Arnaud Debussche

Hendrik Weber

The Schrödinger equation with spatial white noise potential

Keywords:

We consider the linear and nonlinear Schrödinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.

Introduction

In this work we study a linear or nonlinear Schrödinger equation on a periodic domain with a random potential given by a spatial white noise in dimension 2. This equation is important for various purposes. In the linear case, it is used to study Anderson localisation. It is a complex version of the famous PAM model. In the nonlinear case, it describes the evolution of nonlinear dispersive waves in a totally disorder medium (see for instance [START_REF] Conti | Solitonization of the Anderson localization[END_REF], [START_REF] Ghofraniha | Shock waves in disordered media[END_REF] and the references therein).

If u denotes the unknown, the equation is given by:

du dt = ∆u + λ|u| 2 u + uξ, x ∈ T 2 , t ≥ 0,
where T 2 denotes the two dimensional torus, identified with [0, 2π] 2 , and ξ is a real-valued spatial white noise. Of course, λ = 0 for the linear equation. A positive λ corresponds to the focusing case and λ < 0 to the defocusing case. For simplicity, we take λ = ±1 in the nonlinear case. The qualitative properties of the solutions are completely different in these two cases.

We are here interested in the question of existence and uniqueness of solutions. This is a preliminary but important step before studying other phenomena: solitary waves, blow up, Anderson localisation... The main difficulty is of course due to the presence of the rough potential. Recall that in dimension 2, a white noise has a negative regularity which is strictly less than -1. Apparently, since the Schrödinger equation has very few smoothing properties and since this smoothing is very difficult to use, it seems hopeless to regularize it.

However, this equation has many other properties. In particular, it is Hamiltonian and preserves the mass. Using a transformation due to [START_REF] Hairer | A simple construction of the continuum parabolic anderson model on R 2[END_REF] in the context of PAM, we are able to use these invariants and construct solutions which have regularity strictly less than 1. More precisely, we solve a transformed equation in almost H 2 (T 2 ), the standard Sobolev space of functions with derivatives up to order 2 in L 2 (T 2 ). This is rather surprising since the regularity is comparable to what is obtained in the parabolic case, when strong smoothing properties are available.

As in the parabolic case, a renormalization is necessary and at the level of the original equation, the renormalized equation rewrites formally:

du dt = ∆u + λ|u| 2 u + u(ξ -∞), x ∈ T 2 , t ≥ 0.
The transformation u → e i∞t u transforms the original equation into the renormalized one. Therefore, the renormalization amounts to renormalize only the phase. A similar remark was made in [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] in a related but different context. For λ < 0, we obtain global solution for any initial data satisfying some smoothness assumptions. For λ > 0, as in the determinstic case, we need a smallness assumption on the initial data.

We could of course consider the equation with a more general nonlinearity: |u| 2σ u with σ ≤ 1. For σ < 1, no restriction on the size of the initial data is required for λ > 0. Another easy generalization is to consider a general bounded domain and Dirichlet boundary conditions, as long as they are sufficiently smooth and the properties of the Green function of the Laplace operator are sufficiently good so that Lemma 2.1 below holds.

The study of the linear equation is closely related to the understanding of the Schrödinger operator with white noise potential. This is the subject of a recent very interesting article by Allez and Chouk ( [START_REF] Allez | The continuous Anderson Hamiltonian in dimension two[END_REF]) where the paracontrolled calculus is used to study the domain and spectrum of this operator. It is not clear how this can be used for the nonlinear equation.

We use the classical L p = L p (T 2 ) spaces for p ∈ [1, ∞], as well as the L 2 based Sobolev spaces H s = H s (T 2 ) for s ∈ R and the Besov spaces B s p,q = B s p,q (T 2 ), for s ∈ R, p, q ∈ [1, ∞]. These are defined in terms of Fourier series and Littlewood-Paley theory (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]). Recall that H s = B s 2,2 and that, for k ∈ N, s ∈ (0, 1), B k+s ∞,∞ coincide with the Hölder space C k,s (T 2 ).

Throughout the article, c denotes a constant which may change from one line to the next. Also, we use a small parameter 0 < ε < e -1 and K ε is a random constant which can also change but such that EK p ε is uniformly bounded in ε for all p. Similarly, for 0

< ε 1 , ε 2 < e -1 , the random constant K ε 1 ,ε 2 may depend on ε 1 , ε 2 but EK p ε 1 ,ε 2 is uniformly bounded in ε 1 , ε 2 for all p.

Preliminaries

We consider the following nonlinear Schrödinger equation in dimension 2 on the torus, that is periodic boundary conditions are assumed, for the complex-valued unknown u = u(x, t):

i du dt = ∆u + λ|u| 2 u + uξ, x ∈ T 2 , t ≥ 0. (2.1)
It is supplemented with initial data

u(x, 0) = u 0 (x), x ∈ T 2 .
We need some smoothness on the initial data, this will be made precise below. In the focusing case λ > 0 we need an extra assumption on the size of u 0 L 2 which has be small enough (see (4.3) below). The potential ξ is random and is a real valued spatial white noise on T 2 . For simplicity, we assume that it has a zero spatial average. The general case could be recovered by adding an additional Gaussian random potential which is constant in space. This would not change the analysis below.

