε ) H k (R n ) → +∞ as ε → 0 , ∀k ∈ s n 2 -s , s .
This result is in the same spirit as the initial breakthrough by G. Lebeau ([17,[START_REF]Perte de régularité pour les équations d'ondes sur-critiques[END_REF], see also [START_REF] Métivier | Exemples d'instabilités pour des équations d'ondes non linéaires (d'après G. Lebeau)[END_REF]). These former results also rely on geometrical optics (in a super-critical régime; see below for this notion in the case of (1.1)). For (1.3), the above result was first established by M. Christ, J. Colliander and T. Tao [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] in the case k = s (see also [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF]Appendix] and [START_REF]Geometric optics and instability for semi-classical Schrödinger equations[END_REF]Appendix B]). The fact that we can go strictly below the value k = s stems from an analysis of (1.1) in a case where the nonlinearity should be considered as quasilinear, and not semilinear. This result is then a consequence of the original idea of E. Grenier [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF].

The rest of this text is organized as follows. In §2, we introduce the notion of criticality for (1.1) at a formal level. In §3, we explain how to justify this notion, and describe the asymptotic behavior of u ε in different cases. The proof of Theorem 1.1 is given in §4. More instability results are given in §5.

WKB analysis and the notion of criticality

This section remains at a formal level only. It is a preparation to the forthcoming rigorous justifications. In a WKB analysis, one assumes for instance that the initial profile a ε 0 can be expanded as a power series in ε: a ε 0 ∼ a 0 + εa 1 + ε 2 a 2 + . . . Note that the modulation factor ε κ in front of the nonlinearity in (1.1) could be taken equal to one, up to replacing a ε 0 with ε κ/2 a ε 0 . In particular, it is clear that for large κ > 0, the nonlinearity is expected to be negligible in the limit ε → 0 (at least locally in time). WKB analysis consists in seeking an approximation of the form

u ε (t, x) ∼ ε→0 a(t, x)e φ(t,x)/ε ,
where the amplitude a and the phase φ are independent of ε. Plugging this approximate solution into (1.1) and canceling the first powers of ε yields:

O ε 0 : ∂ t φ + 1 2 |∇φ| 2 + V = 0 ; φ |t=0 = φ 0 if κ > 0, (2.1) O ε 1 : ∂ t a + ∇φ • ∇a + 1 2 a∆φ = 0 if κ > 1, -i|a| 2 a if κ = 1. ; a |t=0 = a 0 (2.2)
This shows that the minimal value of κ for which nonlinear effects affect the solution at leading order is κ = 1. We shall consider here the following three cases: κ > 1 (sub-critical case), κ = 1 (critical case), and κ = 0 (a super-critical case). We refer to [START_REF]WKB analysis for nonlinear Schrödinger equations with potential[END_REF] for the case 0 < κ < 1. The first step consists in solving (2.1). The potential V may be time dependent: V = V (t, x). Lemma 2.1. Assume that V and φ 0 are smooth and sub-quadratic: and∂ α x φ 0 ∈ L ∞ (R n ) as soon as |α| 2. Then there exist T > 0 and a unique solution

• V ∈ C ∞ (R t × R n x ), and ∂ α x V ∈ L ∞ loc (R t ; L ∞ (R n x )) as soon as |α| 2. • φ 0 ∈ C ∞ (R n ),
φ eik ∈ C ∞ ([0, T ] × R n ) to (2.1). This solution is sub-quadratic: ∂ α x φ eik ∈ L ∞ ([0, T ] × R n ) as soon as |α| 2.
The proof of this lemma relies on Hamilton-Jacobi theory. Note that the time of existence T is uniform with respect to x ∈ R n : a global inversion theorem is needed, which can be found in [START_REF] Schwartz | Nonlinear functional analysis[END_REF] or [START_REF] Dereziński | Scattering theory of quantum and classical N-particle systems[END_REF]. We refer to [START_REF]WKB analysis for nonlinear Schrödinger equations with potential[END_REF] for the proof of this lemma, and for a discussion on the optimality of the assumptions. In particular, one should not expect the above solution to remain smooth for all time. The appearance of singularities corresponds to the formation of caustics. The aim of WKB analysis is to describe the solution u ε before a caustic is formed. See e.g. [START_REF] Carles | Geometric optics with caustic crossing for some nonlinear Schrödinger equations[END_REF] for the description of a solution to (1.1)-(1.2) beyond a focal point.

To analyze (2.2), we introduce the Hamiltonian flow, on which the proof of Lemma 2.1 relies:

(2.3) ∂ t x(t, y) = ξ (t, y) ; x(0, y) = y, ∂ t ξ(t, y) = -∇ x V (t, x(t, y)) ; ξ(0, y) = ∇φ 0 (y). The time T is such that the map y → x(t, y) is a diffeomorphism of R n for t ∈ [0, T ].
The key observation is that (2.2) is a transport equation, which turns out to be an ordinary differential equation along the classical trajectories. Introduce the Jacobi determinant J t (y) = det∇ y x(t, y). Denote A(t, y) := a (t, x(t, y)) J t (y). For t ∈ [0, T ], (2.2) is equivalent to:

∂ t A = 0 if κ > 1, -iJ t (y) -1 |A| 2 A if κ = 1. ; A(0, y) = a 0 (y).
This ordinary differential equation along the rays of geometrical optics can be solved explicitly: we see that ∂ t |A| 2 = 0, hence, for κ = 1,

A(t, y) = a 0 (y) exp -i t 0 J s (y) -1 |a 0 (y)| 2 ds .
Inverting the map y → x(t, y) yields a(t, x). We see that the critical nonlinear effect is a self-modulation of the amplitude. In the context of laser physics, this phenomenon is known as phase self-modulation (see e.g. [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF][START_REF] Boyd | Nonlinear optics[END_REF][START_REF] Donnat | Quelques contributions mathématiques en optique non linéaire[END_REF]). We now turn to the super-critical case κ = 0. To illustrate the difficulty of this case, seek a more precise asymptotic expansion of u ε :

u ε (t, x) ∼ a 0 (t, x) + εa 1 (t, x) + ε 2 a 2 (t, x) + . . . e iφ(t,x)/ε .
Plugging such an asymptotic expansion into (1.1) yields a shifted cascade of equations:

O ε 0 : ∂ t φ + 1 2 |∇φ| 2 + V + |a 0 | 2 = 0 ; φ |t=0 = φ 0 . O ε 1 : ∂ t a 0 + ∇φ • ∇a 0 + 1 2 a 0 ∆φ = -2iRe (a 0 a 1 ) a 0 ; a 0|t=0 = a 0 .
Two comments are in order. First, we see that there is a strong coupling between the phase and the main amplitude: a 0 is present in the equation for φ. Second, the above system is not closed: φ is determined in function of a 0 , and a 0 is determined in function of a 1 . Even if we pursued the cascade of equations, this phenomenon would remain: no matter how many terms are computed, the system is never closed (see [START_REF] Gérard | Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire[END_REF]). This is a typical feature of super-critical cases in nonlinear geometrical optics (see [START_REF] Cheverry | Cascade of phases in turbulent flows[END_REF][START_REF] Cheverry | Counter-examples to the concentration-cancellation property[END_REF]).

