R Émi

Guillaume Ferriere
email: guillaume.ferriere@inria.fr

LOGARITHMIC GROSS-PITAEVSKII EQUATION

We consider the Schrödinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, which turns out to correspond to the energy space for the standard Gross-Pitaevskii equation with a cubic nonlinearity, in small dimensions. We then characterize the solitary and traveling waves in the one dimensional case.

Introduction

We consider the Schrödinger equation with a logarithmic nonlinearity, whose Cauchy problem was studied in [START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF][START_REF] Gallo | The Cauchy problem for defocusing nonlinear Schrödinger equations with nonvanishing initial data at infinity[END_REF][START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF][START_REF] Gérard | The Gross-Pitaevskii equation in the energy space[END_REF][START_REF] Zhidkov | On the existence of the solution of the Cauchy problem and the stability of kink solutions of the nonlinear Schrödinger equation[END_REF][START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF]. The most complete result regarding this aspect is found in [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF][START_REF] Gérard | The Gross-Pitaevskii equation in the energy space[END_REF], where, for d ≤ 3, (1.1) is proved to be globally well posed in the energy space

E GP := u ∈ H 1 loc (R d ) | E GP (u) < ∞
, where E GP is the Ginzburg-Landau energy

E GP (u) := ∇u 2 L 2 (R d ) + 1 2 R d |u| 2 -1 2 dx.
See also [START_REF] Killip | The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions[END_REF] for an analogous energy-critical problem. The logarithmic nonlinearity was introduced in the context of Schrödinger equations in [START_REF] Bia Lynicki-Birula | Nonlinear wave mechanics[END_REF], as it is the only nonlinearity satisfying the following tensorization property: if u 1 (t, x 1 ), . . . , u d (t, x d ) are solutions to the one-dimensional equation, then u(t, x) := u 1 (t, x 1 ) × • • • × u d (t, x d ) solves the d-dimensional equation. This model has regained interest in various domains of physics: quantum mechanics [START_REF] Yasue | Quantum mechanics of nonconservative systems[END_REF], quantum optics [START_REF] Bia Lynicki-Birula | Nonlinear wave mechanics[END_REF][START_REF] Hansson | Propagation of partially coherent solitons in saturable logarithmic media: A comparative analysis[END_REF][START_REF] Krolikowski | Unified model for partially coherent solitons in logarithmically nonlinear media[END_REF][START_REF] Buljan | Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media[END_REF], nuclear physics [START_REF] Hefter | Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics[END_REF], Bohmian mechanics [START_REF] Martino | Logarithmic Schrödinger-like equation as a model for magma transport[END_REF], effective quantum gravity [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF], theory of superfluidity and Bose-Einstein condensation [START_REF] Avdeenkov | Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent[END_REF]. The logarithmic model may generalize the Gross-Pitaevskii equation, used in the case of two-body interaction, to the case of three-body interaction; see [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF][START_REF] Zloshchastiev | Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory[END_REF].

1.1. Cauchy problem. From the mathematical point of view, the Cauchy problem for (logGP) is more intricate than it may seem at first glance. The difficulty does not lie in the behavior of the logarithm at infinity, of course, but in its singularity at the origin. In the case of vanishing boundary condition at infinity, u 0 ∈ H 1 (R d ), the Cauchy problem was solved in [START_REF] Cazenave | Equations d'évolution avec non linéarité logarithmique[END_REF] (case λ < 0, see also [START_REF] Hayashi | A note on the nonlinear Schrödinger equation in a general domain[END_REF]) and [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF][START_REF] Guerrero | Global solvability of the 3D logarithmic Schrödinger equation[END_REF] (for any λ ∈ R), by constructing solutions of a regularized equation converging to a solution to the exact equation, which turns out to be unique. Even when nontrivial boundary conditions at infinity are considered, like in the present paper, the singularity of the logarithm at the origin is the main difficulty, as the cancellation of the wave function is difficult (if not impossible) to rule out for all (t, x) ∈ R × R d . The nonlinearity fails to be locally Lipschitz continuous, so the fixed point argument employed in [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF] in the case of (1.1) is hard to implement. Similarly, the existence result from [START_REF] Gallo | The Cauchy problem for defocusing nonlinear Schrödinger equations with nonvanishing initial data at infinity[END_REF] holds for more general nonlinearities than in (1.1), of the form f (|u| 2 )u, but the function f is required to be at least C 2 on [0, ∞): in our case, f (r) = ln r is not even continuous at the origin. Even if a solution of the form u ∈ 1 + H 1 is considered in the one dimensional case (where the H 1 -norm control the L ∞ norm), it is not obvious to make a solution global in time, even if u 0 -1 H 1 ≪ 1. Indeed, the conservation of the energy in the context of (logGP) reads dE logGP /dt = 0, where

E logGP (u) := ∇u 2 L 2 (R d ) + λ R d |u| 2 ln |u| 2 -|u| 2 + 1 dx.
Consider F the antiderivative of the logarithm, such that F (1) = 0: F (y) = y ln yy + 1 for y > 0, and F (0) = 1. The above potential energy is the integral in space of F (|u| 2 ). We note that F ≥ 0 on [0, ∞), and Taylor's formula yields 0 ≤ F (y) = y 2 1 0 1 -s 1 + sy ds ≤ y 2 , and so 0 ≤ E logGP (u) ≤ E GP (u). But this is not enough to solve (logGP) in the energy space, neither locally or globally in time. As a byproduct of our analysis, we will see that in low space dimensions, d ≤ 4, the energy spaces for (logGP),

E logGP := v ∈ H 1 loc (R d ) | E logGP (v)
< ∞ , and (1.1) coincide, E logGP = E GP . Note that the energy space E GP was described very accurately in [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF][START_REF] Gérard | The Gross-Pitaevskii equation in the energy space[END_REF].

Since we will distinguish the notions of mild and weak solutions, we clarify these two notions in the next definition. • A mild solution to (logGP) on I if it satisfies Duhamel's formula u(t) = e it∆ u 0 -iλ t 0 e i(t-s)∆ u(s) ln |u(s)| 2 ds, ∀t ∈ I.

As we construct solutions to (logGP) as weak solutions, it is sensible to ask whether or not these solutions are also mild solutions, in the same fashion as in [START_REF] Segal | Non-linear semi-groups[END_REF]. Our main result regarding the Cauchy problem for (logGP) is the following: Theorem 1.2. Let d ≥ 1. For any u 0 ∈ E logGP , there exists a unique weak solution u ∈ L ∞ loc (R, E logGP ) to (logGP). Moreover, the flow of (logGP) enjoys the following properties:

• The energy is conserved, E logGP (u(t)) = E logGP (u 0 ) for all t ∈ R.

• u -u 0 ∈ C 0 (R, L 2 ).

• u is also a mild solution.

• If ∆u 0 ∈ L 2 , then u -u 0 ∈ W 1,∞ loc (R, L 2 ) ∩ L ∞ loc (R, H 2
). We emphasize that due to the singularity of the logarithm at the origin, it is hopeless to look for a direct Picard fixed point argument on Duhamel's formula in order to construct a solution. In addition, like in the case of vanishing boundary conditions at infinity evoked above, it is not clear that Strichartz estimates can help in the construction of (local in time) solutions.

Remark 1.3. The parameter λ > 0 can be set to 1, by considering the unknown function v(t, x) = u t λ , x √ λ . We choose to stick to the general notation λ in order to keep track of the dependence on the nonlinear term more explicitly.

Solitary and traveling waves.

There are many references regarding the existence, description and the stability of solitary and traveling waves for (1.1), as can be seen for instance from the results evoked in [START_REF]Stationary and time dependent Gross-Pitaevskii equations[END_REF]. In order to consider both solitary and traveling waves, we look for solutions of the form

(1.2) u(t, x) = e iωt φ(x -ct),
where ω ∈ R, c ∈ R d and φ ∈ E logGP . In the case c = 0, we classically call the solution a solitary wave, and if c = 0, we call it traveling wave (stationary wave if ω = c = 0). From the mathematical point of view, the study of traveling waves for (1.1) (with ω = 0) goes back to [START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF], and has known many developments since; see e.g. [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevski equation[END_REF][START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF][START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF][START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF][START_REF] Mariş | Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF][START_REF] Mariş | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF] and references therein. We emphasize that in these generalizations, nonlinearities other than cubic are considered, of the form f (|u| 2 )u. However, it is always assumed, among other assumptions, that f is continuous on [0, ∞): in the case considered in the present paper, f (r) = ln r is unbounded at the origin, and some work is needed.

In the one dimensional case, traveling waves for (1.1) are either constant, or such that 0 ≤ |c| < √ 2, and given explicitly by

φ c (x) = 1 - c 2 2 tanh 1 - c 2 2 x √ 2 + i c √ 2 ,
up to space translation and a multiplicative constant of modulus 1; see e.g. [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevski equation[END_REF]. This solution u(t, x) = φ c (x -ct) is usually referred to as dark soliton, and black soliton (or kink solution) in the case c = 0 (the only case where φ has a zero). In the case of (logGP), we do not have such an explicit formula, but somehow a very similar description. In the general case d ≥ 1, we have: This conclusion meets the one from [START_REF] Farina | From Ginzburg-Landau to Gross-Pitaevskii[END_REF], a case where however the assumptions on nonlinearities other than cubic are not written. It is not obvious that the logarithmic case is easily inferred, as [START_REF] Farina | Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in R K[END_REF] (which is central in the argument of [START_REF] Farina | From Ginzburg-Landau to Gross-Pitaevskii[END_REF]) considers only polynomials nonlinearities. We now focus on the one-dimensional case. First, like for (1.1), if |c| is too large, then there is no non-constant traveling wave: Theorem 1.5. Let d = 1 and c 2 ≥ 2λ. Any solution to (logGP) of the form (1.2) is constant.

We now come to the description of non-constant traveling waves: Theorem 1.6. Assume d = 1 and c such that c 2 < 2λ. There exists a unique non-constant traveling wave in the following sense: there exists a traveling wave φ c such that any non-constant traveling wave u of the form (1.2) is such that

φ = e iθ φ c (• -x 0 ),
for some constants θ, x 0 ∈ R. If c = 0, φ c never vanishes. In the case c = 0, φ 0 can be taken real-valued and increasing, and then

lim x→±∞ φ 0 (x) = ±1.
We emphasize that qualitatively, we obtain the same properties as the black soliton, even though it is probably hopeless to get an explicit expression in the case of (logGP). 1.3. Content and notations. In Section 2, we reduce the study of (logGP) in the energy space, by analyzing the potential energy and characterizing the energy space E logGP . In Section 3, we prove that any solution in the energy space is unique, and that it is a mild solution. The heart of Theorem 1.2 is established in Section 4, where a weak solution is constructed by an approximation procedure. In Section 5, we show that H 2 regularity is propagated by the flow, completing the proof of Theorem 1.2. We prove general results regarding solitary and traveling waves in Section 6, in particular Theorem 1.4. In Section 7, we analyze stationary waves in the one-dimensional case (Theorem 1.6 in the case c = 0). One-dimensional traveling waves are studied in Section 8, where we complete the proof of Theorems 1.6 and 1.5. We conclude this paper by open questions in Section 9.

We define the L 2 -bracket by

f, g = R d f g dx.
For k an integer, we denote by C k the class of k times continuously differentiable functions, and if 0 < s < 1, by C k,s the class of k times continuously differentiable functions whose derivative of order k is Hölder continuous with exponent s.

If s = 1, C k,1
is the class of k times continuously differentiable functions whose derivative of order k is Lipschitzian.

Preliminary results

2.1.