Formally equation (2.1) has two invariant quantities. Given a solution u of (2.1), the mass:

N (u(t)) = T 2 |u(x, t)| 2 dx
is constant in time as well as the energy:

H(u(t)) = T 2 1 2 |∇u(x, t)| 2 - λ 4 |u(x, t)| 4 - 1 2 ξ(x)|u(x, t)| 2 dx.
This is formal because the noise ξ is very rough. In dimension 1, the noise has regularity -1/2 -and belongs to B α ∞,∞ for any α < -1/2, therefore the product ξ|u| 2 can be defined rigorously for u ∈ H 1 and this provides a bound in H 1 . Existence and uniqueness through regularization of the noise and a compactness argument can then be obtained.

In dimension 2, the noise lives in any space with regularity -1 -, that is any regularity strictly less than -1, and the solution is not expected to be sufficiently smooth to compensate this. In fact, the product is almost well defined for u ∈ H 1 and we are in a situation similar to the two dimensional nonlinear heat equation with space time white noise. We expect that a renormalization is necessary.

Inspired by [START_REF] Hairer | A simple construction of the continuum parabolic anderson model on R 2[END_REF], we introduce:

Y = ∆ -1 ξ
(note that this is well defined since we consider a zero average noise, we choose Y also with a zero average) and v = ue Y . Then the equation for v reads

i dv dt = ∆v -2∇v • ∇Y + v|∇Y | 2 + λ|v| 2 ve -2Y . (2.2)
Now Y has regularity 1 -and ∇Y is 0 -. Thus this transformation has lowered the roughness of the most irregular term on the right hand side. At this point It is easier to see why we need a renormalization: the term |∇Y | 2 is not well defined since ∇Y is not a function. However, the roughness is mild here and it has been known for long that up to renormalization by a log divergent constant this square term can be defined in the second order Wiener chaos based on ξ.

Let us be more precise. Let ρ ε = ε -2 ρ( • ε ) be a compactly supported smooth mollifier and consider the smooth noise ξ ε = ρ ε * ξ. We denote by Y ε = ∆ -1 ξ ε . Then it is proved in [START_REF] Hairer | A simple construction of the continuum parabolic anderson model on R 2[END_REF] that for every κ > 0, ξ belongs almost surely to B -1-κ ∞,∞ and, as ε → 0, ξ ε converges in probability to ξ in B -1-κ ∞,∞ . Also, denoting by

C ε = E |∇Y ε | 2 the quantity : |∇Y ε | 2 := |∇Y ε | 2 -C ε converges in L p (Ω; B -κ ∞,∞ ) for any p ≥ 1, κ > 0 to a random variable : |∇Y ε | 2 : in the second Wiener chaos associated to ξ. It is easy to see that C ε goes to ∞ as | ln ε| as ε → ∞: E |∇Y ε | 2 ∼ K 0 | ln ε|
for some K 0 > 0. By stationarity this quantity does not depend on x ∈ T 2 .

The precise result, whose proof can be found in [START_REF] Hairer | A simple construction of the continuum parabolic anderson model on R 2[END_REF] in the more difficult case of the space variable in R 2 (see the proofs of Lemma 1.1 and Proposition 1.3 in this work), is the following.

Lemma 2.1. 1 ≥ κ ′ > κ > 0 and any p ≥ 1, there exist a constant c independent of ε such that:

E Y ε -Y p B 1-κ ′ ∞,∞ 1 p ≤ cε κ-2 p and E : |∇Y ε | 2 : -: |∇Y | 2 | p B -κ ′ ∞,∞ 1 p ≤ cε κ-2 p .
Remark 2.2. Using the monotonicity of stochastic L p norms in p, one can drop the exponent -2 p in the right hand side. We state the result in this way because this is the bound that one actually proves. Below, we use this bound with κ -2 p = κ ′ 2 (and κ instead of κ ′ ).

Note that for s < s, p, r ≥ 1, we have B s ∞,∞ ⊂ B s p,r . Thus, bounds in the latter Besov spaces follow.

Instead of solving equation (2.2) for v ε , we consider:

i dv ε dt = ∆v ε -2∇v ε • ∇Y ε + v ε : |∇Y ε | 2 : +λ|v ε | 2 v ε e -2Yε (2.3)
and setting

u ε = v ε e -Yε : i du ε dt = ∆u ε + λ|u ε | 2 u ε + (u ε -C ε )ξ ε , x ∈ T 2 , t ≥ 0.
Since Y ε is smooth, it is classical to prove that these equations have a unique solution in C([0, T ]; H k ) for an initial data in H k , k = 1, 2, provided the L 2 norm is small for λ > 0 (see for instance [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF], Section 3.6). More details are given in Section 4.