In the case when V ≡ 0 and φ 0 ∈ H s , this problem was resolved by E. Grenier [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF], by modifying the usual WKB methods (see §3.2). Note that even though a 1 is not determined by the above system, the pair (ρ, v) := (|a 0 | 2 , ∇φ) solves a compressible Euler equation:

(2.4) ∂ t v + v • ∇v + ∇V + ∇ρ = 0 ; v t=0 = ∇φ 0 ∂ t ρ + ∇ • (ρv) = 0 ; ρ t=0 = |a 0 | 2 .

Rigorous WKB analysis

We outline here some results presented in details in [START_REF]WKB analysis for nonlinear Schrödinger equations with potential[END_REF].

u ε (t, x) = a ε (t, x)e iφ eik (t,x)/ε . Then for t ∈ [0, T ], (1.1)-(1.2) is equivalent to: (3.1) ∂ t a ε + ∇φ eik • ∇a ε + 1 2 a ε ∆φ eik = i ε 2 ∆a ε -iε κ-1 |a ε | 2 a ε , a ε |t=0 = a ε 0 .
Two things must be noticed: first, the potential V and the initial phase φ 0 do not appear in this new problem. Second, the factors involving φ eik have the following features: in view of Lemma 2.1, the term ∆φ eik is in L ∞ ([0, T ] × R n ), and the operator ∂ t +∇φ eik •∇ is a transport operator. We can then obtain energy estimates in Sobolev spaces, and establish: Proposition 3.1. Under the assumptions of Lemma 2.1, assume moreover that

a ε 0 is bounded in H s (R n ) uniformly for ε ∈]0, 1], for all s 0. Let κ 1. Then for all ε ∈]0, 1], (3.1) has a unique solution a ε ∈ C ∞ ([0, T ] × R n ) ∩ C([0, T ]; H s ) for all s > n/2. Moreover, a ε is bounded in L ∞ ([0, T ]; H s ) uniformly in ε ∈]0, 1], for all s 0.
These uniform estimates allow us to neglect the term ε∆a ε on the right hand side of (3.1). Assume moreover the following convergence:

(3.2) a ε 0 → a 0 in H s (R n ), ∀s 0. Corollary 3.2. Let κ 1. Under the above assumptions, a ε -a ε L ∞ ([0,T ];H s ) → 0 as ε → 0, ∀s 0
, where a ε solves:

(3.3) ∂ t a ε + ∇φ eik • ∇ a ε + 1 2 a ε ∆φ eik = -iε κ-1 | a ε | 2 a ε ; a ε |t=0 = a 0 .
Proceeding as in §2, denote

A ε (t, y) := a ε (t, x(t, y)) J t (y).
We see that so long as y → x(t, y) defines a global diffeomorphism (which is guaranteed for t ∈ [0, T ] by construction), (3.3) is equivalent to:

∂ t A ε = -iε κ-1 J t (y) -1 |A ε | 2 A ε ; A ε (0, y) = a 0 (y).
This ordinary differential equation along the rays of geometrical optics can be solved explicitly:

A ε (t, y) = a 0 (y) exp -iε κ-1 t 0 J s (y) -1 |a 0 (y)| 2 ds .
Back to the initial solution u ε , we conclude:

Proposition 3.3. Let κ 1. Under the above assumptions, for all ε ∈]0, 1], (1.1)- (1.2) has a unique solution u ε ∈ C ∞ ([0, T ] × R n ) ∩ C([0, T ]; H s ) for all s > n/2. Moreover, there exist a, G ∈ C ∞ ([0, T ] × R n ), independent of ε ∈]0, 1], where a ∈ C([0, T ]; L 2 ∩ L ∞ ), and G is real-valued with G ∈ C([0, T ]; L ∞ ), such that: u ε -ae iε κ-1 G e iφ eik /ε L ∞ ([0,T ];L 2 ∩L ∞ ) → 0 as ε → 0.
The profile a solves the initial value problem:

(3.4) ∂ t a + ∇φ eik • ∇a + 1 2 a∆φ eik = 0 ; a |t=0 = a 0 ,
and G depends nonlinearly on a:

a(t, x) = 1 J t (y(t, x)) a 0 (y(t, x)) , G(t, x) = - t 0 J s (y(t, x)) -1 |a 0 (y(t, x))| 2 ds.
In particular, if κ > 1, then

u ε -ae iφ eik /ε L ∞ ([0,T ];L 2 ∩L ∞ ) → 0 as ε → 0,
and no nonlinear effect is present in the leading order behavior of u ε . If κ = 1, nonlinear effects are present at leading order, measured by G.

3.2. Super-critical case: κ = 0. In this case, we recall the beautiful idea of E. Grenier, which makes it possible to consider the case V ≡ 0 and φ 0 ∈ H s (R n ) for sufficiently large s. The approach consists in reversing the steps of the WKB analysis: usually, one seeks an approximate solution, and then tries to show that stability arguments imply that the exact solution is well approximated by this process. To overcome the issues mentioned in §2, the idea in [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] consists in first writing the unknown as:

u ε (t, x) = a ε (t, x)e iφ ε (t,x)/ε ,
where the amplitude a ε is complex-valued (even if a ε 0 is real-valued), and φ ε is realvalued. Doing so, one introduces one degree of freedom to rewrite (1.1)-(1.2). The usual approach in physics (see e.g. [START_REF] Landau | Tome III: Mécanique quantique. Théorie non relativiste[END_REF]) consists in writing

     ∂ t φ ε + 1 2 |∇φ ε | 2 + |a ε | 2 = ε 2 ∆a ε 2a ε ; φ ε t=0 = φ 0 , ∂ t a ε + ∇φ ε • ∇a ε + 1 2 a ε ∆φ ε = 0 ; a ε t=0 = a ε 0 .
Obviously, this approach is delicate when a ε has zeroes; see the discussion in [START_REF] Gérard | Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire[END_REF] on this subject. The choice in [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] is to write:

(3.5)      ∂ t φ ε + 1 2 |∇φ ε | 2 + |a ε | 2 = 0 ; φ ε t=0 = φ 0 , ∂ t a ε + ∇φ ε • ∇a ε + 1 2 a ε ∆φ ε = i ε 2 ∆a ε ; a ε t=0 = a ε 0 .
Inspired by the fact that the expected limit is related to the compressible Euler equation (see §2), introduce the "velocity" v ε = ∇φ ε . Then (3.5) yields:

(3.6)    ∂ t v ε + v ε • ∇v ε + 2 Re (a ε ∇a ε ) = 0 ; v ε t=0 = ∇φ 0 , ∂ t a ε + v ε • ∇a ε + 1 2 a ε ∇ • v ε = i ε 2 ∆a ε ; a ε t=0 = a ε 0 .
Separate real and imaginary parts of a ε , a ε = a ε 1 + ia ε 2 . Then we have

(3.7) ∂ t u ε + n j=1 A j (u ε )∂ j u ε = ε 2 Lu ε , with u ε =        a ε 1 a ε 2 v ε 1 . . . v ε n        , L =   0 -∆ 0 . . . 0 ∆ 0 0 . . . 0 0 0 0 n×n   ,
and

A(u, ξ) = n j=1 A j (u)ξ j =   v • ξ 0 a1 2 t ξ 0 v • ξ a2 2 t ξ 2a 1 ξ 2a 2 ξ v • ξI n   .
The matrix A(u, ξ) can be symmetrized by

S = I 2 0 0 1 4 I n .
The important point to notice is that the operator L is skew-symmetric: it is invisible in the energy estimates, so the loss of derivative (it is the only operator of order two in (3.7)) is avoided. Denote H ∞ = ∩ s 0 H s (R n ). Classical theory on symmetric hyperbolic systems yields a solution (v ε , a ε ) to (3.6). Once v ε is known, we note that it is irrotational, so there exists φ ε such that v ε = ∇φ ε . Up to adding a function of time only, (φ ε , a ε ) solves (3.5).