Generalities. First, we recall some properties about the logarithm. The first property was discovered in [START_REF] Cazenave | Equations d'évolution avec non linéarité logarithmique[END_REF], and is crucial for uniqueness issues related to (logGP).

1]

). There holds

Im (z 2 ln|z 2 | 2 -z 1 ln|z 1 | 2 )(z 2 -z 1 ) ≤ 2|z 2 -z 1 | 2 , ∀z 1 , z 2 ∈ C.
The next lemma measures the continuity of the nonlinearity in (logGP):

Lemma 2.2. For all ε ∈ (0, 1) and x, y ∈ C,

x ln |x| 2 -y ln |y| 2 ≤ C |x| ε |ln |x|| + |y| ε |ln |y|| |x -y| 1-ε + 2|x -y|.
Proof. Let x, y ∈ C and assume without loss of generality that |x| ≤ |y|. We also assume x = 0, otherwise the estimate is obvious. Then, we have

x ln |x| 2 -x ln |y| 2 = 2|x||ln |x| -ln |y|| ≤ 2|x| ||x| -|y|| |x| ≤ 2|x -y|.
On the other hand,

x ln |y| 2 -y ln |y| 2 ≤ 2|x -y||ln |y|| ≤ 2 1+ε |x -y| 1-ε |y| ε |ln |y||, hence the conclusion.
Then, we also recall a result about norms on finite dimensional subspace of L 2 .

Lemma 2.3. If {f k } 0≤k≤n is a finite family of L 2 linearly independent functions, then there exists C > 0 such that, for all λ k ∈ C,

max 0≤k≤n |λ k | ≤ C n k=0 λ k f k L 2 .
Proof. As the functions {f k } 0≤k≤n are linearly independent, the map (λ k ) 0≤k≤n → n k=0 λ k f k L 2 defines a norm on C n+1 . The conclusion readily follows, since the left hand side of the conclusion is the ℓ ∞ norm on C n+1 , and all norms are equivalent on finite dimensional spaces.

Potential energy.

We start by a result on the potential energy E pot , given by

E pot (v) := R d |v| 2 ln |v| 2 -|v| 2 + 1 dx. Let E pot (v) := R d (|v| -1) 2 ln (2 + |v|) dx.
We know that both previous quantities are non-negative. The following result shows that they are actually equivalent.

Lemma 2.4. There exists K 0 > 0 such that for all v ∈ E logGP ,

1 K 0 E pot (v) ≤ E pot (v) ≤ K 0 E pot (v).
Proof. Taylor formula yields, for y ≥ 0, y 2 ln y 2 -y 2 + 1 = 4 (y -1)

2 1 0 (1 + ln (1 -s + sy)) (1 -s)ds.
Distinguishing y < 1 and y > 1, we get: ln (2 + y)

ln (2) |v| -1 2 L 2 ≤ E pot (v) ≤ ln (2) |v| -1 2 L 2 + C p |v| -1 p L p . 2.
3. Energy space. As for the energy space, Lemma 2.4 leads to

E logGP = v ∈ H 1 loc (R d ) | ∇v ∈ L 2 (R d
) and E pot (v) < ∞ . We prove an even more explicit description of E logGP : Lemma 2.6. The energy space is characterized by:

E logGP = v ∈ H 1 loc (R d ) | ∇v ∈ L 2 (R d ) and |v| -1 ∈ L 2 (R d ) = v ∈ H 1 loc (R d ) | ∇v ∈ L 2 (R d ) and |v| -1 ∈ H 1 (R d ) .
Moreover, there exists C > 0 such that for all v ∈ E logGP , |v| -

1 2 H 1 ≤ CE logGP (v). Proof. First, we show that E logGP ⊂ v ∈ H 1 loc | ∇v ∈ L 2 and |v| -1 ∈ L 2 . Let v ∈ E logGP : ∇v L 2 < ∞, and E pot (v) < ∞, which is equivalent to E pot (v) < ∞. Lemma 2.5 yields |v| -1 ∈ L 2 . Conversely, let v ∈ H 1 loc such that |v|-1, ∇v ∈ L 2 . Then ∇(|v|-1) = Re v |v| ∇v and |∇(|v| -1)| ≤ |∇v|, which shows that f = |v| -1 ∈ H 1 . Since f ≥ -1, we readily have E pot (v) = R d |f | 2 ln (3 + f ) dx ≤ R d |f | 2 ln (3 + |f |) dx R d |f | 2 (1 + |f | ε ) dx < ∞,
where ε > 0 is arbitrarily small, and where we have used Sobolev embedding to conclude.

From the previous arguments, if v ∈ E logGP , the L 2 -norm of |v| -1 can be estimated thanks to Lemma 2.5 and Lemma 2.4, whereas |∇(|v| -1)| ≤ |∇v|, hence the conclusion.

Corollary 2.7. For all δ ≥ 1, ε > 0, and p such that

(2.1) p, p + ε ∈      [2, ∞] if d = 1, [2, ∞) if d = 2, [2, 2 * ] if d ≥ 3, 2 * := 2d d-2 , there exists C p,ε,δ such that if v ∈ E logGP , then v ln |v| 2 δ ∈ L p (R d ) and v ln |v| 2 δ L p ≤ C p,ε,δ (E logGP (v) 1/2 + E logGP (v) (1+ε)/2 ).
Proof. We know that |v| -1 ∈ H 1 and, since δ ≥ 1,

|v| ln |v| 2 δ ≤ C δ ||v| -1|(ln (2 + ||v| -1|)) δ ≤ C ε,δ (||v| -1| + ||v| -1| 1+ ε p ).
|u| 2 -1 = (|u| -1) (|u| -1 + 2) ,
and the Sobolev embedding

H 1 (R d ) ⊂ L 4 (R d ) for d ≤ 4.
In particular, Gérard [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF] proved that E GP + H 1 ⊂ E GP for d ≥ 1, and therefore the same holds for E logGP when d ≤ 4. This is also true in all dimensions :

Lemma 2.9. Let d ≥ 1. For all v ∈ E logGP and for all f ∈ H 1 , we have v + f ∈ E logGP . In addition, if p > 2 satisfies (2.1), there exists C p such that for every such v and f ,

E logGP (v + f ) ≤ C p E logGP (v) + C p E logGP (v) p + C p f 2 H 1 + C p f p H 1 . Proof. Let v ∈ E logGP and f ∈ H 1 . We know that ∇v, ∇f ∈ L 2 , so ∇(v + f ) ∈ L 2 .
From Lemma 2.6, there only remains to show that |v + f | -1 ∈ L 2 . For this, we have for all x, y ∈ C

|x + y| = |x| + 1 0 x + ry |x + ry| • y dr.
Applying this equality to u and f , we get

|u + f | -1 = |u| -1 + 1 0 u + rf |u + rf | • f dr, hence ||u + f | -1| ≤ ||u| -1| + |f |, and, for any p ∈ [1, ∞], |u + f | -1 L p ≤ |u| -1 L p + f L p .
Moreover, Lemmas 2.4 and 2.5 yield, for any p > 2 satisfying (2.1),

E logGP (v + f ) ≤ ∇v + ∇f 2 L 2 + C |v + f | -1 2 L 2 + C p |v + f | -1 p L p ∇v 2 L 2 + ∇f 2 L 2 + |v| -1 2 L 2 + f 2 L 2 + |v| -1 p L p + f p L p E logGP (v) + |v| -1 p H 1 + f 2 H 1 + f p H 1 E logGP (v) + E logGP (v) p + f 2 H 1 + f p H 1 .
We conclude by adapting [23, Lemma 1]. Introduce the Zhidkov space

X 1 (R d ) = u ∈ L ∞ (R d ), ∇u ∈ L 2 (R d ) . Lemma 2.10. Let d ≥ 1. We have E logGP ⊂ X 1 (R d ) + H 1 (R d ). Proof. Proceeding like in the proof of [23, Lemma 1], consider χ ∈ C ∞ 0 (C) a cutoff function such that 0 ≤ χ ≤ 1, χ(z) = 1 for |z| ≤ 2 and χ(z) = 0 for |z| ≥ 3. Decompose u ∈ E logGP as u = u 1 + u 2 , u 1 = χ(u)u, u 2 = (1 -χ(u)) u.
Then u 1 L ∞ ≤ 3, and since |u| ≥ 2 on the support of u 2 , for any δ > 1, we may find C δ such that

|u 2 | ≤ C δ ||u| -1| δ . Since |u| -1 ∈ H 1 (R d ) from Lemma 2.6, |u| -1 ∈ L 2δ (R d ) by Sobolev embedding (provided that δ ≤ d d-2 if d ≥ 3), hence u 2 ∈ L 2 .
The properties ∇u 1 , ∇u 2 ∈ L 2 are straightforward, and we refer to [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF] for details.

Regularity and uniqueness

We begin by proving the following point in Theorem 1.2:

Lemma 3.1. Let I ∋ 0 be a time interval. If u ∈ L ∞ loc (I, E logGP ) (that is, t → E logGP (u(t)) ∈ L ∞ loc (I)) is a weak solution to (logGP) on I, then u -u 0 ∈ C 0 (I, L 2 ). Proof. Since u ∈ L ∞ loc (I, E logGP ), we know from Corollary 2.7 that u ln |u| 2 ∈ L ∞ loc (I, L 2 ). Moreover, ∆u ∈ L ∞ loc (I, H -1 ). Therefore, (logGP) yields ∂ t u = ∂ t (u-u 0 ) ∈ L ∞ loc (I, H -1 ), with u(0)-u 0 = 0. This proves that u-u 0 ∈ C 0 (I, H -1 ). Since ∇(u -u 0 ) ∈ L ∞ loc (I, L 2 )
, we obtain the result by interpolation.

This result allows us to infer the uniqueness of the solution of (logGP).

Theorem 3.2. Let u 0 ∈ E logGP and I ∋ 0 be an interval. There exists at most one weak solution u ∈ L ∞ loc (I, E logGP ) to (logGP). Proof. Let u, v ∈ L ∞ (I, E logGP ) solve (logGP). Then u(t) -u 0 and v(t) -u 0 are continuous from I to L 2 from Lemma 3.1. Therefore, w := u -v ∈ C 0 (I; L 2 ). This error satisfies (in

D ′ (I × R d )) i∂ t w + ∆w = λ(u ln |u| 2 -v ln |v| 2 ).
Since on the one hand ∆w ∈ L ∞ (I, H -1 ), and on the other hand u ln |u| 2 , v ln |v| 2 ∈ L ∞ (I, L 2 ), the previous equality is also satisfied in L ∞ (I, H -1 ). As w ∈ L ∞ (I, H 1 ), we can take the H -1 × H 1 bracket against w of the previous equality, which yields

(3.1) i ∂ t w, w H -1 ,H 1 -∇w 2 L 2 = λ R d (u ln |u| 2 -v ln |v| 2 )(u -v) dx.
Thanks to Lemma 2.1, we have

Im R d (u ln |u| 2 -v ln |v| 2 )(u -v) dx ≤ 2 u -v 2 L 2 = 2 w 2 L 2 .
Therefore, taking the imaginary part of (3.1) leads to

Re ∂ t w, w H -1 ,H 1 = 1 2 d dt w 2 L 2 ≤ 2λ w 2 L 2 .
Since w(0) = 0, Gronwall lemma concludes the proof.

We end this section with a link between regularity and mild solution.