The mass and energy are transformed into the two following quantities which are invariant under the dynamic for v ε :

Ñε (v ε (t)) = T 2 |v ε (x, t)| 2 e -2Yε(x) dx and Hε (v ε (t)) = T 2 1 2 |∇v ε (x, t)| 2 + 1 2 v 2 ε : |∇Y ε | 2 : - λ 4 |v ε (x, t)| 4 e -2Yε(x) e -2Yε(x) dx.
Since the most irregular term : |∇Y ε | 2 : here is not as rough as ξ, this transformed energy is a much better quantity than the original one. It is possible to give a meaning to it for ε = 0 and use it to get bounds in H 1 . Below, we use the following simple results.

Lemma 2.3. For any κ ∈ (0, 1) and any p ≥ 1, there exist a constant independent on ε such that:

E e -2Yε -e -2Y p B 1-κ ∞,∞ ≤ cε -2 p +κ . Proof: Since B 1-κ ∞,∞ is equal to the Hölder space C 1-κ (T 2 )
we have:

e -2Yε -e -2Y B 1-κ ∞,∞ = e -2Y (e -2(Yε-Y ) -1) B 1-κ ∞,∞ ≤ e -2Y B 1-κ ∞,∞ e -2(Yε-Y ) -1 B 1-κ ∞,∞ .
Then we write:

e -2Y B 1-κ ∞,∞ ≤ 2 e -2Y L ∞ Y B 1-κ ∞,∞ , e -2(Yε-Y ) -1 B 1-κ ∞,∞ ≤ 2 e -2Y L ∞ e -2Yε L ∞ Y ε -Y B 1-κ ∞,∞
The result follows by Hölder inequality, Lemma 2.1 and Gaussianity to bound exponential moments of Y and Y ε .

Lemma 2.4. There exists a contant c independent of ε such that:

E ∇Y ε 4 L 4 ≤ c| ln ε| 2 and E : |∇Y ε | 2 : 4 L 4 ≤ c(| ln ε|) 4 .
Proof: It suffices to write:

E T 2 |∇Y ε (x)| 4 dx = T 2 E |∇Y ε (x)| 4 dx = 12π 2 C 2 ε .
Similarly:

E T 2 | : |∇Y ε (x)| 2 : | 4 dx = E T 2 (|∇Y ε (x)| 2 -C ε ) 4 dx = 51πC 4 ε .

The linear case

In this section, we start with the linear case: λ = 0. Then the equation for v ε reads

i dv ε dt = ∆v ε -2∇v ε • ∇Y ε + v ε : |∇Y ε | 2 : . (3.1)
There exists a unique solution in C([0, T ];

H 2 ) if v ε (0) ∈ H 2 .
We take the initial data

v ε (0) = v 0 = u 0 e Y
and assume below that it belongs to H 2 . The mass and energy of a solution are now:

Ñε (v ε (t)) = T 2 |v ε (x, t)| 2 e -2Yε(x) dx and Hε (v ε (t)) = T 2 1 2 |∇v ε (x, t)| 2 + 1 2 v 2 ε : |∇Y ε | 2 : e -2Yε(x) dx.
They are constant in time under the evolution (3.1). Since Y ε converges in B 1-κ ∞,∞ for any κ > 0 as ε tends to zero, we see that the mass gives a uniform bound in L 2 on v ε . More precisely:

v ε (t) 2 L 2 ≤ e 2Yε L ∞ e -2Yε L ∞ v 0 2 L 2 = K ε v 0 2 L 2 (3.2)
with

K ε = e 2Yε L ∞ e -2Yε L ∞ . (3.3)
The energy enables us to get a bound on the gradient.

Proposition 3.1. Let κ ∈ (0, 1/2), there exists a random constant K ε bounded in L p (Ω) with respect to ε for any p ≥ 1 such that if v 0 ∈ H 1 :

T 2 |∇v ε (x, t)| 2 dx ≤ K ε Hε (v 0 ) + v 0 2 L 2 .
Proof: Since B -κ ∞,2 is in duality with B κ 1,2 we deduce by the standard multiplication rule in Besov spaces (see e.g. [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF], Section 2.8.1)

T 2 v 2 ε : |∇Y ε | 2 : dx ≤ v 2 ε B κ 1,2 : |∇Y ε | 2 : B -κ ∞,2 ≤ K ε v 2 ε B κ 1,2 ≤ K ε v ε 2 B κ 2,2 .
Then we note that v ε

2 B κ 2,2 = v ε 2
H κ so that by interpolation

T 2 v 2 ε : |∇Y ε | 2 : dx ≤ K ε v ε 2(1-κ) L 2 v ε 2κ H 1 . 6
It follows

T 2 |∇v ε (x, t)| 2 dx ≤ K ε H(v ε (t)) + K ε v ε (t) 2(1-κ) L 2 v ε (t) 2κ H 1 ≤ K ε Hε (v 0 ) + K ε v ε (t) 2 L 2 + K ε v ε (t) 2(1-κ) L 2 ∇v ε 2κ L 2 ≤ K ε Hε (v 0 ) + K ε v 0 2 L 2 + 1 2 ∇v ε 2 L 2
and hence, by absorbing the last term in the left hand side,

T 2 |∇v ε (x, t)| 2 dx ≤ K ε Hε (v 0 ) + v 0 2 L 2 .
Since Hε (v 0 ) is bounded for v 0 ∈ H 1 , we obtain a (random) bound on v ε in H 1 using similar arguments as above.