Proposition 3.4 ([14], Th. 1.1). Let κ = 0. Suppose that φ 0 ∈ H ∞ , and that a ε 0 is bounded in H s (R n ) uniformly for ε ∈]0, 1], for all s 0. Let s > 2 + n/2. There exist T s > 0 independent of ε ∈]0, 1] and u ε = a ε e iφ ε /ε solution to

(1.1)-(1.2) on [0, T s ]. Moreover, a ε and φ ε are bounded in L ∞ ([0, T s ]; H s ), uniformly in ε ∈]0, 1].
Assume moreover that (3.2) holds. The solution to (3.5) formally converges to the solution of:

(3.8)      ∂ t φ + 1 2 |∇φ| 2 + |a| 2 = 0 ; φ t=0 = φ 0 , ∂ t a + ∇φ • ∇a + 1 2 a∆φ = 0 ; a t=0 = a 0 .
Under the above assumptions, (3.8) has a unique solution (a, φ) ∈ L ∞ ([0, T * ]; H m ) 2 for all m > 0 for some T * > 0 independent of m (see e.g. [START_REF] Alinhac | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF][START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF]).

Remark 3.5. More general nonlinearity. Suppose that we consider a more general nonlinearity:

iε∂ t u ε + ε 2 2 ∆u ε = f |u ε | 2 u ε .
Then following the same lines as above, the symmetrizer naturally becomes

S = I 2 0 0 1 4f ′ (|a ε | 2 ) I n .
For this matrix to be positive, and to be able to estimate its time derivative, it is natural to assume f ′ > 0. This to the assumption made in [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF], and in [START_REF]WKB analysis for nonlinear Schrödinger equations with potential[END_REF]. For the above analysis to be valid, the nonlinearity has to be defocusing, and cubic at the origin. In particular, the WKB analysis for the quintic defocusing NLS is still an open problem. Note however that it is possible to construct solutions to the limit problem in that case (the analogue of (3.8)), thanks to the result of [START_REF] Makino | Sur la solution à support compact de l'équation d'Euler compressible[END_REF] and the geometrical analysis of §3.1. Yet, the nonlinear change of variable of [START_REF] Makino | Sur la solution à support compact de l'équation d'Euler compressible[END_REF] is apparently incompatible with the above remark that L is skew-symmetric.

If we suppose in addition that there exists a 0 , a 1 ∈ H ∞ such that (3.9) a ε 0 = a 0 + εa 1 + o(ε) in H s , ∀s 0, then we infer: Proposition 3.6. Let s ∈ N. Then T s T * , and there exists C s independent of ε such that for every 0 t T * ,

a ε (t) -a(t) H s C s ε ; φ ε (t) -φ(t) H s C s εt.
Note that this suffices to describe u ε for very small time only:

u ε -ae iφ/ε = a ε e iφ ε /ε -ae iφ/ε = (a ε -a) e iφ ε /ε + 2iae i(φ ε +φ ε )/2ε sin φ ε -φ 2ε .
The first term of the right hand side is of order O(ε) in L 2 ∩ L ∞ , but the second one is of order O(t) only: therefore, we only have u ε (t, x) ∼ a(t, x)e iφ(t,x)/ε for 0 t ≪ 1.

To have a better error estimate, it is necessary to compute the next term in the asymptotic expansion of (φ ε , a ε ) in powers of ε. For times of order O(1), the initial corrector a 1 must be taken into account:

Proposition 3.7. Define (a (1) , φ (1) ) by ∂ t φ (1) + ∇φ • ∇φ (1) + 2 Re aa (1) = 0, ∂ t a (1) + ∇φ • ∇a (1) + ∇φ (1) • ∇a

+ 1 2 a (1) ∆φ + 1 2 a∆φ (1) = i 2 ∆a, φ (1
) t=0 = 0 ; a (1) t=0 = a 1 . Then a (1) , φ (1) ∈ L ∞ ([0, T * ]; H s ) for every s 0, and

a ε -a-εa (1) L ∞ ([0,T * ];H s ) + φ ε -φ-εφ (1) L ∞ ([0,T * ];H s ) C s ε 2 + o(ε) , ∀s 0 .
Despite the notations, it seems unadapted to consider φ (1) as being part of the phase. Indeed, we infer from Proposition 3.7 that Relating this information to the WKB methods presented at the end of §2, we would have: (1) . Since φ (1) depends on a 1 while a does not, we retrieve the fact that in super-critical régimes, the leading order amplitude in WKB methods depends on the initial first corrector a 1 .

u ε -ae iφ (1) e iφ/ε L ∞ ([0,T * ];L 2 ∩L ∞ ) = o(1).
a 0 = ae iφ
Remark 3.9. The term e iφ (1) does not appear in the Wigner measure of ae iφ (1) e iφ/ε . Thus, from the point of view of Wigner measures, the asymptotic behavior of the exact solution is described by the Euler-type system (2.4).

Remark 3.10. If we assume that a 0 is real-valued, then so is a. If moreover a 1 is purely imaginary (for instance, if a 1 = 0), then we see that a (1) is purely imaginary, hence, φ (1) ≡ 0. So far we have assumed V ≡ 0 and φ 0 ∈ H ∞ . If we try to mimic the approach of [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] for a non-trivial external potential for instance, we have to consider:

∂ t φ ε + 1 2 |∇φ ε | 2 + V + |a ε | 2 = 0 ; φ ε t=0 = φ 0 , ∂ t a ε + ∇φ ε • ∇a ε + 1 2 a ε ∆φ ε = i ε 2 ∆a ε ; a ε t=0 = a ε 0 .
The analysis of [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] works in the same way only when ∇V ∈ L ∞ loc (R t ; H s (R n )) for a sufficiently large s. To be able to consider general sub-quadratic potentials (including the harmonic oscillator), resume the assumption of Lemma 2.1, and write

φ ε = φ eik + ϕ ε .
Working with the unknown (ϕ ε , a ε ), we see that we are now rid of the external potential V , and of the possibly unbounded initial phase φ 0 . The price to pay is that extra terms have appeared. The good news however is that these extra terms are semilinear (as in §3.1), and can be treated by perturbative methods in energy estimates. We conclude: Theorem 3.11. Let κ = 0. Under the above assumptions, there exists T * > 0 independent of ε ∈]0, 1] and a unique solution u ε ∈ C ∞ ([0, T * ]×R n )∩C([0, T * ]; H s ) for all s > n/2 to (1.1)-(1.2). Moreover, there exist a, ϕ ∈ C([0, T * ]; H s ) for every s 0, such that:

lim sup ε→0 u ε -ae i(ϕ+φ eik )/ε L 2 ∩L ∞ = O(t) as t → 0.
Here, a and ϕ are nonlinear functions of φ eik and a 0 . Finally, there exists ϕ (1) ∈ C([0, T * ]; H s ) for every s 0, real-valued, such that:

lim sup ε→0 sup 0 t T * u ε -ae iϕ (1) e i(ϕ+φ eik )/ε L 2 ∩L ∞ = 0.
The phase shift ϕ (1) is a nonlinear function of φ eik , a 0 and a 1 .