Lemma 3.3. Let I ∋ 0 be an open interval of R, u 0 ∈ E logGP and u ∈ L ∞ loc (I, E logGP ) be a weak solution to (logGP) on I. Then u is a mild solution, and

t 0 e -is∆ u(s) ln |u(s)| 2 ds ∈ H 1 (R d ) for all t ∈ I.
Proof. From Corollary 2.7, we have u ln |u| 2 ∈ L ∞ loc (I, L 2 ), thus we can define

v NL (t) = -iλ t 0 e -is∆ u(s) ln |u(s)| 2 ds ∈ C 0 (I, L 2 ) ∩ W 1,∞ loc (I, H -2 ).
On the other hand, since ∇u 0 ∈ L 2 and u 0 ∈ L ∞ + L 2 (from Lemma 2.10), we can define e it∆ u 0 and we have e it∆ u 0 -u 0 ∈ C 0 (I, L 2 )∩C 1 (I, H -1 ) (see [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF]). Therefore,

v := e it∆ u 0 + e it∆ v NL (t) ∈ u 0 + C 0 (I, L 2 ) ∩ W 1,∞ loc (I, H -2
), and we can compute

∂ t v = i∆e it∆ u 0 -iλ t 0 e -is∆ u(s) ln |u(s)| 2 ds + e it∆ ∂ t u 0 -iλ t 0 e -is∆ u(s) ln |u(s)| 2 ds = i∆v + e it∆ -iλe -it∆ u(t) ln |u(t)| 2 = i∆v -iλu(t) ln |u(t)| 2 ,
where the equality is to be understood in

L ∞ loc (I, H -2 ). Let w = u -v.
Then, w ∈ C 0 (I, L 2 ) from Lemma 3.1 along with the previous arguments, and with the previous equality, we have, in

L ∞ loc (I, H -2 ), ∂ t w -i∆w = 0.
Since w(0) = 0 by construction, we get w = 0, which proves the mild formulation. Last, we know that ∇(u(t) -u 0 ) ∈ L 2 and ∇(e it∆ u 0 -u 0 ) ∈ L 2 for all t ∈ I, therefore ∇v NL (t) ∈ L 2 , ∀t ∈ I. Since we already know that v NL (t) ∈ L 2 , we get the conclusion.

Construction of a solution

In this section, we construct a global solution u ∈ L ∞ (R, E logNLS ) to (logGP). We adapt the method used by Ginibre and Velo [START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF] to construct global weak solutions to non-linear Schrödinger equations (NLS) by compactness. Here, the framework is different since the solution is not in L 2 . However, we know that we should have u(t) -u 0 ∈ H 1 . Therefore, we will approximate the solution on u 0 + X m with X m a sequence of finite dimensional linear subspaces approximating H 1 . Remark 4.1. In [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF], for vanishing boundary condition at infinity, another approximation is considered, consisting in removing the singularity of the logarithm by saturating the nonlinearity, for ε > 0, (4.1)

i∂ t u ε + ∆u ε = λu ε ln |u ε | 2 + ε , u ε |t=0 = u 0
, and letting ε → 0. This approximation has the advantage of working whichever the sign of λ is, while the approach introduced initially in [START_REF] Cazenave | Equations d'évolution avec non linéarité logarithmique[END_REF] (see also [START_REF] Hayashi | A note on the nonlinear Schrödinger equation in a general domain[END_REF]) seems to be bound to the nondispersive case λ < 0. In the case of Gross-Pitaevskii equation, integrability constraints are different, and the corresponding potential energy is given by

λ R d (|u ε | 2 + ε) ln |u ε | 2 + ε -(1 + ln(1 + ε)) |u ε | 2 -1 -(1 + ε) ln(1 + ε) .
We believe that this strategy should work, possibly with long computations; we rather choose an approach involving an energy independent of the approximation. 4.1. Finite dimensional approximation. Let {w j } j∈N a Hilbert basis of L 2 with all w j ∈ H 1 and m ∈ N. One may think for instance of Hermite functions, all the more since in Lemma 5.1 below, in order to prove the propagation of Ḣ2 regularity, we further require w j ∈ H 2 . We take X m := Vect(w j ) j≤m and look for an approximation u m of the form

(4.2) u m (t, x) = u 0 (x) + m k=0 g m,k (t)w k (x) =: u 0 (x) + ϕ m (t, x), satisfying (4.3) w j , i∂ t u m + ∆u m -λu m ln |u m | 2 H 1 ,H -1 = 0, 0 ≤ j ≤ m, ∀t ∈ R, where ϕ, ψ H 1 ,H -1 = R d ϕ ψ for Schwartz functions, with initial condition u m (t) = u 0 , which is equivalent to g m,k (0) = 0 for all 0 ≤ k ≤ m. By substitution, (4.3) is equivalent to (4.4) i ġm,j -∇w j , ∇u 0 - m k=0 ḡm,k (t) ∇w j , ∇w k = λ w j , f u 0 + m k=0 g m,k (t)w k ,
where f (x) = x ln |x| 2 . First, we study the last term.

Lemma 4.2. The map ϕ → w j , f (ϕ) is well defined and continuous from u 0 +H 1 into C for any j ∈ N. More precisely, it is C 0,ε (u 0 + H 1 , C) for all ε ∈ (0, 1).

Proof. By Corollary 2.7 along with Lemma 2.9, we know that the right-hand side is well defined. Moreover, at fixed j, taking ϕ, ψ ∈ u 0 + H 1 , we get

| w j , f (ϕ) -w j , f (ψ) | = | w j , f (ϕ) -f (ψ) | ≤ R d |w j ||f (ϕ) -f (ψ)| dx ≤ C R d |w j | |ϕ| ε |ln |ϕ|| + |ψ| ε |ln |ψ|| |ϕ -ψ| 1-ε dx + 2 R d |w j ||ϕ -ψ| dx,
for any ε ∈ (0, 1), where we have used Lemma 2.2. The second term of the last inequality is obviously Lipschitzian with respect to the L 2 -norm of the difference.

As for the first term, by Hölder inequality with exponents 2, 2 ε and 2 1-ε , we control it by

w j L 2 ϕ -ψ 1-ε L 2 ϕ|ln |ϕ|| 1 ε ε L 2 + ψ|ln |ψ|| 1 ε ε L 2 . Now, ϕ|ln |ϕ|| 1 ε L 2 and ψ|ln |ψ|| 1 ε
L 2 are locally bounded thanks to Corollary 2.7 and Lemma 2.9 once again. These estimates yield the conclusion. (And similarly for T m .) Moreover, the g m,k 's are C 1,1-ε (-T m , T m ) for all ε ∈ (0, 1).

Proof. The last term of (4.4) is continuous (and even C 0,ε for all ε ∈ (0, 1)) with respect to the g m,k 's thanks to Lemma 4.2, and so are obviously the other terms. The conclusion comes from Peano Theorem and the fact that (4.4) is an autonomous ODE. Note that we invoke Peano Theorem, and not Cauchy-Lipschitz Theorem, since the nonlinearity f is continuous, but not locally Lipschitzian, due to the singularity of the logarithm at the origin.

We will then show that such a solution is global. First, we prove an intermediate result.

E logGP (u m (t)) = E logGP (u 0 ), and ∇ϕ m (t) L 2 ≤ 2 E logGP (u 0 ) for all t ∈ R.
Proof. For this, we only have to prove that all the g m,k s are (locally) bounded. Come back to (4.3): by multiplying by ġm,j and summing over j ≤ m, we get, in view of (4.2),

∂ t u m , i∂ t u m + ∆u m -λu m ln |u m | 2 = 0.
By taking the real part of this equation, we obtain

d dt E logGP (u m (t)) = 0.
Therefore, E logGP (u m (t)) = E logGP (u 0 ) for all t ∈ (-T m , T m ). In particular, ∇u m (t) L 2 is uniformly bounded, and thus so is

∇ϕ m (t) L 2 : for t ∈ (-T m , T m ), ∇ϕ m (t) L 2 ≤ ∇u 0 L 2 + ∇u m (t) L 2 ≤ ∇u 0 L 2 + E logGP (u m (t)) ≤ 2 E logGP (u 0 ).
We know that ∇ϕ m (t) = m k=0 g m,k (t)∇w k , thus Lemma 4.4 shows that we can apply Lemma 2.3, which gives

max 0≤k≤m |g m,k (t)| ≤ C m ∇ϕ m (t) L 2 ≤ 2C m E logGP (u 0 ) < ∞, ∀t ∈ (-T m , T m ).
This proves that the g m,k 's are actually globally bounded and therefore the solution is global.

L 2 = Im ∇ϕ m , ∇u 0 + λ Im ϕ m , u m ln |u m | 2 ,
since Im ϕ m , ∆ϕ m H 1 ,H -1 = 0. Then, we can estimate the right-hand side :

| ∇ϕ m , ∇u 0 | ≤ ∇ϕ m L 2 ∇u 0 L 2 ≤ 2E logGP (u 0 ),
thanks to Lemma 4.5, and

ϕ m , u m ln |u m | 2 ≤ ϕ m L 2 u m ln |u m | 2 L 2 ϕ m L 2 E logGP (u 0 ) 1/2 + E logGP (u 0 ) ,
by Corollary 2.7. Since ϕ m (0) = 0, Gronwall lemma gives the uniform boundedness of the L 2 norm on every bounded interval I. Define

ξ m (t) := -∆ϕ m -∆u 0 + λu m ln |u m | 2 .
Then ξ m is bounded in L ∞ (I, H -1 ) uniformly in m for every bounded interval I. Moreover, for every t, s ∈ I,

(4.5) ϕ m (t) -ϕ m (s) 2 L 2 = 2 t s ϕ m (τ ) -ϕ m (s), ∂ t ϕ m (τ ) H 1 ,H -1 dτ.
By multiplying again (4.3) at time τ by g m,j (τ ) -g m,j (s) and summing over j, there holds

ϕ m (τ ) -ϕ m (s), ∂ t ϕ m (τ ) H 1 ,H -1 = ϕ m (τ ) -ϕ m (s), ξ m (τ ) H 1 ,H -1 .
Since we know that the H 1 -norms of ϕ m (τ ) and ϕ m (s) are bounded uniformly in m for τ, s ∈ I, we get that

ϕ m (τ ) -ϕ m (s), ξ m (τ ) H 1 ,H -1 is bounded uniformly in m ∈ N,
ϕ ∈ C 0,1 (R, H -1 ) ∩ C 0, 1 2 (R, L 2 ) ∩ L ∞ loc (R, H 1
) such that ϕ m converges to ϕ as m → ∞ in the following sense:

• ϕ m * ⇀ ϕ in L ∞ (I, H 1 )
for its weak- * topology, for every bounded interval I,

• ϕ m (t) ⇀ ϕ(t) in L 2 for its weak topology, for every t ∈ R.

Moreover, λu m ln |u m | 2 converges to i∂ t ϕ + ∆ϕ + ∆u 0 for the weak- * topology of L ∞ (R, L 2 ).

Proof. We know by Lemmas 4.5 and 4.6 that ϕ m is uniformly bounded in L ∞ (I, H 1 ), which is the dual of L 1 (I, H -1 ), for every bounded interval I. Then, this sequence is relatively compact for the weak- * topology of L ∞ (I, H 1 ). By diagonal extraction, there is a subsequence (still denoted by ϕ m ) which converges to some ϕ ∈ L ∞ loc (R, H 1 ) for the weak- * topology of L ∞ (I, H 1 ), for every bounded interval I.