Corollary 3.2. There exists a random constant K ε bounded in L p (Ω) with respect to ε for any p ≥ 1 such for any v 0 ∈ H 1 :

v ε (t) H 1 ≤ K ε v 0 H 1 , t ≥ 0.
Unfortunately, this regularity is not sufficient to control the product ∇v ε • ∇Y ε on the right hand side of (3.1).

The next observation is that, if v 0 is smooth enough, v ε is time differentiable and setting w ε = dvε dt it is a solution of:

i dw ε dt = ∆w ε -2∇w ε • ∇Y ε + w ε : |∇Y ε | 2 : . (3.4)
Since w ε satisfies the same equation as v ε , it has the same invariant quantities. We use in particular the mass: Ñ (w ε (t)) = Ñ (w ε (0)).

Hence:

w ε (t) L 2 ≤ K ε Ñ (w ε (0)) ≤ K ε w ε (0) L 2 .
This is still true under the assumption that w ε (0

) ∈ L 2 , which is equivalent to v 0 ∈ H 2 . Proposition 3.3. There exist a random constant K ε bounded in L p (Ω) with respect to ε for any p ≥ 1 such that if v 0 ∈ H 2 : v ε H 2 ≤ cK ε v 0 H 2 + v 0 L 2 | ln ε| 2 .
Proof: From (3.1), we have:

w ε (0) = -i(∆v 0 -2∇v 0 • ∇Y ε + v 0 : |∇Y ε | 2 :),
so that, thanks to the embedding

H 1/2 ⊂ L 4 , w ε (0) L 2 ≤ c v 0 H 2 + v 0 H 3/2 ∇Y ε L 4 + v 0 H 1/2 : |∇Y ε | 2 : L 4 .
By interpolation we deduce:

w ε (0) L 2 ≤ c v 0 H 2 + v 0 3/4 H 2 v 0 1/4 L 2 ∇Y ε L 4 + v 0 1/4 H 2 v 0 3/4 L 2 : |∇Y ε | 2 : L 4 ≤ c v 0 H 2 + v 0 L 2 ∇Y ε L 4 + v 0 L 2 : |∇Y ε | 2 : 4/3 L 4 ≤ c v 0 H 2 + K ε v 0 L 2 | ln ε| 2 , (3.5)
where:

K ε = ∇Y ε 4 L 4 | ln ε| -2 + : |∇Y ε | 2 : 4/3 L 4 | ln ε| -2
. By Lemma 2.4 and gaussianity, we know that the moments of this random variable are bounded with respect to ε.

It follows

w ε (t) L 2 ≤ K ε v 0 H 2 + v 0 L 2 | ln ε| 2 .
This in turn allows us to control v ε H 2 . Indeed, from (3.1),

∆v ε L 2 ≤ w ε (t) L 2 + 2 ∇v ε • ∇Y ε L 2 + v ε : |∇Y ε | 2 : L 2
and by similar arguments as above

∆v ε L 2 ≤ w ε (t) L 2 + 1 2 ∆v ε L 2 + cK ε v ε (t) L 2 | ln ε| 2 and finally ∆v ε L 2 ≤ K ε v 0 H 2 + v 0 L 2 | ln ε| 2 .
The result follows thanks to (3.2). This bound does not seem to be very useful since it explodes as ε → 0. To use it, we consider the difference of two solutions.

Proposition 3.4. Let ε 2 > ε 1 > 0 then for κ ∈ (0, 1], p ≥ 1 there exist a random constant K ε 1 ,ε 2 bounded in L p (Ω) with respect to ε 1 , ε 2 for any p ≥ 1 such that if v 0 ∈ H 2 sup t∈[0,T ] v ε 1 (t) -v ε 2 (t) 2 L 2 ≤ K ε 1 ,ε 2 ε κ/2 2 | ln ε 2 | 1+2κ v 0 2 H 2 Proof: We set r = v ε 1 -v ε 2 and write: i dr dt = ∆r -2r • ∇Y ε 1 + r : |∇Y ε 1 | 2 : -2∇v ε 2 • ∇(Y ε 1 -Y ε 2 ) + v ε 2 (: |∇Y ε 1 | 2 : -: |∇Y ε 2 | 2 :).
By standard computations, we deduce:

1 2 d dt T 2 |r(x, t)| 2 e -2Yε 1 (x) dx = Im T 2 -2∇v ε 2 • ∇(Y ε 1 -Y ε 2 ) + v ε 2 (: |∇Y ε 1 | 2 : -: |∇Y ε 1 | 2 :) re -2Yε 1 (x) dx ≤ 2 ∇v ε 2 re -2Yε 1 B κ 1,2 ∇(Y ε 1 -Y ε 2 ) B -κ ∞,2 + v ε 2 re -2Yε 1 B κ 1,2 : |∇Y ε 1 | 2 : -: |∇Y ε 2 | 2 : B -κ ∞,2
the first term of the right hand side is bounded thanks to interpolation and paraproduct inequalities (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]) and we have