Remark 3.12. In [START_REF]WKB analysis for nonlinear Schrödinger equations with potential[END_REF], some assumptions on the momentum of a ε 0 are made, and not here. This is due to the fact that here, the nonlinearity that we consider is exactly cubic. When it is cubic at the origin only (see Remark 3.5), extra estimates are needed, which apparently impose some extra decay at infinity for a ε 0 , a 0 and a 1 .

Proof of Theorem 1.1

Theorem 1.1 is a straightforward consequence of Proposition 3.7. For a 0 ∈ S(R n ), let

u 0 (x) = λ -n 2 +s a 0 x λ .
Let ε = λ n 2 -1-s : ε and λ go simultaneously to zero, since s < s c . Define

ψ ε (t, x) = u λ (εt, x) = λ n 2 -s u λ n 2 +1-s t, λx .
It solves:

(4.1) iε∂ t ψ ε + ε 2 2 ∆ψ ε = |ψ ε | 2 ψ ε ; ψ ε |t=0 = a 0 (x) .
The idea of the proof is that for times of order O(1), ψ ε has become ε-oscillatory. We infer from Proposition 3.7 that there exist T > 0 independent of ε ∈]0, 1], and a, φ, φ 1 ∈ C([0, T ]; H m ) for any m 0, such that:

ψ ε -ae iφ1 e iφ/ε L ∞ ([0,T ];H m ) C m ε 1-m .
Since the Ḣm -norm of ae iφ1 e iφ/ε is of order ε -m (when φ is not stationary), we deduce that there exists t ∈]0, T ] such that for any m 0:

ψ ε (t) Ḣm ≈ ε -m .
This implies:

u λ n 2 +1-s t Ḣk ≈ λ s-k ψ ε (t) Ḣk ≈ λ s-k ε -k = λ s-k-k( n 2 -1-s) .
The result then follows when considering the limit λ → 0. We get exactly the statement of the theorem by replacing a 0 by | log λ| -1 a 0 for instance.

Remark 4.1. The proof of ill-posed presented in [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] (see also [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF]Appendix], [6, Appendix B]) consists in neglecting the Laplacian in (4.1) for very small times, and integrating explicitly an ordinary differential equation. Proving that the Laplacian is negligible stems from Gronwall lemma. Essentially, the error satisfies an inequality of the form

w ε (t) X ε + 1 ε t 0 w ε (s) X ds,
for some space X that we do not describe. The singular factor ε -1 is due to the ε in front of the time derivative, and to the fact that no power of ε is present in front of the nonlinearity. Therefore, Gronwall lemma yields no better than:

w ε (t) X εe Ct/ε ,
for some C > 0. The error is small on an interval of the form [0, ε| log ε| θ ] for some θ > 0. This is enough to prove Theorem 1.1 for k = s. This analysis considers (4.1) as a semilinear equation, since the nonlinearity is viewed as a perturbation of the linear equation. To prove Theorem 1.1 for k < s, it seems necessary to consider (4.1) as a quasilinear equation, as was done by E. Grenier. Note also that the quasilinear approach shows that the Laplacian in (4.1) is negligible for 0 < t ε ≪ ε 1/3 , that is a "much larger" interval than [0, ε| log ε| θ ] (but still very small!). See the next section.

Remark 4.2. On the other hand, Theorem 1.1 is valid only for cubic, defocusing nonlinear Schrödinger equations, while the results in [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] are valid for more general equations. This is due to the fact that the justification of super-critical nonlinear geometric optics for times O( 1) is available only for nonlinearities which are defocusing, and cubic at the origin (see Remark 3.5). Since the proofs of ill-posedness rely on an homogeneous change of unknown function, we are left with the only possibility of an exactly cubic, defocusing nonlinearity. However, it is very likely that (an analogue of) Theorem 1.1 should be true under more general assumptions.

Instability for the semi-classical equation

The results we present in the paragraph are taken from [START_REF]Geometric optics and instability for semi-classical Schrödinger equations[END_REF]. We assume V = φ 0 = 0 for the sake of concision. We first fix some notations. Notation. Let (α ε ) 0<ε 1 and (β ε ) 0<ε 1 be two families of positive real numbers.

• We write [START_REF]Geometric optics and instability for semi-classical Schrödinger equations[END_REF] is the following:

α ε ≪ β ε if lim sup ε→0 α ε /β ε = 0. • We write α ε β ε if lim sup ε→0 α ε /β ε < ∞. • We write α ε ≈ β ε if α ε β ε and β ε α ε . A typical result of
Theorem 5.1. Let n 1, a 0 , a ε 0 ∈ S(R n )
, where a 0 is independent of ε. Let u ε and v ε solve the initial value problems:

iε∂ t u ε + ε 2 2 ∆u ε = |u ε | 2 u ε ; u ε t=0 = a 0 . iε∂ t v ε + ε 2 2 ∆v ε = |v ε | 2 v ε ; v ε t=0 = a ε 0 .
Assume that there exists N ∈ N and ε 1-1 N ≪ δ ε ≪ 1 such that:

(5.1)

a 0 -a ε 0 H s ≈ δ ε , ∀s 0 ; lim sup ε→0 Re(a 0 -a ε 0 )a 0 δ ε L ∞ (R n ) = 0.
Then we can find 0 < t ε ≪ 1 such that:

u ε (t ε ) -v ε (t ε ) L 2 1.
More precisely, this mechanism occurs as soon as t ε δ ε ε. In particular, for all s 0,

u ε -v ε L ∞ ([0,t ε ];L 2 ) u ε |t=0 -v ε |t=0 H s → +∞ as ε → 0 .
Example. Consider a 0 , b 0 ∈ S(R n ) independent of ε, such that Re(a 0 b 0 ) ≡ 0, and take

a ε 0 = a 0 + δ ε b 0 . Example. Consider a 0 ∈ S(R n ) independent of ε and x ε ∈ R n . We can take a ε 0 (x) = a 0 (x -x ε ), provided that |x ε | = δ ε and lim sup ε→0 x ε |x ε | • ∇ |a 0 | 2 L ∞ = 0.
This example and the analysis of [START_REF]Geometric optics and instability for semi-classical Schrödinger equations[END_REF] make it possible to refine some results of [START_REF] Burq | Instability for the semiclassical non-linear Schrödinger equation[END_REF].