Besides,

u m ln |u m | 2 is also uniformly bounded in L ∞ (R, L 2 ) = L 1 (R, L 2 )
′ thanks to Corollary 2.7 and the conservation of the energy. Therefore, it is also relatively compact for the weak- * topology of

L ∞ (R, L 2 ). Let θ ∈ C ∞ c (R) and j ≤ m: λ θ(τ ) w j , u m ln |u m | 2 dτ = θ(τ ) w j , iϕ m (τ ) dτ -θ(t) ∇w j , ∇ϕ m (τ ) dτ -θ(t) ∇w j , ∇u 0 dτ -→ m→∞ θ(τ ) w j , iϕ(τ ) dτ -θ(t) ∇w j , ∇ϕ(τ ) dτ
-θ(t) ∇w j , ∇u 0 dτ.

Since (w j ) j is a Hilbert basis of L 2 , this proves that λu m ln |u m | 2 also converges to i∂ t ϕ + ∆ϕ + ∆u 0 in D ′ (R, L 2 ). Therefore, the convergence is also in the weak- * topology of L ∞ (R, L 2 ), thus we get i∂ t ϕ + ∆ϕ + ∆u 0 ∈ L ∞ (R, L 2 ).

From this and the fact that ∆ϕ ∈ L ∞ loc (R, H -1 ), we get

∂ t ϕ ∈ L ∞ loc (R, H -1 ). Therefore, ϕ ∈ W 1,∞ loc (R, H -1 ) and thus ϕ ∈ C 0,1 loc (R, H -1 ) ∩ L ∞ loc (R, H 1 ). By inter- polation, this leads to ϕ ∈ C 0, 1 2 loc (R, L 2 ).
As for the convergence in the second item, let t ∈ R. Since ϕ m (t) is bounded in L 2 uniformly in m, the sequence is relatively compact for the weak topology of L 2 . Let χ be the limit of a subsequence (still denoted by ϕ m ). It is then enough to prove that χ = ϕ(t). Let γ ∈ (0, 1) Then,

ϕ(t) -χ 2 L 2 = ϕ(t) -χ, ϕ m (t) -χ + 1 2γ t+γ t-γ ϕ(t) -χ, ϕ(t) -ϕ m (t) dτ = ϕ(t) -χ, ϕ m (t) -χ + 1 2γ t+γ t-γ ϕ(t) -χ, ϕ(t) -ϕ(τ ) + ϕ(τ ) -ϕ m (τ ) + ϕ m (τ ) -ϕ m (t) dτ. Since ϕ, ϕ m ∈ C 0, 1 2 ([t -1, t + 1], L 2 ) uniformly in m, we have for all τ ∈ [t -1, t + 1] | ϕ(t) -χ, ϕ(t) -ϕ(τ ) | ≤ C t |t -τ | 1 2 ϕ(t) -χ L 2 , | ϕ(t) -χ, ϕ m (t) -ϕ m (τ ) | ≤ C t |t -τ | 1 2 ϕ(t) -χ L 2 , hence (4.6) ϕ(t) -χ 2 L 2 ≤ ϕ(t) -χ, ϕ m (t) -χ + C t 2γ ϕ(t) -χ L 2 t+γ t-γ |t -τ | 1 2 dτ + 1 2γ t+γ t-γ ϕ(t) -χ, ϕ(τ ) -ϕ m (τ ) dτ ≤ ϕ(t) -χ, ϕ m (t) -χ + C t γ 1 2 ϕ(t) -χ L 2 + 1 2γ t+γ t-γ ϕ(t) -χ, ϕ(τ ) -ϕ m (τ ) dτ.
Moreover, we know that

• ϕ(t) -χ, ϕ m (t) -χ -→ m→∞ 0 by weak convergence in L 2 of ϕ m (t) to χ, • t+γ t-γ ϕ(t) -χ, ϕ(τ ) -ϕ m (τ ) dτ -→ m→∞ 0 by the weak- * convergence in L ∞ ((t -1, t + 1), L 2
) of ϕ m to ϕ. Therefore, taking the limit m → ∞ in (4.6), we get

ϕ(t) -χ 2 L 2 ≤ C t γ 1 2 ϕ(t) -χ L 2 .
Since this is true for all γ ∈ (0, 1), we get ϕ(t) -χ L 2 = 0, hence the conclusion. 4.4. Equation and initial data for the limit. We are now able to prove that u = u 0 + ϕ is indeed a solution to (logGP).

Lemma 4.8. Let ϕ ∈ C 0,1 (R, H -1 )∩C 0, 1 2 (R, L 2 )∩L ∞ loc (R, H 1
) defined in Lemma 4.7. Then u := u 0 + ϕ is a weak solution to (logGP).

Proof. First, since ϕ m (0) ⇀ ϕ(0) and ϕ m (0) = 0 by construction, we get ϕ(0) = 0 and thus u |t=0 = u 0 .

We also know from Lemma 4.7 that λu m ln |u m | 2 converges to i∂ t ϕ+∆ϕ+∆u 0 for the weak- * topology of L ∞ (R, L 2 ), and thus also for the weak topology of L 2 (I × Ω) for every bounded interval I and every bounded open subset Ω of R d .

We also know that ϕ m is 1 2 -Hölder continuous in time with values in L 2 (R d ) on I, uniformly in m from Lemma 4.6, and thus also on L 2 (Ω). Last, ϕ m (t) ∈ H 1 for all t ∈ I along with a uniform bound in m by Lemmas 4.5 and 4.6. Thus, Arzela-Ascoli Theorem for ϕ m |Ω yields its relative compactness in C(I, L 2 (Ω)), which shows that ϕ m|Ω converges strongly to ϕ |Ω in C(I, L 2 (Ω)), and thus also in L 2 (I × Ω). Up to a further subsequence, ϕ m converges a.e. to ϕ in I × Ω, and so does u m to u. By continuity of z → z ln |z| 2 on C, we obtain the convergence of Therefore, by uniqueness of the limit, we get λu ln |u| 2 = i∂ t ϕ+∆ϕ+∆u 0 , which gives the conclusion. 4.5. Conservation law. We now prove that the energy of the solution that we have just constructed is independent of time. Note that this is necessarily the solution to (logGP), in view of Theorem 3.2. Lemma 4.9. Let ϕ ∈ C 0,1 (R, H -1 )∩C 0, 1 2 (R, L 2 )∩L ∞ loc (R, H 1 ) defined in Lemma 4.7 and u := u 0 + ϕ. Then u satisfies E logGP (u(t)) = E logGP (u 0 ) for all t ∈ R.

u m (t, x) ln |u m (t, x)| 2 to u(t, x) ln |u(t, x)| 2 a.
Proof. In view of the construction of the solution, Fatou's lemma yields

E logGP (u(t)) ≤ E logGP (u 0 ), ∀t ∈ R.
Arguing like in the proof of [13, Theorem 3.3.9], for t 0 ∈ R, we denote by v the solution to (logGP) with v |t=0 = u |t=t0 . We have like above,

E logGP (v(t)) ≤ E logGP (v(0)) = E logGP (u(t 0 )), ∀t ∈ R.
By uniqueness (Theorem 3.2), we infer v(t, x) = u(t + t 0 , x), and, taking t = -t 0 in the previous inequality,

E logGP (u 0 ) ≤ E logGP (u(t 0 )), ∀t 0 ∈ R.
We conclude E logGP (u(t)) = E logGP (u 0 ) for all t ∈ R.

Higher regularity

In this section, we prove the propagation of the Ḣ2 regularity by the flow of (logGP). We would like to proceed like in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] (see also [START_REF] Cazenave | Equations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]) directly on the solution u constructed, i.e. differentiating (logGP) with respect to time and get an L 2 -energy estimate for ∂ t ϕ. A direct formal computation would lead to the local boundedness of this L 2 -norm with the additional argument that ∂ t u(0) ∈ L 2 as soon as ∆u 0 ∈ L 2 . However, this computation cannot directly be made rigorous since u ln |u| 2 cannot be differentiated due to the lack of regularity of the logarithm. To overcome this difficulty, we shall work on the approximate solutions u m . In [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF], the authors worked on the approximate solutions to prove the propagation of the H 2 regularity, since the above mentioned energy estimate is licit in the case of (4.1) (and provides bounds which are uniform in ε). However, we cannot reproduce the same proof here: our approximate solutions do not satisfy an equation with a regularized nonlinearity. Indeed, the equations (4.3) can be put under the form

i∂ t ϕ m + P m ∆u m -λu m ln |u m | 2 = 0,
where P m is the orthogonal projector from L 2 onto X m (we can assume w j ∈ H 2 to make this rigorous).

Thus, the nonlinearity is still not smooth enough to make the computation rigorous. However, by assuming w j ∈ H 2 (for all j), we automatically have ϕ m (t) ∈ H 2 for all t ∈ R and m ∈ N. Even more, we have Lemma 4.3). Those crude bounds are a priori not uniform in m, but we show that we can improve them. Lemma 5.1. Let u m given by Lemma 4.3 and assume w j ∈ H 2 for all j ∈ N. Then, for all bounded interval I, ∂ t u m is bounded in C 0 (I, L 2 ) uniformly in m.

ϕ m ∈ C 1 (R, H 2 ) since the g m,k 's belong to C 1 (R) (see
Proof. From Lemma 4.3, we know that the g m,j 's are C 1,ε for all ε ∈ (0, 1), therefore

∂ t u m ∈ C 0,ε (R, H 2 ). Let τ > 0 and ψ m,τ (t) := ∂ t ϕ m (t + τ ) 2 L 2 -∂ t ϕ m (t) 2 L 2 τ ,
which is well defined for all t ∈ R and any τ > 0. Our goal is to prove a bound on ψ m,τ (t) independent of τ ≤ 1. For this, we first rewrite

ψ m,τ (t) = 1 τ Re ∂ t ϕ m (t + τ ) + ∂ t ϕ m (t), ∂ t ϕ m (t + τ ) -∂ t ϕ m (t) .
Since ∂ t ϕ m (t + τ ) and ∂ t ϕ m (t) are in Vect(w j ) j≤m , we can use (4.3) and get

τ ψ m,τ (t) = -Im ∂ t ϕ m (t + τ ) + ∂ t ϕ m (t), ∆ϕ m (t + τ ) -∆ϕ m (t) + λ Im ∂ t ϕ m (t + τ ) + ∂ t ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t)| 2 .
For the first term, we get

∂ t ϕ m (t), ∆ϕ m (t + τ ) -∆ϕ m (t) = ∂ t ϕ m (t), ∆ϕ m (t + τ ) -∆ϕ m (t) -τ ∂ t ∆ϕ m (t) + τ ∂ t ϕ m (t), ∂ t ∆ϕ m (t) . Since ∂ t ϕ m (t), ∂ t ∆ϕ m (t) = -∂ t ∇ϕ m 2 L 2
, the last term vanishes when we take the imaginary part. On the other hand, we have shown that ∆ϕ m ∈ C 1,ε (R, L 2 ). Hence, for all τ ≤ 1,

∆ϕ m (t + τ ) -∆ϕ m (t) -τ ∂ t ∆ϕ m (t) L 2 ≤ C m,ε,t τ 1+ε .
Therefore, for all t ∈ R, τ, ε ∈ (0, 1),

|Im ∂ t ϕ m (t), ∆ϕ m (t + τ ) -∆ϕ m (t) | ≤ C m,ε,t τ 1+ε ,
and similarly for Im ∂ t ϕ m (t + τ ), ∆ϕ m (t + τ ) -∆ϕ m (t) . Now, we also have

∂ t ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t)| 2 = 1 τ ϕ m (t + τ ) -ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 - 1 τ ϕ m (t + τ ) -ϕ m (t) -τ ∂ t ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 .
For the first term, we can use Lemma 2.1 since ϕ m (t+τ )-ϕ m (t) = u m (t+τ )-u m (t), so that