∇v ε 2 re -2Yε 1 B κ 1,2 ≤ c v ε 2 H 1+κ ( v ε 1 H κ + v ε 2 H κ ) e -2Yε 1 B κ ∞,2
. Then, by Proposition 3.3 and interpolation:

v ε 2 H 1+κ ≤ c v ε 2 (1+κ)/2 H 2 v ε 2 (1-κ)/2 L 2 ≤ K ε 2 ( v 0 H 2 + v 0 L 2 | ln ǫ 2 | 2 ) (1+κ)/2 v 0 (1-κ)/2 L 2
and, by Corollary 3.2, for i = 1, 2

v ε i H κ ≤ v ε i κ H 1 v ε i 1-κ L 2 ≤ K ε i v 0 κ H 1 v 0 1-κ L 2 ≤ K ε i v 0 κ/2 H 2 v 0 1-κ/2 L 2 . It follows ∇v ε 2 re -2Yε 1 B κ 1,2 ≤ K ǫ 2 e -2Yε 1 B κ ∞,2 v 0 3 2 -κ L 2 ( v 0 H 2 + v 0 L 2 | ln ǫ 2 | 2 ) 1 2 +κ
The second term is bounded by the same quantity and we deduce:

1 2 d dt T 2 |r(x, t)| 2 e -2Yε 1 (x) dx ≤ K ǫ 2 e -2Yε 1 B κ ∞,2 v 0 3 2 -κ L 2 ( v 0 H 2 + v 0 L 2 | ln ǫ 2 | 2 ) 1 2 +κ × ∇(Y ε 1 -Y ε 2 ) B -κ ∞,2 + : |∇Y ε 1 | 2 : -: |∇Y ε 2 | 2 : B -κ ∞,2
.

The result follows thanks to integration in time and Lemma 2.1. By interpolation, we deduce from Proposition 3.3 and 3.4 the following result.

Corollary 3.5. Let ε 2 > ε 1 > 0 then for κ ∈ (0, 1], γ ∈ [0, 2), p ≥ 1 there exist a random constant K ε 1 ,ε 2 bounded in L p (Ω) with respect to ε 1 , ε 2 for any p ≥ 1 such that if v 0 ∈ H 2 : sup t∈[0,T ] v ε 1 (t) -v ε 2 (t) 2 H γ ≤ K ε 1 ,ε 2 ε κ 2 (1-γ 2 ) 2 | ln ε 1 | 4 v 0 2 H 2
We are now ready to state and prove the main result of this section.

Theorem 3.6. Assume that v 0 = u 0 e Y ∈ L p (Ω; H 2 ) for some p > 1. For any T ≥ 0, q < p, γ ∈ (1, 2), when ε → 0, the solution v ǫ of (3.1) satisfying v ε (0) = v 0 converges in L q (Ω; C([0, T ]; H γ )) to v which is the unique solution to i dv dt = ∆v -2∇v • ∇Y + v : |∇Y | 2 : (3.6) in C([0, T ]; H γ ) such that v(0) = v 0 .
Proof: Pathwise uniqueness is clear. Indeed if v ∈ C([0, T ]; H γ ) is a solution of (3.6), it is not difficult to use a regularization argument to justify

d dt T 2 |v(x, t)| 2 e -2Y (x) dx = 0. Now let ε k = 2 -k , Corollary 3.5 implies that (v ε k ) is Cauchy in L q (Ω; C([0, T ]; H γ ))
. It is not difficult to prove that the limit v is a solution of (3.6) and

E sup t∈[0,T ] v ε k (t) q H γ ≤ cE v 0 p H 2 q/p , E sup t∈[0,T ] v(t) q H γ ≤ cE v 0 p H 2 q/p .
Then, by interpolation with γ < γ < 2:

sup t∈[0,T ] v ε (t) -v ε k (t) 2 H γ ≤ c sup t∈[0,T ] v ε (t) -v ε k (t) 2(1-γ/γ ) L 2 sup t∈[0,T ] v ε (t) -v ε k (t) 2γ/γ H γ ≤ c sup t∈[0,T ] v ε (t) -v ε k (t) 2(1-γ/γ ) L 2 sup t∈[0,T ] ( v ε (t) H 2 + v ε k (t) H γ ) 2γ/γ .
By Proposition 3.4, Proposition 3.3 and the above inequality, we deduce:

E sup t∈[0,T ] v ε (t) -v ε k (t) q H γ ≤ cε κ 2 (2-γ/γ)) | ln ε| 4 E v 0 p H 2 q/p .
Letting k → ∞, we deduce that the whole family (v ε ) ε>0 converges to v in L q (Ω; C([0, T ]; H γ )).

The nonlinear equation

We now study the nonlinear equation (2.2) and consider its approximation

(2.3) with initial condition v ε (0) = v 0 = u 0 e Y
and assume below that it belongs to H 2 . Again the mass gives a uniform bound in L 2 on v ε :

v ε (t) 2 L 2 ≤ K ε v 0 2 L 2 . (4.1)
The estimate on the H 1 norm using the energy is similar to the linear case. Recall that the energy is given by:

Hε (v ε (t)) = T 2 1 2 |∇v ε (x, t)| 2 + 1 2 v 2 ε : |∇Y ε | 2 : - λ 4 |v ε (x, t)| 4 e -2Yε(x) e -2Yε(x) dx
and it can be checked that for all t ≥ 0 we have Hε (v ε (t)) = Hε (v 0 ).