The general idea consists in using the WKB analysis in this super-critical case. Roughly speaking, we have seen that (3.8) provides a good approximation of u ε for very small time only. Since we are interested in instabilities occurring for very small time, this is not a problem for us now. The coupling in (3.8) shows that a small perturbation of a 0 yields a small perturbation of φ. But when we write u ε (t, x) ∼ a(t, x)e iφ(t,x)/ε , we see that this small perturbation is divided by ε, which goes to zero. The result may not be small. . . Technically, our approach consists in resuming the result provided by Proposition 3.4. Instead of letting ε → 0 in the initial data of (3.5), just neglect the skew-symmetric term (recall that we assume φ 0 = 0):

(5.2)

∂ t Φ ε + 1 2 |∇Φ ε | 2 + |a ε | 2 = 0 ; Φ ε t=0 = 0 , ∂ t a ε + ∇Φ ε • ∇a ε + 1 2 a ε ∆Φ ε = 0 ; a ε t=0 = a ε 0 .
Assuming that a ε 0 is bounded in H s for all s 0, (5.2) has a unique solution (Φ ε , a ε ) ∈ L ∞ ([0, T * ]; H m ) 2 for all m > 0 for some T * > 0 independent of ε and m. Proposition 5.2. Let s ∈ N. Then T s T * , and there exists C s independent of ε such that for every 0 t T * ,

a ε (t) -a ε (t) H s C s εt ; φ ε (t) -Φ ε (t) H s C s εt 2 .
The second idea consists in considering the Taylor expansion in time of (Φ ε , a ε ):

Φ ε (t, x) ∼ j 1 t 2j-1 Φ ε j (x) ; a ε (t, x) ∼ j 1 t 2j a ε j (x).
Note that only odd powers of t are present in the expansion of Φ ε , and even powers in that of a ε . This is because we have assumed φ 0 = 0. Plugging these expansions into (5.2), we get formally:

a ε 0 = a ε 0 ; Φ ε 1 = -|a ε 0 | 2 .
We can then check that a perturbation of order δ ε of a ε 0 yields a perturbation of order δ ε of Φ ε 1 , provided that the polarization condition (5.1) is satisfied. By induction, we see that this perturbs the other Φ ε j 's and a ε j 's by a O(δ ε ). Consider the approximate solution defined by

u ε K (t, x) = a ε 0 (x) exp   i K j=1 t 2j-1 Φ ε j (x)/ε   .
Formally, we have:

a ε (t,
x)e iΦ ε (t,x)/ε -u ε K (t, x) = (a ε (t, x) -a ε 0 (x)) e iΦ ε (t,x)/ε +a ε 0 (x) exp iΦ ε (t, x)/ε -exp i

K j=1 t 2j-1 Φ ε j (x)/ε = O(t 2 ) + O Φ ε (t, x) - K j=1 t 2j-1 Φ ε j (x) /ε = O(t 2 ) + O t 2K+1 /ε .
We infer that the above quantity is small for times such that t ε ≪ 1 and t ε ≪ ε 1 2K+1 . On the other hand, Proposition 5.2 shows that a ε e iΦ ε /ε is a good approximation of u ε for times such that t ε ≪ 1. Therefore, we expect

u ε (t ε , •) -u ε K (t ε , •) L 2 ∩L ∞ ≪ 1 for t ε ≪ ε 1 2K+1 .
This can be proved by the analysis presented in §3.2.

The case K = 1 is of special interest. Indeed, the Laplacian plays no role in the definition of u ε 1 , and we check that it solves:

iε∂ t u ε 1 = |u ε 1 | 2 u ε 1 ; u ε 1|t=0 = a ε 0 .
This is the solution of the ordinary differential equation considered in [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] and [START_REF] Burq | Instability for the semiclassical non-linear Schrödinger equation[END_REF]. The above analysis shows that it is a reasonable approximation of u ε for 0 < t ε ≪ ε 1/3 : see Remark 4.1.

In view of Theorem 5.1, we define u ε K and v ε K in an obvious way, and we have:

u ε (t ε , •) -u ε K (t ε , •) L 2 ∩L ∞ + v ε (t ε , •) -v ε K (t ε , •) L 2 ∩L ∞ ≪ 1 for t ε ≪ ε 1 2K+1 .
So to prove Theorem 5.1, we just have to compare u ε K and v ε K :

u ε K (t, x) -v ε K (t, x) =a 0 (x) exp i K j=1 t 2j-1 Φ j (x)/ε -a ε 0 (x) exp i K j=1 t 2j-1 Φ ε j (x)/ε = (a 0 (x) -a ε 0 (x)) exp i K j=1 t 2j-1 Φ j (x)/ε -a ε 0 (x) exp i K j=1 t 2j-1 Φ j (x)/ε -exp i K j=1 t 2j-1 Φ ε j (x)/ε . (5.3)
The first term is of order δ ε by assumption. To estimate the second term, examine:

K j=1 t 2j-1 Φ j (x) -Φ ε j (x) /ε.
Since we consider times such that t ε ≪ 1 and that we have seen that Φ j -Φ ε j = O(δ ε ) for j 2, the leading order term is simply:

t Φ 1 (x) -Φ ε 1 (x) /ε = t ε | a ε 0 (x)| 2 -|a ε 0 (x)| 2 = t ε Re (a 0 -a ε 0 )a 0 ) + O (δ ε ) 2 .
By assumption, the modulus of (5.3) behaves like

a 0 (x) sin tδ ε ε f (x) ,
for some non-trivial function f . The conclusion of Theorem 5.1 follows easily.

3. 1 .

 1 Sub-critical and critical cases. Let κ 1. Under the assumptions of Lemma 2.1, we change the Cauchy problem (1.1)-(1.2): define a ε by

Remark 3 . 8 .

 38 If the term o(ε) in (3.9) is controlled more precisely as a O(ε 2 ), then the above o(1) becomes a O(ε).

Future affiliation (starting September

2007): UMR CNRS 5149, Montpellier, France


Alba Málaga Sabogal

Serge Troubetzkoy
email: serge.troubetzkoy@univ-amu.fr

Alba Málaga
email: alba.malaga@polytechnique.edu

WEAK-MIXING POLYGONAL BILLIARDS

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

Introduction

A polygon is called rational if all the angles between pairs of sides are rational multiples of π. The billiard flow in a rational polygon is quasi-intergrable. Namely if we fix the initial direction of the flow, then the billiard orbit is contained in a finite set of directions, this set of directions is the integral. it has a natural first integral, the direction of the flow. A major achievement in the theory of polygonal billiards is the Kerchoff, Masur, Smillie theorem: the billiard flow in any rational polygon is ergodic, in fact uniquely ergodic, for almost every value of the first integral [KMS]. One of the major unsolved questions about rational polygonal billiards is to understand if for almost all values of the integral the billiard flow is in fact weakly mixing.

There is a well known construction, known as unfolding, which associates a translation surface to each rational polygon. Ávila and Forni have shown that the typical translation flow on a typical surface is weakly mixing [AF], but this result tells us nothing about the behavior of polygonal billiards since the translation surface formed by unfolding are a set of measure zero in the space of all translation surfaces. Recently Ávila and Delecroix have shown that the same holds for the billiard in a regular polygon (except for the lattice cases: the equilateral triangle, the square, or the regular hexagon) [AD]. Other tham this, to our knowledge, there is only one result about weak-mixing for the billiard flow in a rational polygon. Consider polygons for which all sides are vertical or horizontal (VH-tables). Fix the combinatorics of such a polygon (see the next section for an exact definition) and fix an angle θ ∈ S 1 , then Gutkin and Katok have shown that there is a dense G δ set of VH-pollygons with this combinatorics, such that for each polygon in this dense G δ set the billiard flow in the direction θ is weakly mining [GK]. Their result also holds for tables of fixed combinatorics for which all sides are parallel to the sides of a fixed equilateral triangle.

In this article we strengthen this last result. We show that for any fixed combinatorics of VH-tables, there is a dense G δ set of VHpollygons with this combinatorics, such that for each polygon in this dense G δ set the billiard flow is weakly mining is weakly-mixing in almost every direction. The method of proof is the same as in [GK], their article contains several of the necessary ingredients for our proof.