Im ϕ m (t + τ ) -ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 ≤ 2 u m (t + τ ) -u m (t) 2 L 2 = 2 ϕ m (t + τ ) -ϕ m (t) 2 L 2 .
For the second term, we can use Lemma 2.2 so that

ϕ m (t + τ ) -ϕ m (t) -τ ∂ t ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 ≤ ϕ m (t + τ ) -ϕ m (t) -τ ∂ t ϕ m (t) L 2 u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 L 2
The first factor can be estimated like previously with the fact that ϕ m ∈ C 1,ε (R; L 2 ):

ϕ m (t + τ ) -ϕ m (t) -τ ∂ t ∂ t ϕ m (t) L 2 ≤ C m,t,ε τ 1+ε .
As for the other factor, we use Lemma 2.2 and then Corollary 2.7 to obtain

u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 L 2 |u m (t + τ )| ε 2 |ln |u m (t + τ )|| + |u m (t)| ε 2 |ln |u m (t)|| |u m (t + τ ) -u m (t)| 1-ε 2 L 2 + u m (t + τ ) -u m (t) L 2 |u m (t + τ )| ε 2 |ln |u m (t + τ )|| + |u m (t)| ε 2 |ln |u m (t)|| L 4 ε |u m (t + τ ) -u m (t)| 1-ε L 2 1-ε 2 + u m (t + τ ) -u m (t) L 2 |u m (t + τ )||ln |u m (t + τ )|| 2 ε ε L 2 + |u m (t)||ln |u m (t)|| 2 ε ε L 2 u m (t + τ ) -u m (t) 1-ε 2 L 2 + u m (t + τ ) -u m (t) L 2 E logGP (u 0 ) ε/2 + E logGP (u 0 ) (ε+ε 2 )/2 u m (t + τ ) -u m (t) 1-ε 2 L 2 + u m (t + τ ) -u m (t) L 2 . Therefore, we obtain ϕ m (t + τ ) -ϕ m (t) -τ ∂ t ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t + τ )| 2 ≤ C m,t,ε τ 2+ ε 2 E logGP (u 0 ) ε/2 + E logGP (u 0 ) ε δ τ ϕ m 1-ε 2 L 2 + τ 2+ε δ τ ϕ m L 2 , where δ τ ϕ m := 1 τ (ϕ m (t + τ ) -ϕ m (t)
). Thus, we get

Im ∂ t ϕ m (t), u m (t + τ ) ln |u m (t + τ )| 2 -u m (t) ln |u m (t)| 2 ≤ (2τ + C m,t,ε τ 1+ε ) δ τ ϕ m 2 L 2 + C m,t,ε τ 1+ ε 2 E logGP (u 0 ) ε/2 + E logGP (u 0 ) ε δ τ ϕ m 1-ε 2 L 2 .

Similar computations can be done for Im

∂ t ϕ m (t + τ ), u m (t + τ ) ln |u m (t + τ )| 2 - u m (t) ln |u m (t)| 2 , so we get (5.1) |ψ m,τ (t)| ≤ C m,t,ε τ ε + (4λ + C m,t,ε τ ε ) δ τ ϕ m 2 L 2 + C m,t,ε τ 1+ ε 2 E logGP (u 0 ) ε/2 + E logGP (u 0 ) ε δ τ ϕ m 1-ε 2 L 2 . At m fixed, since ϕ m ∈ C 1,ε (R; L 2 ), we know that as τ → 0, δ τ ϕ m (t) converges strongly to ∂ t ϕ m (t) in L 2 for any t ∈ R, but also that, for t ∈ R fixed, δ τ ϕ m L 2 is uniformly bounded in τ ≤ 1. The previous estimate shows that ψ m,τ (t) is uniformly bounded in τ ≤ 1 at m ∈ N and t ∈ R fixed, which means that t → ∂ t ϕ m (t) 2 L 2 ∈ W 1,∞ loc (R, R). The limit τ → 0 in (5.1) then yields d dt ∂ t ϕ m (t) 2 L 2 ≤ 4λ ∂ t ϕ m (t) 2 L 2 .
On the other hand, we also have i∂ t ϕ m (0) = P m λu 0 ln |u 0 | 2 -∆u 0 , and since P m is an orthogonal projector, we get

∂ t ϕ m (0) L 2 ≤ λ u 0 ln |u 0 | 2 L 2 + ∆u 0 L 2 ≤ C ε λ(E logGP (u 0 ) ε/2 + E logGP (u 0 ) ε ) + ∆u 0 L 2 .
Thanks to Gronwall Lemma, we get a bound which is independent of m: for all m ∈ N and t ∈ R,

∂ t ϕ m (t) 2 L 2 ≤ C ε λ(E logGP (u 0 ) ε/2 + E logGP (u 0 ) ε ) + ∆u 0 L 2 2 e 4λt .
Corollary 5.2. Let u given by Lemma 4.7. Then, for all bounded interval I, ∂ t u is bounded in L ∞ (I; L 2 ), and so is ∆u.

Proof. We want to take the limit m → ∞ in Lemma 5.1. For this, we use the fact that ϕ m * ⇀ ϕ in L ∞ (I; H 1 ) for its weak- * topology for any bounded interval I by Lemma 4.7. Therefore,

∂ t ϕ m ⇀ ∂ t ϕ in D ′ (I × R d ). As the L ∞ (I; L 2 ) norm of ∂ t ϕ m is uniformly bounded in m and since C c (I × R d ) is dense in L ∞ (I; L 2 ) = L 1 (I; L 2 ) ′ , this limit shows that ∂ t ϕ = ∂ t u belongs to L ∞ (I; L 2 ).
To conclude, we show that u ln |u| 2 ∈ L ∞ (I; L 2 ): in view of Corollary 2.7, for t ∈ I,

u(t) ln |u(t)| 2 L 2 E logGP (u 0 ) 1/2 + E logGP (u 0 ).
Hence, using (logGP), ∆u ∈ L ∞ (I; L 2 ).

On stationary and traveling waves

In this section, we prove some rather general results regarding solitary and traveling waves. First, plugging (1.2) into (logGP), we get the equation for φ:

(6.1)
-ωφ -ic • ∇φ + ∆φ = λφ ln |φ| 2 .

6.1. The only possible value for ω.

Proof of Theorem 1.4. u satisfies (logGP) with initial data φ and u ∈ L ∞ (R; E logGP ). From Lemma 3.1, we thus know that u -φ ∈ C 0 (R; L 2 ). On the other hand, we know that, for any t ∈ R, x ∈ R d and any f smooth enough,

f (x -tc) -f (x) = -t 1 0 ∇f (x -τ c) • c dτ, which leads to f (x -tc) -f (x) L 2 ≤ |t| 1 0 ∇f (x -τ c) • c L 2 dτ ≤ |t| 1 0 ∇f (x -τ c) L 2 |c| dτ ≤ |t||c| 1 0 ∇f L 2 dτ ≤ |t||c| ∇f L 2 .
Since ∇φ ∈ L 2 , we can also apply this inequality to φ, which proves that for all t ∈ R, φ(.

-ct) -φ ∈ L 2 .
The above two claims imply that u -φ(• -ct) ∈ L 2 for all t ∈ R. In view of (1.2), this means that (e iωt -1)φ ∈ L 2 for all t ∈ R. Suppose that ω = 0: considering t = π ω , we get -2φ ∈ L 2 , which is in contradiction with the fact that |φ| -1 ∈ L 2 from Lemma 2.6. Therefore, ω = 0. 6.2. Regularity and limits of a traveling wave in dimension d = 1. In this section, we prove a result about the regularity and limits at infinity of traveling waves of the form (1.2) solution to (logGP) in dimension d = 1. In view of Theorem 1.4, such traveling waves satisfy (6.2)

-icφ ′ + φ ′′ = λφ ln |φ| 2 .
Lemma 6.1. A traveling wave φ solution to (6.2) satisfies

• φ ∈ C 2 b and lim |x|→∞ |φ(x)| = 1, • φ ′ ∈ H 1 and lim |x|→∞ φ ′ (x) = 0.
Proof. Since φ ∈ E logGP , we have φ ∈ H 1 loc and in particular φ ∈ C 0, 1 2 (R) by Sobolev embedding in dimension d = 1. Actually, φ satisfies (6.2) with ω = 0 and φ ln |φ| 2 ∈ L 2 ∩ C 0 from Corollary 2.7. Moreover, φ ′ ∈ L 2 since φ ∈ E logGP . Therefore, we get φ ′′ ∈ L 2 , which leads to φ ′ ∈ H 1 . By Sobolev embedding, this leads in particular to φ ′ ∈ C 0 b and, coming back to (6.2) once again, we get φ ′′ ∈ C 0 . This leads to φ ∈ H 2 loc ∩ C 2 . φ ′ ∈ H 1 also gives lim |x|→∞ φ ′ (x) = 0. Furthermore, with Lemma 2.6, |φ| -1 ∈ H 1 , which means that lim |x|→∞ |φ(x)| = 1 and that φ is bounded, which shows that φ ′′ is also bounded by (6.2).

Solitary waves in the one-dimensional case

In this section, we prove Theorem 1.6 in the case c = 0. By Theorem 1.4, we know that ω = 0. Thus, the equation we address in this section is:

(7.1) φ ′′ = λφ ln |φ| 2 .
7.1. Properties of a solitary wave in dimension d = 1. We first assume that there exists a non-constant traveling wave u of the form (1.2) with c = 0, solution to (logGP) in the energy space, and we gather some of its properties. First, we show that φ satisfies an interesting energy equality.

Lemma 7.1. If φ ∈ E logGP , then (7.2) |φ ′ | 2 = λ |φ| 2 ln |φ| 2 -|φ| 2 + 1 .
Proof. Multiplying (7.1) by 2 φ′ (x), and taking the real value, we get

d dx |φ ′ | 2 = λ d dx |φ| 2 ln |φ| 2 -|φ| 2 + 1 .
By the facts that φ ∈ E logGP and

|φ| 2 ln |φ| 2 -|φ| 2 + 1 ≥ 0, we get |φ ′ | 2 ∈ L 1 and |φ| 2 ln |φ| 2 -|φ| 2 +1 ∈ L 1
, so we can integrate the above equation, and no additional constant of integration appears.

Corollary 7.2. If φ ∈ E logGP is real-valued on some unbounded interval I, then either φ ′ never vanishes and does not change sign on I, or φ is constant on I (equals to ±1).

Proof. If there is some y ∈ I such that φ ′ (y) = 0, then Lemma 7.1 shows that φ(y) = ±1 as x → x ln x -x + 1 vanishes only for x = 1 on R + , and thus the uniqueness part of the Cauchy theorem applied to (7.1) gives (as long as it does not vanish) φ ≡ ±1 on I. On the other hand, if there is no such y, φ ′ cannot change sign on I since it is continuous and never vanishes.

Then, we show that φ can be taken real-valued up to a gauge on some neighborhood of +∞. Lemma 7.3. There exists x -∈ R ∪ {-∞} and θ ∈ [0, 2π) such that φ(x)e -iθ is real-valued and positive for all x ≥ x -, and, either x -= -∞, or lim x→x -φ(x) = 0.

Proof. We can write (7.1) as a first order differential system,

(7.3) φ ψ ′ = ψ λφ ln |φ| 2 =: F φ ψ .
F is a continuous function on C 2 and is of class C 1 on (C\{0})×C. From Lemma 6.1, there exists

x 0 ∈ R such that 3 2 ≥ |φ(x)| ≥ 1 2 and |φ ′ (x)| ≤ 1
2 for all x ≥ x 0 . Therefore, by Cauchy-Lipschitz Theorem, there exists a maximal interval I + = (x -, x + ) such that x 0 ∈ I + and (φ, φ ′ ) is the unique solution of (7.3) on I + with initial data φ(x 0 ) and ψ(x

0 ) = φ ′ (x 0 ), with values in C 2 \ ({0} × C). Moreover, if |x + | < ∞, then either lim x→x + φ(x) = 0 or lim x→x + (|φ(x)| + |ψ(x)|) = ∞
, and similarly for the limit x → x -.