Proposition 4.1. There exists a constant K ε bounded in L p (Ω) for any p ≥ 1 such that if v 0 ∈ H 1 and

e -2Yε 3 L ∞ e 2Yε L ∞ v 0 L 2 ≤ 1 if λ = 1 (4.2) then v ε (t) 2 H 1 ≤ K ε v 0 2 H 1 + v 0 2 L 2 v 0 2 H 1 . Moreover, the sequence (K k ) = (K 2 -k ) is bounded almost surely. Remark 4.2.
It is classical that the equation for u ε is locally well posed in H 1 , see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] Section 3.6. The bound obtained here gives a global bound in H 1 for u ε so that global existence and uniqueness hold for u ε and thus for v ε .

Proof: We proceed as in the proof of Proposition 3.1. We first have

T 2 v 2 ε : |∇Y ε | 2 : dx ≤ K ε v ε 2(1-κ) L 2 v ε 2κ H 1 and T 2 |∇v ε (x, t)| 2 dx ≤ K ε H(v ε (t)) + K ε v ε (t) 2(1-κ) L 2 v ε (t) 2κ H 1 + λ 4 T 2 |v ε (x, t)| 4 e -2Yε(x) e -2Yε(x) dx ≤ K ε Hε (v 0 ) + K ε v 0 2 L 2 + 1 2 ∇v ε 2 L 2 + λ 4 T 2 |v ε (x, t)| 4 e -2Yε(
x) e -2Yε(x) dx.

For λ = -1, the result follows after dropping the last term and using

T 2 |v 0 (x)| 4 e -4Yε(x) dx ≤ K ε v 0 4 L 4 ≤ K ε v 0 4 H 1/2 ≤ K ε v 0 2 L 2 v 0 2 H 1
thanks to the Sobolev embedding H 1/2 ⊂ L 4 and interpolation. For λ = 1, Gagliardo-Nirenberg inequality (see for instance [START_REF] Brezis | Gallouet Nonlinear Schrödinger evolution equations[END_REF] for a simple proof with the constant 1/2 used below):

T 2 |v ε (x, t)| 4 e -4Yε(x) dx ≤ e -2Yε 2 L ∞ T 2 |v ε (x, t)| 4 dx ≤ 1 2 e -2Yε 2
L ∞ T 2 |v ε (x, t)| 2 dx T 2 |∇v ε (x, t)| 2 dx ≤ 1 2 e -2Yε 3 L ∞ e 2Yε L ∞ v 0 2 L 2 T 2 |∇v ε (x, t)| 2 dx,
where we have made use of

v ε (t) 2 L 2 ≤ e 2Yε L ∞ e -2Yε L ∞ v 0 2 L 2 according to (3.

2). The result follows easily under assumption (4.2).

The constant K ε is a polynomial in :

|∇Y ε | 2 : B -κ ∞,2
, e -2Yε L ∞ and e 2Yε L ∞ . By Lemma 2.1 and Lemma 2.3 and Borel-Cantelli, we know that :

|∇Y 2 -k | 2 : and Y 2 -k converge almost surely in B -κ ∞,∞ and B 1-κ
∞,∞ so that K k is indeed bounded almost surely. We now proceed with the H 2 bound. Proposition 4.3. There exist a random constants K ε bounded in L p (Ω) with respect to ε for any p ≥ 1 such that if v 0 = u 0 e -Y ∈ H 2 and (4.2) holds:

v ε (t) H 2 ≤ cK ε 1 + v 0 H 2 + v 0 L 2 | ln ε| 4 + v 0 3 H 1 + v 0 3 L 2 v 0 3 H 1 exp(Kε v 0 2 H 1 + v 0 2 L 2 v 0 2 H 1 t) . Moreover K k = K 2 -k is bounded almost surely.
Remark 4.4. We know that if u ε (0) ∈ H 2 , the solution u ε lives in H 2 (see [START_REF] Brezis | Gallouet Nonlinear Schrödinger evolution equations[END_REF]). Since u ε (0) = v 0 e -Yε , we can apply this result.

Proof: As in Section 3, we set w ε = dvε dt which now satisfies:

i dw ε dt = ∆w ε -2∇w ε • ∇Y ε + w ε : |∇Y | 2 : +λ |v ε | 2 w ε + 2Re(v ε wε )v ε e -2Yε .
From (2.3), we have:

w ε (0) = -i(∆v 0 -2∇v 0 • ∇Y ε + v 0 : |∇Y ε | 2 :) + λ|v 0 | 2 v 0 e -2Yε ,
and as in Proposition 3.3 and using the embedding H 1 ⊂ L 6 :

w ε (0) L 2 ≤ c v 0 H 2 + v 0 L 2 ∇Y ε 4 L 4 + : |∇Y ε | 2 : 4/3 L 4 + e -2Yε L ∞ v 0 3 H 1 .
By Lemma 2.4:

P( ∇Y ε 4 L 4 + : |∇Y ε | 2 : 4/3 L 4 ≥ | ln ε| 4 ) ≤ c| ln ε| -2 it follows that w ε (0) L 2 ≤ c v 0 H 2 + K ε v 0 L 2 | ln ε| 4 + K ε v 0 3 H 1
with K ε having all moments finite and such that K 2 -k is almost finite by Borel-Cantelli.We have taken | ln ε| 4 instead of | ln ε| 2 in the estimate above in order to have this latter property. Recall that Y 2 -k converges a.s. in L ∞ . We do not have preservation of the L 2 norm but:

1 2 d dt T 2 |w ε (x, t)| 2 e -2Yε(x) dx = 2λ T 2 Re(v ε (x, t) wε (x, t))Im(v ε (x, t) wε (x, t))e -4Yε(x) dx ≤ K ε v ε (t) 2 L ∞ T 2 |w ε (x, t)| 2 e -2Yε(x) dx ≤ K ε v ε (t) 2 H 1 (1 + ln(1 + v ε (t) H 2 )) T 2 |w ε (x, t)| 2 e -2Yε(x) dx ≤ K ε v 0 2 H 1 + v 0 2 L 2 v 0 2 H 1 (1 + ln(1 + v ε (t) H 2 )) T 2 |w ε (x, t)| 2 e -2Yε(x) dx
thanks to the Brezis-Gallouet inequality ( [START_REF] Brezis | Gallouet Nonlinear Schrödinger evolution equations[END_REF]) and to Proposition 4.1. Then as above we have:

∆v ε (t) L 2 ≤ 2 w ε (t) L 2 + cK ε v ε (t) L 2 | ln ε| 4 + λ |v ε (t)| 2 v ε e -2Yε L 2 ≤ 2 w ε (t) L 2 + K ε v ε (t) L 2 | ln ε| 4 + K ε v ε (t) 3 L 6 .
By the embedding H 1 ⊂ L 6 and Proposition 4.1, we deduce

v ε (t) H 2 ≤ c w ε (t) L 2 + K ε v 0 L 2 | ln ε| 4 + v 0 3 H 1 + v 0 3 L 2 v 0 3 H 1 .
Again, the almost sure boundedness of the different constant K k is obtained thanks to Lemma 2.3, Lemma 2.4 and Borel-Cantelli.

To lighten the following computation, we use the temporary notations:

wε = w ε e -Yε 2 L 2 , α ε = K ε v 0 2 H 1 + v 0 2 L 2 v 0 2 H 1 , β ε = K ε v 0 L 2 | ln ε| 4 + v 0 3 H 1 + v 0 3 L 2 v 0 3 H 1 .
Then we have:

d dt wε ≤ α ε (1 + ln(1 + v ε H 2 )) wε ≤ α ε (1 + ln(1 + c w ε L 2 + β ε )) wε ≤ α ε (1 + ln(1 + K ε wε + β ε )) wε . Hence d dt (1 + ln(1 + K ε wε (t) + β ε )) ≤ α ε (1 + ln(1 + K ε wε (t) + β ε )).
By Gronwall's Lemma we deduce:

1 + ln(1 + K ε wε (t) + β ε ) ≤ (1 + ln(1 + K ε wε (0) + β ε )) exp(α ε t)
and taking the exponential

w ε (t) L 2 ≤ K ε wε (t) ≤ cK ε (1 + K ε wε (0) + β ε ) exp(αεt) ≤ K ε (1 + K ε w ε (0) L 2 + β ε ) exp(αεt) ≤ K ε (1 + K ε v 0 H 2 + β ε ) exp(αεt) .
The result follows. We see that this H 2 bound is not as good as in the linear case. Due to the double exponential, we do not have moments here. However, this is sufficient to prove existence and uniqueness. We now state the main result of this section.

-2Y 3 L ∞ e 2Y L ∞ v 0 L 2 < 1 if λ = 1. (4.3)
For any T ≥ 0, p ≥ 1, γ ∈ (1, 2), when ε → 0, the solution v ǫ of (3.1) satisfying v ε (0) = v 0 converges in probability in C([0, T ]; H γ )) to v which is the unique solution to

i dv dt = ∆v -2∇v • ∇Y + v : |∇Y 2 : +λ|v| 2 ve -2Y (4.4)
with paths in C([0, T ]; H γ ) such that v(0) = v 0 .

Proof: Again, pathwise uniqueness is easy.

Under assumption (4.3), we know that (4.2) holds for ε small enough. Thus we may use Propositions 4.1 and 4.3.