Definitions and main results

For details on polygonal billiards see [MT]. A VH-polygon is a simply connected polygon all of whose sides are either vertical or horizontal. The smallest rectangle congaing a VH-polygon is called is called its bounding box. We orient the border of B in such a way that the interior of the table is always to the left of the border, thus we associate a finite word in four letters W , S, E, N to the VH-polygon P . This word is assumed to start at the side of the table on the lower side of the bounding box, if there are several we take the left most segment. This word is called the combinatorics of P . The location (x, y) of the lower left corner of the bounding box, the combinatorics, and the lengths (ℓ 1 , . . . , ℓ n ) of sides of P naturally parametrize the set of all VH-polygons. The set of VH-polygons with a fixed combinatorics is naturally parametrized by an open subset of R n . To see this let B be a table with a given combinatorics. Let ℓ 1 , . . . , ℓ n be the lengths of the sides of B. The total displacement to the right must be equal to the total displacement to the left, and the same for up and down. So there are two dependent length parameters for each word, but we need to also specify the coordinates (x, y) which compensate for this.

A VH-table is a VH-polygon from whose interior we remove a finite number of disjoint VH-polygons. The combinatorics of a VH-table is the collection of these words, with the word associated to the exterior component distinguished. We want to parametrized the set of VHtables with fixed combinatorics and fixed area for which the outer most bounding box has lower left corner at the point (1, 1). As before we consider the lengths (ℓ 1 , . . . , ℓ n ) of all the sides of the table, the the set of VH-tables with a fixed combinatorics and fixed area is then naturally parametrized by an open subset O of R n-3 since we additionally fixed the area and the point (1, 1). Some coordinates of points in this space are lengths, others are positions of corners of bounding boxes, thus we will use the letters a or b to denote points in O. We will often confound the parameter a with the table B a which it determines.

If all the entries of a are rational, the table can be tiled by rectangles of size ( 1 p , 1 q ) with p, q positive integers. The number of rectangles N(p, q) in the tiling is pq times the area of the table. We call a table in O a (p, q)-VH table if it is tiled with rectangles of size ( 1 p , 1 q ) with p, q minimal. Fix a VH-table a ∈ O and a direction θ ∈ S 1 . The billiard flow φ a,θ t on B a in the direction θ or simply φ θ t is the free motion on the interior of B a with elastic collision from the boundary of B a . Once launched in the direction θ, the billiard direction can only achieve four directions [θ] := {±θ, ±(π -θ)}; thus the phase space X a,θ of the billiard flow in the direction θ is a subset of the cartesian product of B a with these four directions. Note that in this notation φ θ t , φ -θ t , φ π-θ t and φ θ-π t are all the same.

For each direction θ, the billiard flow φ θ t preserves the area measure µ on B a times a discrete measure on [θ], we will also call this measure µ. The billiard flow on the full phase space preserves the volume measure µ × λ with λ the length measure on S 1 .

A flow ψ t : M → M preserving a measure m is called weakly-mixing if for any f ∈ L 2 (M, m) the function t → U t f, f strongly converges in the sense of Cesaro to | f, 1 | 2 as |t| → ∞, where 1 is the constant function taking the value 1 on M and U t f = f • ψ t [P].

Theorem 1. The set of VH-tables with a fixed area and fixed combinatorics (with at least 6 sides) contains a dense G δ set of tables G such that for each table a ∈ G there exists a full measure G δ -dense set Θ a of directions such that the billiard flow on the table a ∈ G is weakly mixing in every direction θ ∈ Θ a .

In fact, the conclusion holds if we restrict ourselves to any closed subset of the set of VH-tables with fixed combinatorics which for each Q ≥ 1 contains a dense set of (p, q)-VH tables with min(p, q) ≥ Q.

Proof of generic weak mixing

Proof of Theorem 1. Let us consider a ∈ O. The table B a can be unfolded [MT] to a translation surface which in fact will be obtained as a gluing of four copies of B a along corresponding pairs of parallel sides. Without loss of generality we assume that the area of the table is 1 4 , and thus the area of the unfolded surface is 1. For our proof we look at these copies as embedded disjointly in R 2 in such a way that this embedding varies continuously with f . The bounding box of one copy of the table is always placed with the inferior left corner to be (1, 1) and the other copies to be the images under reflections with respect to the axis, thus their respective bounding boxes having a corner placed at (±1, ±1).

Thus, modulo the boundary of B a which is a set of zero-measure, the phase space of the billiard on the table B a is a cartesian product of a compact subset of R 2 with S 1 . Thus any function in L 2 (R 2 × S 1 ) is projected to a function in L 2 (B a × S 1 ) by restriction and reciprocally, any function in L 2 (B a × S 1 ) can be seen as a function in L 2 (R 2 × S 1 ).

We extend the flow φ a t to all of R 2 by setting φ a t ( z, θ) = ( z, θ) on the exterior of B a . For each t let U a t be the unitary operator on L 2 (R 2 ×S 1 ) defined by U a t h( z, θ) = h(φ a,θ t ( z, θ)). Suppose additionally that a is a (p, q)-VH table. Let χ a be the characteristic function of B a × S 1 , and for any h : R 2 × S 1 → R let h a := χ a • h be the restriction of h to B a × S 1 . Let h d : R 2 × S 1 → R be the function defined by 1 N(p, q) i,j h a (s + (i, j), θ).

It follows from definition that

(1)

B a ×S 1 h d d(µ × λ) = B a ×S 1 h d(µ × λ).
Note that h a ) a = h a , and thus

h a d = h d . Furthermore h d a = h a d however h d a = h a d a .
Note that h d is ( 1 p , 1 q )-periodic on R 2 × {θ}, for any θ. Thus we can think of h d as descending to a function in L 2 ([0, 1 p ]×[0, 1 q ]×S 1 ), we will use the same symbol h d for both functions, but will differentiate by noting the variable ( z, θ) ∈ R 2 ×S 1 , and (u, θ) ∈ [0, 1 p ] × [0, 1 q ] × S 1 . By Theorem 3 of [GK], if θ is irrational, i.e. non commensurable with π, then the function h d a is the projection of h a on the discrete spectrum of the flow φ t . Let

h c := h -h d . Then h a = h d a + h c a , thus h c
a is the projection of h a on the continuous spectrum of φ t . For any

h 1 , h 2 in L 2 (R 2 × S 1 ), let h 1 , h 2 θ = R 2 h 1 ( z, θ)h 2 ( z, θ) dµ and ||h 1 || = h 1 , h 1 θ .
For any h ∈ L 2 (R 2 ), and any fixed direction θ, (2)

U a t h a , h a θ -h a , χ a θ 2 = U a t (h d a -h a , χ a θ χ a ) + h a , χ a θ χ a + h c a , h a θ -h a , χ a θ 2 = U a t h d a -h a , χ a θ χ a , h a θ + U a t h c a , h a θ = U a t h d a -h a , χ a θ χ a , h a θ + U a t h c a , h c a θ
. By the definition of the continuous spectrum for any ε > 0 there is a set T ⊂ R of density one such that if t ∈ T then

(3) U a t h c a , h c a θ < ε. Suppose that h is additionally continuous; thus for any ε > 0 there is a δ(ε, h) > 0 such that |h( z, θ) -h(s ′ , θ)| < ε/||h|| if |s -s ′ | < δ(ε, h). Suppose that max( 1 p , 1 q ) < δ(ε, h). Then (4) |h d ( z, θ) -h d (s ′ , θ)| = |h d (u, θ) -h d (u ′ , θ)| ≤ 1 pq i,j |h(u + (i, j), θ) -h(u ′ + (i, j), θ)| < ε ||h||pq × the number of rectangular tiles ≤ ε
||h|| . Using the Cauchy-Schwartz inequality, and then applying Formulas (1) and ( 4) for any t ∈ R we have (5)

U a t h d a -h a , χ a θ χ a , h a θ ≤ U a t [h d a -h a , χ a θ χ a ] • ||h a || = ||h d a -h a , χ a θ χ a || • ||h a || = ||h d a -h d a , χ a θ χ a || • ||h a || < ε ||h|| • ||h a || ≤ ε.
Combining Equations ( 2),( 3) and ( 5) we conclude that for any t ∈ T U a t h a , h a θ -h a , χ a θ 2 < 2ε.