However, since we already know that 3 2 ≥ |φ(x)| ≥ 1 2 and |φ ′ (x)| ≤ 1 2 for all x ≥ x 0 , we can already conclude that x + = ∞. On the other hand, since φ ∈ C 2 (R) by Lemma 6.1, we know that lim

x→x -(|φ(x)| + |ψ(x)|) = ∞ cannot be true if x -> -∞.
On I + , φ does not vanish. Hence, we can use a polar decomposition: φ = ρe iθ , with ρ and θ real-valued and defined as

(7.4) ρ = |φ|, θ ′ = Im(φ ′ φ) ρ 2 , θ(x 0 ) ∈ [0, 2π) such that e iθ(x0) = φ(x 0 ) ρ(x 0 ) .
The regularity of φ and the fact that it does not vanish on I + ensure that ρ and θ are well defined and C 2 on I + . Thus, we can compute φ ′′ in terms of ρ, θ, and their derivatives:

φ ′′ = ρ ′′ -ρ(θ ′ ) 2 + 2iρ ′ θ ′ e iθ .
Substituting into (7.1), we get (after simplification by e iθ ) (7.5)

ρ ′′ -ρ(θ ′ ) 2 + 2iρ ′ θ ′ + iρθ ′′ = λρ ln ρ 2 .
By taking the imaginary part, we get ρθ ′′ + 2ρ ′ θ ′ = 0. Since ρ does not vanish on I + , we get for all x ∈ I + : 2 , where c 0 is a constant. Then, we can compute for all

θ ′ (x) = c 0 ρ(x)
x ∈ I + φ ′ (x) = (ρ ′ (x) + iρ(x)θ ′ (x)) e iθ(x) = ρ ′ (x) + i c 0 ρ(x)
e iθ(x) , which leads to c0 ρ ∈ L 2 (I + ) and at the same time lim +∞ c0 ρ = c 0 since we know that lim x→+∞ ρ(x) = 1. Combining these two arguments yields c 0 = 0.

Hence, θ is constant on I + , which gives the conclusion.

From now on, without loss of generality, we can assume that θ = 0 in Lemma 7.3, up to changing φ to e -iθ φ.

Lemma 7.4. Let x -∈ R ∪ {-∞} from Lemma 7.3. Then x -> -∞. Moreover, φ(x) ∈ (0, 1) and φ ′ (x) > 0 for all x > x -. Last, φ(x -) = 0 and φ ′ (x -) = 1.

Proof. From Corollary 7.2, we know that φ ′ never vanishes and does not change sign on (x -, ∞). By contradiction, if φ(x 2 ) > 1 for some x 2 > x -, then

• Either φ ′ > 0 on (x -, ∞), and thus φ(x) ≥ φ(x 2 ) for all x ≥ x 2 , which is in contradiction with the fact that lim ∞ |φ| = 1. • Or φ ′ < 0 on (x -, ∞), which means that φ(x) ≥ φ(x 2 ) > 1 for all x -< x ≤ x 2 . Therefore, x -= -∞, but then we have the same contradiction as in the previous case, for -∞. Therefore, φ(x) ∈ (0, 1) for all x > x -, and this leads to 2 is decreasing, which shows that, taking x 0 ∈ (x -, ∞), we have φ ′ (x) 2 ≥ φ ′ (x 0 ) 2 for all x -< x ≤ x 0 , and thus φ ′ (x) ≥ φ ′ (x 0 ). By integration, we get for all x ∈ (x -, x 0 ],

φ ′ > 0 on (x -, ∞). Now, remark that f (x) = x 2 ln x 2 -x 2 + 1 is decreasing on [0, 1]. Thus, λf (φ) = (φ ′ )
φ(x) ≤ φ(x 0 ) + (x -x 0 )φ ′ (x 0 ).
Since the right hand side goes to -∞ as x → -∞, and since φ > 0 on (x -, ∞), this yields x -> -∞. Thus, we are in the second case of Lemma 7.3, and since φ is continuous, we get φ(x -) = 0. Last, (7.2) holds for x = x -, and we know by continuity that φ ′ (x -) ≥ 0, which gives φ ′ (x -) = 1.

Lemma 7.5. Let x -∈ R from Lemma 7.3. Then, for all x < x -, φ(x) is realvalued, negative and increasing.

Proof. From Lemma 7.4, we know that φ(x -) = 0 and φ ′ (x -) = 1. Therefore, there exists δ > 0 such that φ = 0 on [x --δ, x -). Define

x - 0 := inf y < x -| ∀x ∈ (y, x -), φ(x) = 0 < x --δ. On I 2 := (x - 0 ,
x -), we can use a polar factorization in the same way as in (7.4): φ = ρe iθ , where ρ and θ also satisfy (7.5) on I 2 . Taking again the imaginary part and integrating the ODE, we have, like before,

θ ′ (x) = c 0 ρ(x) 2 , ∀x ∈ I 2 .
We still have then

φ ′ (x) = ρ ′ (x) + i c 0 ρ(x) e iθ(x) ,
and once again we must have c0 ρ(x) ∈ L 2 (I 2 ) since φ ′ ∈ L 2 . However, we know that φ(x -) = 0 and φ ′ (x -) = 1, which means that φ(x) ∼ x -x -as x → x -. In terms of ρ, this implies ρ(x) ∼ |x -x -| as x → x -, and c0 ρ(x) ∈ L 2 (I 2 ) if and only if c 0 = 0. Once again, θ is constant on I 2 , and therefore so is φ ρ . With the previous asymptotics, we know that it tends to -1 at x -. Thus, we get

φ = -ρ.
This shows that φ is real-valued and negative on I 2 . Then, we can apply Lemma 7.1 on (x - 0 , ∞), which shows that φ ′ does not vanish on this interval, and thus φ ′ does not change sign once again, which proves that φ ′ > 0 on I 2 . Therefore, φ(x) ≤ φ(x --δ) < 0 for all x ∈ (x - 0 , x --δ), which proves that x - 0 = -∞. It only remains to analyze the limit of φ at -∞. Lemma 7.6. We have lim

x→-∞ φ(x) = -1.
Proof. We have seen that φ is increasing, thus it has a limit at -∞. From Lemma 7.5, we know it is negative. Moreover, |φ| -1 ∈ L 2 , which means that φ + 1 ∈ L 2 ((-∞, 0)). The conclusion easily follows.

The conclusion of all the previous lemmas is the following:

Corollary 7.7. If u is a non-constant traveling wave solution to (logGP) of the form (1.2) with c = 0, then there exists θ ∈ R such that e -iθ φ is a real-valued, increasing, C 2 function with values in (-1, 1) which vanishes at a unique point x 0 . Moreover, φ 0 (x) = e -iθ φ(x + x 0 ) also satisfies (logGP),

lim x→±∞ φ 0 (x) = ±1, and
(7.6) (φ ′ 0 ) 2 = λ φ 2 0 ln φ 2 0 -φ 2 0 + 1 on R.
7.2. Analysis of the new ODE and proof of Theorem 1.6. From Corollary 7.7, φ 0 vanishes at x = 0, satisfies (7.6) and is strictly increasing, i.e. φ ′ 0 > 0. Thus, φ 0 satisfies the ODE (7.7)

φ ′ 0 = √ λ φ 2 0 ln φ 2 0 -φ 2 0 + 1 on R.
The uniqueness of this function is the topic of the following lemma. Moreover, φ 0 ∈ E logGP .

Proof. We already know that f (x) = x ln x -x + 1 ≥ 0 for all x ≥ 0, and f vanishes only at x = 1. Moreover, by simple computations, we check that g(x) = f (x 2 ) is C 1 on R, and C 2 on R \ {0}. We also compute f ′ (1) = 0 and f ′′ (1) = 1, hence

g(x) ∼ x→±1 1 2 x 2 -1 2 .
From these facts, we deduce that

h = √ g is a C 0 (R)∩C 1 (R\{-1, 1}) function such that h(x) ∼ x→±1 x 2 -1 / √ 2.
Therefore, it is C 0,1 locally on R, and we may invoke Cauchy-Lipschitz Theorem: there exists a unique φ 0 satisfying φ ′ 0 = √ λh(φ 0 ) with the initial condition φ 0 (0) = 0, on a maximal interval I ∋ 0 of existence.

Then, φ 0 cannot reach 1 or -1, because the constant function 1 and -1 are both solutions to this ODE, and it would contradict the uniqueness in the Cauchy-Lipschitz Theorem. Since φ 0 is continuous, we infer that φ 0 (x) ∈ (-1, 1) for all x ∈ R. Moreover, we can easily prove that there exists c 1 > 0 such that

h(y) ≥ c 1 (y + 1), ∀y ∈ [-1, 0], h(y) ≥ c 1 (1 -y), ∀y ∈ [0, 1].
Since we know that φ 0 (x) ∈ (-1, 0] for all x ≤ 0, we can estimate

φ ′ 0 (x) ≥ c 1 √ λ(φ 0 (x) + 1), ∀x ≤ 0.
By integrating backward, we get for all x ≤ 0,

-1 < φ 0 (x) ≤ -1 + e √ λc1x , 0 < φ ′ 0 (x) ≤ c 1 √ λe √ λc1x .
Similarly, we also have for all x ≥ 0

1 -e - √ λc1x ≤ φ 0 (x) < -1, 0 < φ ′ 0 (x) ≤ c 1 √ λe √ λc1x .
Those estimates prove the expected limits at ±∞, and also show that φ ′ 0 ∈ L 2 together with |φ 0 | -1 ∈ L 2 . We conclude that φ 0 ∈ E logGP by Lemma 2.6.

The previous result obviously proves that we have a set of solitary waves for (logGP).

Corollary 7.9. For any θ ∈ R and any x 0 ∈ R, e iθ φ 0 (. -x 0 ) ∈ E logGP is a solitary (stationary) wave of (logGP).

It remains to show that they are the only possible solitary waves in order to prove Theorem 1.6 in the case c = 0.

Proof of Theorem 1.6: case c = 0. Let φ ∈ E logGP and u be a traveling wave of the form (1.2) with c = 0. From Corollary 7.7, we can find θ ∈ R and x 0 ∈ R such that e -iθ φ(. + x 0 ) is a real-valued, increasing, C 2 function with values in (-1, 1), which vanishes at 0, and satisfies (7.6), i.e. (7.7) by positivity of its derivative. It is therefore φ 0 , defined in Lemma 7.8 by uniqueness given by the same lemma: e -iθ φ(. + x 0 ) = φ 0 , i.e. φ = e iθ φ 0 (. -x 0 ).

8.

Traveling waves in the one-dimensional case 8.1. Admissible velocities. The goal of this section is to prove Theorem 1.5, and characterize velocities of nontrivial (that is, nonconstant) traveling waves. We assume that there exists a traveling wave of the form (1.2) solution to (logGP), i.e. φ ∈ E logGP solution to (6.2).