We take ε 2 > ε 1 > 0, set r = v ε 1 -v ε 2 and write:

i dr dt = ∆r -2r • ∇Y ε 1 + r : |∇Y ε 1 | 2 : -2∇v ε 2 • ∇(Y ε 1 -Y ε 2 ) + v ε 2 (: |∇Y ε 1 | 2 : -: |∇Y ε 2 | 2 :) +λ|v ε 1 | 2 re -2Yε 1 -λ(|v ε 2 | 2 -|v ε 1 | 2 )v ε 2 e -2Yε 1 + λ|v ε 2 | 2 v ε 2 (e -2Yε 1 -e -2Yε 2 ).
By the same arguments as in Section 3 and standard estimates we obtain;

1 2 d dt T 2 |r(x, t)| 2 e -2Yε 1 (x) dx ≤ c e -2Yε 1 B κ ∞,2 v ε 2 (t) H 1+κ ( v ε 1 (t) H κ + v ε 1 (t) H κ ) × ∇(Y ε 1 -Y ε 2 ) B -κ ∞,2 + : |∇Y ε 1 | 2 : -: |∇Y ε 2 | 2 : B -κ ∞,2 +K ε 1 ( v ε 1 L ∞ + v ε 2 L ∞ ) v ε 2 L ∞ T 2 |r(x, t)| 2 e -2Yε 1 (x) dx +K ε 1 v ε 2 3 L ∞ ( v ε 1 L 2 + v ε 2 L 2 ) e -2Yε 1 -e -2Yε 2 L 2 ≤ K ε 1 ,ε 2 v ε 2 (t) κ H 2 ( v ε 1 (t) H 1 + v ε 1 (t) H 1 ) 2-κ ε κ/2 2 +K ε 1 ( v ε 1 H 1 + v ε 2 H 1 ) 2 (1 + ln(1 + v ε 1 H 2 ) + ln(1 + v ε 2 H 2 )) × T 2 |r(x, t)| 2 e -2Yε 1 (x) dx +K ε 1 ,ε 2 v ε 2 3 H 1 (1 + ln(1 + v ε 1 H 2 )) 3/2 ( v ε 1 L 2 + v ε 2 L 2 )ε κ 2 ,
by the Brezis-Gallouet inequality. We have used for κ ∈ (0, 1):

e -2Yε 1 -e -2Yε 1 L 2 ≤ 2 Y ε 1 -Y ε 2 L 2 ( e -2Yε 1 L ∞ + e -2Yε 2 L ∞ ) ≤ c|ε 2 | κ Y B κ ∞,∞ ( e -2Yε 1 L ∞ + e -2Yε 2 L ∞ ) ≤ K ε 1 ,ε 2 |ε 2 | κ .
Again, the constants K ε 1 , K ε 1 ,ε 2 above have all moments bounded independently of ε 1 , ε 2 are almost surely bounded in k for ε 1 = 2 -(k+1) , ε 2 = 2 -k . We deduce from Gronwall's Lemma: ln r L 2 ≤ K ε 1 ,ε 2 P (v 0 )(1 + ln P (v 0 ) + ln | ln ε 1 |) exp(Kε

= 2 -(k+1) , ε 2 = 2 -k so that v 2 -(k+1) -v 2 -k H γ ≤ C(v 0 )| ln 2 -(k+1) | C(v 0 ) 2 -κ 2 (k+1) ,
where now C(v 0 ) is a random constant depending on v 0 H 2 . It follows that v 2 -k is Cauchy in C([0, T ], H γ ). Finally, we reproduce the estimate above for v ε (t) -v 2 -k L 2 but bound v 2 -k L ∞ by v 2 -k H γ instead of using Brezis-Gallouet inequality. We obtain: Remark 4.6. Condition (4.3) is probably not optimal. It can easily be weakened to λK 3 2,ε K 1,ε v 0 L 2 < 2, but this is probably not optimal either.

ln v ε (t) -v 2 -k L 2 ≤ Kε,

Theorem 4 . 5 .

 45 Assume that v 0 = u 0 e Y ∈ H 2 and e

2 | 2 |

 22 2 -k P (v 0 )(1 + ln P (v 0 ) + ln | ln ε|) exp( Kε,2 -k P (v 0 )t) +κ exp( Kε,2 -k P (v 0 )t)(ln Kε,2 -k + ln P (v 0 ) + ln | ln ε 1 |) -κ ln ε|,where Kε,2 -k are constants with moments bounded independently on ε and k. Again, a similar bound holds for the H γ norm thanks to an interpolation argument. Letting k → ∞ yields:ln v ε (t) -v L 2 ≤ Kε P (v 0 )(1 + ln P (v 0 ) + ln | ln ε|) exp( KεP (v 0 )t)+κ exp( Kε P (v 0 )t)(ln Kε + lnP (v 0 ) + ln | ln ε 1 |) -κ ln ε|,where again Kε are constants with bounded moments. The conclusion follows.

  1 ,ε 2 P (v 0 )t) +κ exp(K ε 1 ,ε 2 P (v 0 )t)(ln K ε 1 ,ε 2 + ln P (v 0 ) + ln | ln ε 1 |) -κ 2 | ln ε 2 |,where P (v 0 ) denotes a polynomial in v 0 H 2 . By interpolation, we have ln rH γ ≤ c + (1 -γ 2 ) ln r L 2 + γ 2 ln r H 2so that a similar estimate holds for ln r H γ , γ < 2.The constants K ε 1 ,ε 2 are bounded almost surely when ε 1

Acknowledgement

This work was started during the Fall semester 2015, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by the National Science Foundation under Grant No. DMS-1440140. A. Debussche benefits from the support of the French government "Investissements d'Avenir" program ANR-11-LABX-0020-01. H. Weber is supported by the Royal Society through the University Research Fellowship UF140187.