For a given VH-table b in O and every τ > N, we consider the set of directions

Θ( b, τ, h, N) := θ : ∃t ∈ (N, τ ), U b t h b , h b θ -| h b , χ b θ | 2 < 1 N .
From Formula (3) it follows that for any any (p, q)-VH-table a and any θ irrational there exists τ 0 > 0 such that θ is in Θ( a, τ 0 , h, N). Moreover, θ is in Θ( a, τ, h, N) for any τ ≥ τ 0 as these sets are increasing in τ . Thus, the measure of Θ( a, τ, h, N) tends to 1 as τ goes to infinity.

For any θ ∈ Θ( a, τ, h, N), let η θ > 0 be as large as possible such that the

η θ -neighborhood U( a, η θ ) is included in b ∈ O : ∃t < τ, U b t h b , h b θ -| h b , χ b θ | 2 ≤ 1 2N
So, for any N we can choose τ ( a, h, N) and η( a, h, N) > 0 such that Θ( a, h, N) := Θ( a, τ ( a, h, N), h, N) ∩ {θ : η θ > η( a, h, N)} has measure at least 1 -1 N 2 . Thus, any b in η-neighborhood U( a, η( a, h, N)) satisfies

U b t h b , h b θ -| h b , χ b θ | 2 < 1 N
for any θ in Θ( a, h, N) and some N < t < τ ( a, h, N).

Let {h j } be a countable basis of continuous functions in L 2 (R 2 ). Under the identification h j ( z, θ) = h j (s), this is a countable basis of L 2 (R 2 ×{θ}) for all θ. Let { a i } be a dense collection of (p, q)-VH-tables. Let us consider the following dense G δ -set

G := ∞ N =1 ∞ i=N N j=1 U a i , η( a i , h j , N) . Let b ∈ G. Then for each N there is i N ≥ N such that b and a i N are η N -close, where η N := min η( a i N , h j , N) : 1 ≤ j ≤ N .
The set N j=1 Θ( a i N , h j , N) is of measure at least 1 -1 N and for every direction θ in this set, and every j ≤ N, there exists t N (j) ≥ N such that

U b t N (j) h b , h b θ -| h b , χ b θ | 2 < 1 N , where h = h j . Let Θ b := ∞ n=1 ∞ N =n N j=1
Θ( a i N , h j , N). This is a dense G δ -set of full measure. For every θ ∈ Θ b and any j ≥ 1 there exist sequences

N k = N k (j, θ, g) → ∞ and t k = t k (j, θ, g) ≥ N k , such that (6) U b t k h b , h b θ -| h b , χ b θ | 2 < 1 N k
where h = h j .

We will now show that for every θ ∈ Θ b the flow

φ θ t on B b is weak- mixing. Fix θ ∈ Θ b .
Since θ is fixed, we will denote by U b t the operator U b t restricted to the direction θ. Suppose φ θ t is not weak-mixing, then there exists an eigenfunction h ∈ L 2 (B b × {θ}) with an eigenvalue e ia ; thus, U b t h = e iat h. We can assume that : ||h|| θ = 1 and B b h dµ = 0.

Let ε > 0, and let h j be such that ||h -h j || θ < ε. Then for any t ∈ R we have

U b t h j = U b t (h j -h) + U b t h = U b t (h j -h) + e iat h.
Therefore

U b t h j , (h j ) b θ = U b t (h j -h) + U b t h, (h j ) b θ = U b t (h j -h) + e iat h, (h j ) b -h + h θ = U b t (h j -h), (h j ) b -h θ + 2ℜ h, (h j ) b -h θ + e iat which implies the estimate U b t h j , (h j ) b θ -e iat < ε 2 + 2ε. Since B b (h j ) b dµ = B b ((h j ) b -h) dµ ≤ ||(h j ) b -h|| θ ||χ b || θ (by Cauchy-Schwartz inequality) = ||(h j ) b -h|| θ µ(B b ) = ||(h j ) b -h|| θ < ε,
we have for any t (by the triangular inequality):

U b t h j , (h j ) b θ -| h j , χ b θ | 2 ≥ | U b t h j , (h j ) b θ | -| h j , χ b θ | 2 > (|e iat | -ε 2 -2ε) -ε 2 > 1 -2ε -2ε 2 .
Thus, for ε small enough, h j will satisfy

U b t h j , (h j ) b θ -| h j , χ b θ | 2 > 1 2
for all t. This contradicts Equation ( 6). Thus such an h does not exist and the direction θ is weak-mixing.

Generalizations

The theorem also holds for billiards in polygons with fixed combinatorics (and at least 4 sides) all of whose sides are parallel to a fixed equilateral triangle. In Figure 2 we give two examples of fixed combinatoric: there is a dense G δ of trapezoids with angles π/3 and 2π/3 (of fixed area) for which the billiard flow is weakly mixing in almost every direction, and the same for parallelograms of fixed area with these angles.

Our method also proves topological genericity of the set of translation surfaces within a given stratum for which the translation flow is weakly mixing in almost every direction; the result in [AF] only showed the measure theoretic genericity of such translation surfaces. Indeed, it is sufficient to check that in each stratum the set of rectangle tiled surfaces which are (p, q)-tiled with min(p, q) ≥ Q is dense for each Q.

There are two well known coordinate systems on the set of of translation surfaces: local coordinates for the ablian differential or the zippered rectangle representation; the density holds for both.

Consider first local coordinates for Abelian differentials, they are given by the relative periods of the holomorpic one form, i.e. a collection of planar vectors corresponding to certain saddle connections (see [Z] section 6.4). Lemma 18 and Remark 7 of [KZ] show that the set of compact surfaces with single-cylinder directions is dense in each stratum. The constructed surfaces have the vertical direction as singlecylinder. By making arbitrarily small changes to the vectors of the vertical saddle connections, we ensure that the lengths of all vertical saddle connections are rational multiples of the width of the cylinder without changing the fact that the vertical flow is a single-cylinder direction. By rescaling the width by an arbitrarily small amount, we may ensure all such lengths are in fact rational. We then need only translate all the charts of M by an arbitrarily small amount to ensure that all singularities project to rational coordinates; the surface is now square-tiled with a vertical single-cylinder direction. If additionally we require that all these rational number have a fixed prime denominator q, then the surface is (q, q) tiled by squares, and is not square tiled by larger squares. For any Q ≥ 1 the set of surfaces of this type with q ≥ Q is dense.

Next we describe the density in the zippered rectangle representation of translation surfaces (see [Z] sections 5.5 and 5.7 for a description of this representation). The zippered rectangle coordinates are given by an interval exchange, thus a permutation and some lengths of intervals, the heights of the rectangles, as well as the heights of the singularities. If each of the length and height parameters is rational with common prime denominator q then the surface is (q, q)-tiled by square and is not square tiled by larger squares. For any Q ≥ 1 the set of surfaces of this type with q ≥ Q is dense.

Figure 1 .

 1 Figure 1. A VH-polygon and its bounding rectangle.


Laurence Catanese

Olivier Commowick

Christian Barillot

AUTOMATIC GRAPH CUT SEGMENTATION OF MULTIPLE SCLEROSIS LESIONS

Keywords: Graph Cut, Expectation-Maximization, multiple sclerosis, tissue classification

A fully automated segmentation algorithm for Multiple Sclerosis (MS) lesions is presented. Our method includes two main steps: the detection of lesions by graph cut initialized with a robust Expectation-Maximization (EM) algorithm and the application of rules to remove false positives. Our algorithm will be tested on the ISBI 2015 challenge longitudinal data. For each patient, a unique parameter set is used to run the algorithm.

INTRODUCTION

Manual and semi-automatic segmentation methods are very time consuming and can show a high variability among manual delineations, especially on longitudinal data. To solve this issue, we present a fully automated method for MS lesions segmentation based on the combination of graph cut and robust EM tissues segmentation using multiple sequences of Magnetic Resonance Imaging (MRI). The algorithm is based on several previous segmentation algorithms. A fully automated method implies that it does not include user interactions. Our process is applied to the ISBI 2015 challenge data for longitudinal MS lesion segmentation. Only one parameter set per patient is used. No training steps are involved in the workflow and we do not use the longitudinal information to obtain the segmentation of a given time point.

DATA AND PRE-PROCESSING

The challenge dataset includes T1-w, T2-w, PD and FLAIR sequences. The challenge data will not be described further in this paper, more details can be found on the challenge website 1 . In order to reduce the dependency of the segmentation results on pre-processing performance, the data provided for the challenge is already rigidly co-registered in the same space, skull-stripped and the inhomogeneities are corrected. Therefore, in the following the described method will focus only on the MS lesions segmentation itself. Our algorithm requires a map of the brain to reduce the region of interest. As the brain extraction has already been performed, we only add a pre-processing step and threshold the T2-w image to use it as a brain mask.

MS LESIONS SEGMENTATION WORKFLOW

Lesions detection using graph cut

The segmentation algorithm relies on a graph cut previously presented in [START_REF] Garcia-Lorenzo | Multiple sclerosis lesion segmentation using an automatic multimodal graph cuts[END_REF]. This algorithm requires 3 MR sequences. In our workflow, we choose to use T1-w, T2-w and FLAIR sequences and to discard PD as it generally shows less MS lesion contrast than T2-w and FLAIR. An example of segmentation result is shown figure 1.

Graph cut principle:

The image is represented as a graph where each node is a voxel. Each node is connected to two particular nodes called terminal source and sink which respectively represent the object class for MS lesions and the background class for normal appearing brain tissues (NABT). Based on both contour and regional information, the graph cut allows computing an optimal segmentation of the object from a set of seed points. Spatially neighboring nodes are connected by n-links weighted by boundary values that reflect the similarity of the two considered voxels. The contour information contained in the n-links weights is computed using a spectral gradient [START_REF] Garcia-Lorenzo | Multiple sclerosis lesion segmentation using an automatic multimodal graph cuts[END_REF]. The regional term represents how the voxel fits into given models of the object and background. The edges between a node of the image and the terminal source and sink nodes are called t-links. Normally these models are estimated using seeds given by the user. Instead, we use an automated version of the graph cut where the object and background seeds for the initialization are computed from the images. To do so we use a 3-class multivariate Gaussian mixture model (GMM), each class representing a tissue of the brain: Cerebrospinal Fluid (CSF), Grey Matter (GM) and White Matter (WM). MS lesions will not be a new class but considered as the outliers of this NABT model.

Seeds computation:

The NABT model is estimated using a robust EM algorithm [START_REF] Garcia-Lorenzo | Trimmed-likelihood estimation for focal lesions and tissue segmentation in multisequence mri for multiple sclerosis[END_REF] which optimizes a trimmed likelihood in order to be robust to outliers, controlled by the rejection ratio h that represents the portion of voxels that are removed from the estimation. The algorithm then alternates between the computation of the GMM parameters and the h% outlier voxels. This parameter needs to be high enough to take into account that MS lesions need to be rejected from the model as well as other outliers like veins or skull stripping errors. To process the challenge data, we set h to 20% of the total number of voxels. From the GMM NABT parameters, we then compute a distance of each voxel to each class of the NABT model as a Mahalanobis distance. From this distance, a p-value can be computed, representing the probability not to fit into each of the 3 classes. For each voxel i, we keep the lowest p-value among the three classes, denoted p i . Sinks represent voxels that are close to the NABT model, therefore they should have a high value when not outliers. The sinks t-links weights W bi are then computed as:

W bi = 1 -p i (1)
All voxels that do not fit in the NABT model have a high p-value. To differentiate MS lesions from other outliers (vessels,...), we use a priori knowledge: MS lesions are usually hyperintense compared to the WM in T2-w and FLAIR images. Instead of using a clear threshold, we define fuzzy weights between 0 and 1, based respectively on T2-w hyperintensities (W T 2 ) and FLAIR hyper-intensities (W f lair ) (see [START_REF] Garcia-Lorenzo | Multiple sclerosis lesion segmentation using an automatic multimodal graph cuts[END_REF] for more details). We compute the final sources weights W oi by taking the minimum value between the p-value and the fuzzy weights W T 2 and W f lair :

W oi = AND(p i , W T 2 , W f lair ) (2)

Refinement of the classification

After the detection of candidates lesions, some post-processing steps are performed as false positives may still remain in the detected outliers. Intensity rules may not be enough to discard false positives, therefore we also use localization information.

Considering that MS lesions are typically located in WM, we remove the detected ones that do not sufficiently achieve this condition. Those touching the brain mask border are also eliminated since they are probably false positives due to vessels or errors in the skull stripping. Finally, all candidate lesions having a size lower than 3mm 3 are discarded.

Contour adjustment

Even if a lesion is correctly detected, it does not mean that the contour of the lesion fits the ground truth since MS lesions do not always have a clear border and differ according to the sequence. This may lead to variability on the lesions load and the consideration of the evolution of lesions. When a lesion occurs, it is generally represented by a cavity that has very clear contours inside the WM mask. Thus we consider the whole cavity as a lesion if at least one voxel overlaps with a lesion voxel on the previously computed segmentation mask.

Computation time

The algorithm benefices of a multi-threaded implementation. The total computation time to process each time point of the dataset on a laptop with an Intel Core i7 CPU 2.40GHz (8 cores) is approximately 13 minutes.

CONCLUSION

We presented a fully automated method for MS lesions segmentation using a graph cut initialized with a robust EM, removing all the dependencies to human interactions. However, MS lesion segmentation is a complicated task since the definition of MS lesion is not very precise. There is a high variability in the detection of lesions even in between manual delineations of experts (the Dice scores between two ground truth given for the training data vary from 0,59 to 0,88). Some of the MR acquisitions can have a low initial resolution, complicating the computation of a good segmentation. Therefore setting rules for the segmentation can be difficult as well as setting the input parameters.

Fig. 1 .

 1 Fig. 1. Dataset training 02 time point 1: Top, from left to right: T1-w, T2-w and FLAIR. Bottom, from left to right: ground truth 1, ground truth 2 and automatic segmentation.

http://iacl.ece.jhu.edu/MSChallenge