Lemma 8.1. The function η = 1 -|φ| 2 satisfies, on R,

1 2 (η ′ ) 2 = h c (η),
where

h c (y) := λ (1 -y) 2 ln(1 -y) 2 -(1 -y) 2 + 1 - 2λ + c 2 2 y 2 .
Proof. We follow the same lines as in [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevski equation[END_REF]. Writing φ = φ 1 + iφ 2 , (6.2) becomes (8.1)

φ ′′ 1 + cφ ′ 2 = λφ 1 ln φ 2 1 + φ 2 2 , φ ′′ 2 -cφ ′ 1 = λφ 2 ln φ 2 1 + φ 2 2 .
We multiply the first equation by φ 2 and the second one by φ 1 , and we subtract in order to get:

(φ ′ 1 φ 2 -φ 1 φ ′ 2 ) ′ = -c(φ 2 φ ′ 2 + φ 1 φ ′ 1 ) = c 2 η ′ , η := 1 -|φ| 2 .
Moreover, we know that φ is bounded by Lemma 6.1, thus φ ′ 1 φ 2 , φ 1 φ ′ 2 ∈ L 2 , and η ∈ L 2 by using Lemma 2.6. Thus, integrating the above identity yields

(8.2) φ ′ 1 φ 2 -φ 1 φ ′ 2 = c 2 η.
Now, we multiply the first equation of (8.1) by φ ′ 1 , the second by φ ′ 2 and we sum in order to get:

φ ′ 1 φ ′′ 1 + φ ′ 2 φ ′′ 2 = λ (φ 1 φ ′ 1 + φ 2 φ ′ 2 ) ln φ 2 1 + φ 2 2
, which can be written as

|φ ′ | 2 ′ = λ |φ| 2 ln |φ| 2 -|φ| 2 + 1 ′ .
We know that both |φ ′ | 2 and |φ| 2 ln |φ| 2 -|φ| 2 + 1 (which is positive and whose integral over R is bounded by E logGP (φ)) are in L 1 , hence:

|φ ′ | 2 = λ |φ| 2 ln |φ| 2 -|φ| 2 + 1 on R.
We deduce then

η ′′ = -2|φ ′ | 2 -2(φ 1 φ ′′ 1 + φ 2 φ ′′ 2 ) = -2|φ ′ | 2 -2φ 1 -cφ ′ 2 + λφ 1 ln |φ| 2 -2φ 2 cφ ′ 1 + λφ 2 ln |φ| 2 = -4λ(1 -η) ln(1 -η) -2λη -c 2 η = h ′ c (η). Multiplying by η ′ , we obtain 1 2 (η ′ ) 2 ′ = (h c (η)) ′ .
Since lim ±∞ η ′ = lim ±∞ η = 0 from Lemma 6.1, we conclude by integrating.

Remark 8.2. This lemma may be extended to the case of any non-linearity F (u) with F a continuous function on C of the form F (u) = u f (|u| 2 ) with f (1) = 0.

Corollary 8.3. If c = 0, then φ never vanishes.

Proof. It comes from the fact that h

c (1) = -c 2 2 < 0 if c = 0, while h c (η) = 1 2 (η ′ ) 2 ≥ 0. Therefore, η(x) = 1, that is φ(x) = 0, for all x ∈ R. Corollary 8.4. If c 2 ≥ 2λ, then η ≡ 0.
Proof. The limiting case c 2 = 2λ is treated like in [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevski equation[END_REF]: if η is not identically zero, then in view of Lemma 6.1, by translation invariance, we may assume

|η(0)| = max {|η(x)|, x ∈ R} > 0, η ′ (0) = 0.
In view of Lemma 8.1, h c (η(0)) = 0. On the other hand, Corollary 8.3 implies that 1 -η(x) > 0 for all x ∈ R, and direct computation yields

h ′′ c (y) = 4λ ln(1 -y), 0 ≤ y < 1. Therefore, h ′′ c (y) is positive for y < 0, negative for 0 < y < 1. As h ′ c (0) = 0, h ′ c
is negative on (-∞, 0) ∪ (0, 1): h c is decreasing on (-∞, 1). Since h c (0) = 0, the only zero of h c on [0, 1) is the origin, and so η(0) = 0, hence a contradiction: η is identically zero.

In particular, g c is increasing on (0, y 1 ), from -∞ to a positive value, and there exists a unique y 0 ∈ (0, y 1 ) such that g c (y 0 ) = 0, and the lemma follows easily.

Lemma 8.7. If c is such that 0 < c 2 < 2λ, then ρ = |φ| satisfies (8.3) ρ ′′ = f c (ρ), (8.4) (ρ ′ ) 2 = g c (ρ).
Moreover, there exists x 0 ∈ R such that ρ(x 0 ) = y 0 , where y 0 is defined in Lemma 8.6, and θ ∈ C 2 , defined so that φ = ρe iθ , satisfies

(8.5) θ ′ = c 2 1 - 1 ρ 2 .
Proof. By Corollary 8.3, φ never vanishes. It is also a C 2 b function. Therefore, we can define ρ and θ as in (7.4) so that φ = ρe iθ . They satisfy

(8.6) -icρ ′ + cρθ ′ + ρ ′′ -ρ(θ ′ ) 2 + 2iρ ′ θ ′ + iρθ ′′ = λρ ln ρ 2 .
By taking the imaginary part, we get

(8.7) -cρ ′ + ρθ ′′ + 2ρ ′ θ ′ = 0.
As ρ never vanishes, the solution of this ODE on θ ′ takes the form

θ ′ (x) = c 2 + c 0 ρ(x) 2 ,
where c 0 is a real constant. Then, there holds

φ ′ = ρ ′ + i c 2 ρ + c 0 ρ e iθ ∈ L 2 .
On the other hand, we already know that lim ±∞ ρ = 1, thus we must have c 0 = -c 2 , so that

θ ′ (x) = c 2 1 - 1 ρ(x) 2 . Now, (8.6) reads ρ ′′ = - c 2 2 ρ - 1 ρ + c 2 4 ρ 1 - 1 ρ 2 2 + λρ ln ρ 2 = - c 2 2 ρ - 1 ρ + c 2 4 ρ - 2 ρ + 1 ρ 3 + λρ ln ρ 2 = c 2 4 1 ρ 3 -ρ + λρ ln ρ 2 .
After multiplication by ρ ′ and integration, we find:

(ρ ′ ) 2 = - c 2 4 (1 -ρ 2 ) 2 ρ 2 + λ(ρ 2 ln ρ 2 -ρ 2 + 1).
We also know that if ρ ≡ 1, then θ ′ ≡ 0 and thus φ is constant. On the contrary, if φ is not constant, then ρ is not either, and thus has an extremum since lim ±∞ ρ = 1. Let x 0 be extremal. Then ρ ′ (x 0 ) = 0, and therefore 0 = g c (ρ(x 0 )), with ρ(x 0 ) = 1. Lemma 8.6 implies ρ(x 0 ) = y 0 .

Lemma 8.8. There exists a unique ρ c ∈ C 2 satisfying (8.3), ρ c (0) = y 0 and ρ ′ c (0) = 0. Moreover, ρ c ≥ y 0 > 0, lim ±∞ ρ c = 1 and ρ c -1 ∈ H 1 .

Proof. Since f c is C ∞ on (0, ∞), we can apply Cauchy-Lipschitz Theorem to get a local solution. By multiplying (8.3) by ρ ′ c and integrating, this solution satisfies also (8.4). Lemma 8.6 then yields ρ c ≥ y 0 , which proves that this solution is global.

Then, by (8.3) and Lemma 8.6 again, we get ρ ′′ c (0) > 0. Hence there exists ε > 0 such that ρ ′ c < 0 on (-ε, 0) and ρ ′ c > 0 on (0, ε), and ρ c > y 0 on (-ε, ε) \ {0}. Let

x + := sup {x > 0 | ρ ′ c (y) > 0 for all y ∈ (0, x)} , x -:= inf {x < 0 | ρ ′ c (
y) < 0 for all y ∈ (x, 0)} . From (8.4) and Lemma 8.6 along with the fact that ρ c and ρ ′ c are continuous, we know that either x + = +∞ or ρ c (x + ) = 1, and similarly either x -= -∞ or ρ c (x -) = 1. However, if ρ(x 1 ) = 1 at some point x 1 ∈ R, then (8.4) gives ρ ′ c (x 1 ) = 0 and the uniqueness in Cauchy-Lipschitz Theorem for (8.3) leads to ρ c ≡ 1, hence a contradiction. Therefore, x + = +∞ and x -= -∞, which implies that ρ c is decreasing on (-∞, 0), increasing on (0, ∞) and ρ c (x) ∈ [y 0 , 1) for all x ∈ R. Thus, it has limits ℓ ± > y 0 at ±∞ and must also satisfy lim ±∞ ρ ′ c = 0 in view of (8.4). Therefore, the limit must satisfy g c (ℓ ± ) = 0, which leads to ℓ ± = 1.

Last, since ρ ′ c > 0 on (0, ∞), (8.4) reads on this interval ρ ′ c = g c (ρ c ). By Lemma 8.6 and since ρ c (x) ≥ y 1 for x large enough, we get

ρ ′ c ≥ C c (1 -ρ c
). Thus ρ c converges exponentially fast to 1 at +∞ by Gronwall lemma, and therefore ρ ′ c converges exponentially fast to 0. The same holds as x → -∞, hence the conclusion ρ c -1 ∈ H 1 . Lemma 8.9. For c such that 0 < c 2 < 2λ and θ 0 ∈ R, define Θ c by

Θ ′ c = c 2 1 - 1 ρ 2 c , Θ c (0) = θ 0 ,
where ρ c is given by Lemma 8.8. Then φ c := ρ c e iΘc ∈ E logGP and u defined by (1.2) is a traveling wave for (logGP).

Proof. Direct computations show that Θ c satisfies (8.7), and thus (ρ c , Θ c ) satisfies (8.6), which is equivalent to φ c satisfying (6.2) or u satisfying (logGP). Therefore, we only have to prove that φ c ∈ E logGP . By Lemma 8.8, we already know that

|φ c | -1 = ρ c -1 ∈ H 1 . Moreover, φ ′ c = (ρ ′ c + iρ c Θ ′ c ) e iΘc = ρ ′ c + i c 2 ρ 2 c -1 ρ c e iΘc .
By Lemma 8.8 again, ρ ′ c ∈ L 2 and ρ c is bounded, and bounded away from 0, so that

ρ 2 c -1 ρc ∈ L 2 .
We conclude φ c ∈ E logGP thanks to Lemma 2.6.

End of the proof of Theorem 1.6. Let φ ∈ E logGP and u a traveling wave of the form (1.2). By applying Lemma 8.7, we get x 0 ∈ R such that ρ(x 0 ) = y 0 where ρ = |φ|, which yields ρ ′ (x 0 ) = 0 through (8.4). Since ρ(. + x 0 ) also satisfies (8.3), this function satisfies the assumptions of Lemma 8.8. Thus ρ(. + x 0 ) = ρ c , and since θ satisfies (8.5), φ is of the form φ c defined in Lemma 8.9 for some θ 0 .

Some open questions

Multidimensional solitary and traveling waves. We have classified solitary and traveling waves in the one-dimensional setting, using ODE techniques. The picture is of course rather different for d ≥ 2, see typically the case of traveling waves for the Gross-Pitaevskii equation (1.1), in [START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF][START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF], where the role of hydrodynamical formulation (via Madelung transform) is decisive. See also [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevski equation[END_REF][START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF] and references therein.

We note that contrary to the case with vanishing boundary condition at infinity, one cannot easily construct multidimensional waves from one-dimensional ones by using the tensorization property evoked in the introduction. Recall that if u 1 (t, x 1 ), . . . , u d (t, x d ) are solutions to the one-dimensional logarithmic Schrödinger equation, then u(t, x) := u 1 (t, x 1 )× • • •× u d (t, x d ) solves the d-dimensional logarithmic Schrödinger equation. In the case of vanishing boundary condition at infinity, u j (t, •) ∈ L 2 (R), hence u(t, •) ∈ L 2 (R d ); see e.g. [START_REF] Carles | Logarithmic Schrödinger equation with quadratic potential[END_REF] for some applications. On the other hand, if u j (t, •) ∈ E logGP = E GP (the sets are the same in the one-dimensional case), then even for d = 2, (x 1 , x 2 ) → u 1 (t, x 1 )u 2 (t, x 2 ) does not belong to the energy space E logGP = E GP (case of R 2 ), as the spatial derivatives are not in L 2 (R 2 ).

Orbital stability. In the case of the one-dimensional Gross-Pitaevskii equation, the orbital stability of traveling waves was proven in [START_REF] Gérard | Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation[END_REF] by using the fact that this equation is integrable, in the sense that it admits a Lax pair, and solutions are analyzed thanks to Zakharov-Shabat's inverse scattering method. It seems unlikely that even for d = 1, (logGP) is completely integrable, so the stability of traveling waves will require a different approach.

In [START_REF] Lin | Stability and instability of traveling solitonic bubbles[END_REF][START_REF] Béthuel | Orbital stability of the black soliton for the Gross-Pitaevskii equation[END_REF], the orbital stability of the black soliton is studied without using the integrable structure, but variational arguments. In [START_REF] Béthuel | Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation[END_REF] (dark soliton) and [START_REF] Gravejat | Asymptotic stability of the black soliton for the Gross-Pitaevskii equation[END_REF] (black soliton), the orbital stability is improved to asymptotic stability (the modulation factors, which consist of a translation parameter and a phase shift, are analyzed). These approaches may be more suited to (logGP), but new difficulties due to the special form of the nonlinearity arise, starting with the control of the presence of vacuum, {u = 0}.

Scattering. Another way to study stability properties consists in establishing a scattering theory, considering the plane wave solution, u(t, x) = e ik•x-i|k| 2 t , k ∈ R d , as a reference solution. By Galilean invariance, the analysis is reduced to the case k = 0, and the question is the behavior of the solution as t → +∞ when the initial datum u 0 ≡ 1 is perturbed. In such a framework, a crucial object is the linearized Schrödinger operator about the constant 1. Using in addition normal form techniques, a scattering theory for (1.1) was developed in [START_REF] Gustafson | Scattering for the Gross-Pitaevskii equation[END_REF][START_REF] Gustafson | Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions[END_REF][START_REF] Gustafson | Scattering theory for the Gross-Pitaevskii equation in three dimensions[END_REF] for d ≥ 2, and resumed in [START_REF] Killip | The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions[END_REF][START_REF] Killip | The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions[END_REF] in the case of the three-dimensional cubic-quintic Schrödinger equation (the quintic term introduces the new difficulty of an energycritical factor).

In the case of (logGP), the new difficulty is due to the fact that the logarithmic nonlinearity is not multilinear in (u, ū), reminding of the difficulty of giving sense to linearization in such a context (see e.g. [START_REF] Ferriere | Existence of multi-solitons for the focusing logarithmic non-linear Schrödinger equation[END_REF][START_REF] Pelinovsky | On the linearized log-KdV equation[END_REF]).

Convergence toward other models. It is well-established that in the long wave régime, the Gross-Pitaevskii equation converges to the KdV equation in the onedimensional case, and to the KP-I equation in the multidimensional case; see e.g. [START_REF] Béthuel | On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation[END_REF][START_REF] Béthuel | On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II[END_REF][START_REF] Chiron | The KdV/KP-I limit of the nonlinear Schrödinger equation[END_REF]. More precisely, we may resume the general presentation from [START_REF] Chiron | The KdV/KP-I limit of the nonlinear Schrödinger equation[END_REF]: considering the nonlinear Schrödinger equation

i∂ t u + ∆u = f |u| 2 u,
where the nonlinearity f is such that f (1) = 0 and f ′ (1) > 0, the KdV and KP-I equations appear in the limit ε → 0, after the rescaling t = cε 3 τ, X 1 = ε(x 1 -ct), X j = ε 2 x j , j ∈ {2, . . . , d} .

Writing

u(t, X) = 1 + ε 2 A ε (t, X) e iεϕ ε (t,X) , and plugging this expression into the equation solved by u, the formal limit ε → 0 in the obtained system provides a limit A solution to the KdV equation if d = 1, and to the KP-I equation if d ≥ 2. We emphasize that in the usual Gross-Pitaevskii equation (1.1), f (y) = y -1, in the logarithmic case (logGP) with λ = 1, f (y) = ln y, so we always have f (1) = 0 and f ′ (1) > 0. The general case considered in [START_REF] Chiron | The KdV/KP-I limit of the nonlinear Schrödinger equation[END_REF] is analyzed under the assumption f ∈ C ∞ (R, R), which is not satisfied in the logarithmic case. This is probably not the sharp assumption in order for the arguments presented in [START_REF] Chiron | The KdV/KP-I limit of the nonlinear Schrödinger equation[END_REF] to remain valid, but it is most likely that the proof uses a C 0 regularity on [0, +∞), like in [START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF][START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF][START_REF] Mariş | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF] (these papers are devoted to the existence of traveling waves in the same framework; C 1 regularity of f is required only near 1). Again, the main difficulty in the case of the logarithmic nonlinearity lies in the control of vacuum, as f fails to be continuous at the origin.

  (logGP) i∂ t u + ∆u = λu ln |u| 2 , u |t=0 = u 0 , where x ∈ R d , d ≥ 1, λ > 0, and with the boundary condition at infinity |u(t, x)| → 1 as |x| → ∞. Such boundary condition is reminiscent of the standard Gross-Pitaevskii equation (1.1) i∂ t u + ∆u = |u| 2 -1 u, u |t=0 = u 0 ,

Definition 1 . 1 .

 11 Let u 0 ∈ E logGP , and0 ∈ I ⊂ R. We say that u ∈ L ∞ loc (I, E logGP ) is: • A weak solution to (logGP) on I if it satisfies i∂ t u + ∆u = λu ln |u| 2 in D ′ (I × R d ),and for any ψ ∈ C ∞ 0 (R d ), lim t→0 R d u(t, x)ψ(x) dx = R d u 0 (x)ψ(x) dx.

Theorem 1 . 4 .

 14 Let d ≥ 1. If φ ∈ E logGP and u, solution to (logGP), is a traveling wave of the form (1.2), then ω = 0.

Lemma 2 . 1 ([ 14 ,

 2114 Lemma 1.1.

Lemma 4 . 3 .

 43 There exists a solution {g m,k } k≤m to (4.4) on a maximal time interval (-T m , T m ), meaning that T m < ∞ if and only if lim sup t→T m sup j≤m |g m,j (t)| = ∞.

Lemma 4 . 4 .

 44 For all m ∈ N, (∇w k ) k≤m are linearly independent.Proof. Let λ k ∈ C such that m k=0 λ k ∇w k = 0. Then ∇ψ = 0, with ψ := m k=0 λ k w k ∈ H 1 .Therefore, ψ = 0, and we conclude by the fact that {w k } is a Hilbert basis of L 2 . Lemma 4.5. The solution given by Lemma 4.3 is global. Moreover, it satisfies

2 L 2 ≤

 22 for s, τ ∈ I. Then, we get by (4.5) ϕ m (t) -ϕ m (s) C I |t -s|, for all t, s ∈ I, with C I depending on I but not on m, hence the conclusion. 4.3. Convergence. Lemma 4.7. Let u m = u 0 + ϕ m the solution to (4.3) given by Lemma 4.3. There exists a subsequence of (ϕ m ) m (still denoted by ϕ m ) and

2 in L 2 (

 22 e. in I × Ω. Besides, we also know thatu m ln |u m | 2 is bounded in L ∞ (I, L 2) uniformly in m, and thus also in L 2 (I × Ω). Therefore, along this further subsequence, u m ln |u m | 2 converges weakly to u ln |u| I × Ω).

Lemma 7 . 8 .

 78 There exists a unique function φ 0 satisfying (7.7) with the initial data φ 0 (0) = 0. It is defined on R and satisfies lim x→±∞ φ 0 (x) = ±1.

  This second functional E pot appears to be more convenient than E pot . For instance, since ln 2≤ ln (2 + |x|) ≤ ln 2 + C δ ||x| -1|δ for all δ > 0, we can relate E pot to the L 2 -norm of |v| -1, but also to any L p -norm, for p > 2 as follows:Lemma 2.5. Let p > 2. There exists C p such that for all v ∈ E logGP ,

	1
	0	(1 + ln (1 -s + sy)) (1 -s)ds ln (2 + y) ,
	and the result follows, by considering y = |v|.

  4.2. Uniform estimates. We have already shown the conservation of the energy and an estimate of the L 2 -norm of ∇ϕ m , uniform both in t and m. Now, we proceed with its L 2 -norm. Lemma 4.6. Let u m = u 0 + ϕ m the solution to (4.3) given by Lemma 4.3. Then ϕ m is bounded in C 0,1 2 (I, L 2 ) uniformly in m, for every bounded interval I. Proof. Since all g m,k 's are continuous, we already know that ϕ m is continuous in time with values in L 2 . By multiplying (4.3) by g m,j (t) and summing over j, we obtain ϕ m , i∂ t ϕ m + ∆u 0 + ∆ϕ m -λu m ln |u m |

				2	H 1 ,H -1 = 0,
	and the imaginary part gives
	1 2	d dt	ϕ m	2

RC is supported by Centre Henri Lebesgue, program ANR-11-LABX-0020-0.

Suppose now c 2 > 2λ. Since h c (0) = h ′ c (0) = 0, and h ′′ c (0) = 2λ -c 2 , there exists ε > 0 such that h c (y) < 0 for all y ∈ [-ε, ε] \ {0}. On the other hand, h c (η) is continuous, nonnegative (from Lemma 8.1) and tends to 0 at ±∞. The conclusion follows easily. Lemma 8.5. If c 2 ≥ 2λ, then φ is constant.

Proof. By Corollary 8.4, we know that |φ(x)| = 1 for all x ∈ R. Since φ is a C 2 b (R) function, there exists a real-valued function θ ∈ C 2 such that φ = e iθ (defined like in (7.4) for instance). By substitution in (6.2), θ satisfies then

By taking the imaginary part, we get θ ′′ = 0. Since we must have φ ′ = iθ ′ e iθ ∈ L 2 , we get θ ′ ∈ L 2 and thus θ ′ = 0. 8.2. Nontrivial traveling waves. We now consider the case 0 < c 2 < 2λ. Define, for all y > 0,

The following lemma is established by direct calculations:

Lemma 8.6. Let c such that 0 < c 2 < 2λ. There exists 0 < y 0 < y 1 < 1 such that the following holds:

• f c has exactly two zeroes on (0, +∞): y 1 and 1.

• f c is positive on (0, y 1 ) ∪ (1, +∞), negative on (y 1 , 1).

• g c has exactly two zeroes on (0, +∞): y 0 and 1.

• g c is negative on (0, y 0 ), positive on (y 0 , +∞) \ {1}.

• There exists 2 for all y ∈ (y 1 , 1).

Proof. We compute

As lim

c has a unique zero on (0, +∞), which we denote by y 2 > 0. Since f ′ c (1) = 2λ -c 2 > 0, we know that 0 < y 2 < 1: f c (1) = 0, hence f c (y 2 ) < 0, and there exists a unique 0 < y 1 < y 2 such that f c (y 1 ) = 0.

We note that f c = 1 2 g ′ c , and in view of the above pieces of information, we can draw: