Ludovick Gagnon

Ground state solitary waves local controllability for the nonlinear focusing Schrödinger equation in the mass critical and slightly mass subcritical regime

Keywords: Nonlinear focusing Schrödinger equation, controllability, solitary waves MSC2020 : 93B05 35Q51 35Q55 93C10

In this paper, we prove the exact local controllability around different ground state solitary wave for the slightly subcritical mass and mass critical nonlinear Schrödinger equation. More precisely, if Q c1 and Q c2 denotes the ground states with two different scaling, we prove the exact local controllability from Q c1 to Q c2 in a minimal time depending on c 1 and c 2 . The results presented relies on the blow-up profile in the mass slightly supercritical case and mass critical case.

Introduction

We study in this article the controllability of the ground state solitary wave in the mass critical and slightly subcritical mass regime of the nonlinear focusing Schrödinger equation,

iψ t + ∆ψ + ψ|ψ| p-1 = 0, (x, t) ∈ R d × R, ψ(x, 0) = ψ 0 , x ∈ R d , (1)
with ψ 0 ∈ H1 (R d ) and p > 1. Equation (1) has many physical applications for p = 3 and d = 1, 2, as it serves as a model for signal propagation in nonlinear optic for optic fibers and self-focusing laser beams in hollow core fibers ( [START_REF] Hasegawa | Optical Solitons in Fibers[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF]). The nonlinear focusing Schrödinger equation ( 1) is also completely integrable if d = 1 and p = 3 ( [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). The solutions to (1) conserve three quantities over time,

Mass: ψ(., t) L 2 (R d ) = ψ 0 L 2 (R d ) ,
Energy: E(ψ(., t)) := 1 2 ∇ψ(., t) 2 L 2 -

1 p + 1 ψ(., t) p+1 L p+1 = E(ψ 0 ),
Momentum: Im ∇ψ(x, t)ψ(x, t)dx = Im ∇ψ 0 (x)ψ 0 (x)dx .

Bound states

The bound states, or solitary waves, are stationary solutions to (1) of the form ψ(x, t) = e iλt W λ (x) where W λ (x) = λ

1 p-1 W ( √ λx
) is solution to the nonlinear elliptic equation,

∆W λ + |W λ | p-1 W λ = λW λ , x ∈ R d . (3)
Non-trivial solutions to (3) exist if the energy is subcritical and if and only if λ > 0. The solutions W λ to (3) belongs to W 3,q (R d ) for any 2 ≤ q < ∞ and numerous properties of bound states have been established. It is well-known that solutions to (3) are not unique, but unicity is recovered under the additional constraint of seeking a positive and radially symmetric solution to (3) [START_REF] Tao | Nonlinear dispersive equations[END_REF][START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF]. This unique solution denoted Q λ (x) = λ 1 p-1 Q( √ λx), the ground state solitary wave, belongs to the Schwarz class S(R d ) and has an explicit representation in dimension one,

Q(x) = p + 1 2 sech 2 p -1 2 x 1 p-1 .
In higher dimensions, the exponential decay of the ground state is characterized by the following estimate for r ≥ 1 ( [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF]), Q(r) -κr -(d-1)/2 e -r + Q (r) -κr -(d-1)/2 e -r ≤ Cr -(d+1)/2 e -r ,

where r = |x| and where κ, C > 0 are two constants, depending on d ≥ 1. Here and below, the constant κ will always refer to the constant appearing in the inequality [START_REF] Burq | Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes[END_REF]. Finally, Q λ satisfies E(Q λ ) = λE(Q) = 0 in the mass critical case thanks to Pohozaev's identity [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. We refer to [START_REF] Tao | Nonlinear dispersive equations[END_REF]Appendix B] for additional properties of the ground state.

for d = 1, 2 ( [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Ginibre | On a class of nonlinear schrödinger equations. i. the cauchy problem, general case[END_REF]). Such solutions belongs to ψ ∈ C((-T min , T max ); H 1 (R d )), with T min = T min (ψ 0 ), T max = T max (ψ 0 ) and T min , T max ∈ [0, ∞). Moreover, there is a blow-up alternative ( [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]), that is to say, either T min = T max = ∞ and the solution is defined globally, or, if T max < ∞ (resp. T min < ∞), then lim t → T - max ψ(t) H 1 (R d ) = +∞ (resp. lim t → T - min ψ(-t) H 1 (R d ) = +∞). In the mass subcritical regime s c < 0, the conserved quantities as well as the Gagliardo-Nirenberg inequality allows to extend the time existence, T min and T max , of all solutions to [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF], implying the global well-posedness in H 1 (R d ) ( [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]).

The mass critical regime s c = 0 is the first regime to exhibit blow-up phenomenon. Weinstein proved in [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF] that the estimate ψ 0 L 2 (R d ) < Q L 2 (R d ) ensures the global existence of solutions to [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF], thanks to the conservation of mass, energy and the Gagliardo-Nirenberg inequality

E(ψ) ≥ 1 2 ψ 2 Ḣ1 (R d )   1 - ψ 2 L 2 (R d ) Q 2 L 2 (R d ) 4/d   , ∀ψ ∈ H 1 (R d ).
This estimate was proved to be sharp by Merle [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF] in the following sense. Assume

ψ 0 L 2 (R d ) = Q L 2 (R d )
and assume that the solution ψ(., t) of (1) blows up in finite time T > 0. Then the solution ψ is given (with an initial data at t = -T for simplicity) by the pseudo conformal transformation profile

S(x, t) = 1 |t| d/2 Q x t e -i |x| 2 4t + i t . (5)
This profile lies in the set Σ = H 1 (R d ) ∩ {xψ 0 ∈ L 2 (R d )}, invariant by the flow, and blows up at the speed ∇S(., t) L 2 (R d ) 1/|t|, t → 0 -. A general blow-up phenomenon occurs in Σ in the range 0 ≤ s c < 1 due to the so-called virial identity. Indeed, assume ψ 0 ∈ Σ = H 1 (R d ) ∩ {xψ 0 ∈ L 2 (R d )} and E(ψ 0 ) < 0. Then, the associated solution ψ(., t) of (1) belongs to Σ and d 2 dt 2 xψ(., t) 2 L 2 (R d ) = 4d(p -1)E(ψ 0 ) < 0, leading to a contradiction on the infinite time of existence of the solution.

The blow-up phenomenon in the mass critical regime was further investigated in a serie of works [START_REF] Merle | Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF][START_REF] Raphaël | Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation[END_REF]. The blow-up phenomenon was studied in the set

B α * = ψ 0 ∈ H 1 (R d ) Q 2 L 2 < ψ 0 2 L 2 < Q 2 L 2 + α *
for α * > 0 sufficiently small. Assuming the spectral property 1 , it was shown that if E(ψ 0 ) > 0, then blow-up occurs at rate 1/|t|, the same rate as the pseudo conformal transformation [START_REF] Burq | Geometric control in the presence of a black box[END_REF] and if E(ψ 0 ) < 0, then blow-up occurs at the rate

∇ψ(., t) L 2 (R) log | log(T -t)| T -t .
We shall denote here the concentration factor in the mass critical regime

λ * (t) = T -t log | log(T -t)|
, t < T.

1 Without going into details, the spectral property is related to the coercivity of linear form coming from the linearization around state Q along with H 1 orthogonality properties. It was proven in dimension d = 1 in [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF] and in dimensions d = 2, 3, 4 using rigorously verified computing [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF]. It is conjectured that the spectral property holds in any dimension. We refer to these works for a precise definition. This rate is not self-similar as

lim t→T - λ * (t) √ T -t = 0.
In fact, more can be said about the blow-up profile in the case E(ψ 0 ) < 0. Indeed, in [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF], it is shown that the solution ψ(., t) coming from the initial data ψ 0 ∈ B α * such that E(ψ 0 ) < 0 decomposes as

ψ(x, t) = e iγ(t) λ(t) (Q + ) x -x(t) λ(t) , t , with H 1 (R d ) → 0 as α * → 0.
An important remark that we shall use in the present article is that, if the initial data ψ 0 is radial, then translation parameter x(t) is equal to 0 ( [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF]).

Let us underline here that the mass subcritical case s c < 0 is in sharp contrast with the mass critical and supercritical regime. Not only the solutions are always globally defined in H 1 (R d ), but the ground state is in fact orbitally stable with respect to H 1 (R d ) pertubations, that is, stable up to translation in space and phase shift [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF]. More precisely, for any > 0, there exists δ > 0 such that if, inf

(γ,x 0 )∈[0,2π]×R ψ 0 (. -x 0 ) -Q(.)e iγ H 1 (R d ) < δ, then inf (γ,x 0 )∈[0,2π]×R ψ(. -x 0 , t) -e i(t+γ) Q(.) H 1 (R d ) < .
Finally, the mass intercritical case 0 < s c < 1 is conjectured ( [START_REF] Sulem | Focusing nonlinear Schrödinger equation and wave-packet collapse[END_REF][START_REF] Zakharov | Collapse of self-focusing of langmuir waves[END_REF]) to exhibit self-similar blow-up in finite time for initial data close to the ground state Q in Ḣ1 (R d ). The existence and stability of the self-similar blow-up was proved in [START_REF] Merle | Stable self-similar blow-up dynamics for slightly L 2 super-critical NLS equations[END_REF] for 0 < s c 1 and a precise description of the blow-up profile was obtained in a recent work of Bahri, Martel and Raphaël [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF].

Before stating our main results, we present a short overview of the controllability of solitary waves as well as the controllability of the Schrödinger equation, as it will ease the presentation and allow us to state more precisely the nature of our results.

Literature overview

Despite the extensive literature on solitary waves, it seems that little is known so far on its controllability properties. If the solitary waves are defined on the whole space, only one result exists on the controllability of the ground state for the generalized KdV equation (where the ground state is also solution to (3)) due to Muñoz. In [START_REF] Muñoz | On approximate controllability of generalized KdV solitons[END_REF], considering the ground state on the whole line and using a moving distributed control, Muñoz proved the approximate controllability in large time from

Q c 1 to Q c 2 for c 1 , c 2 ∈ R + .
More precisely, the nonlinear interaction between the control and the solitary wave allows Muñoz to add (or remove, depending on the sign of c 2 -c 1 ) an important mass to the ground state with a slow varying distributed control. It is important to highlight that the control strategy is not destructive, meaning that the solution over time remains close to the ground state for t ∈ [0, T ]. We also note a second result on linking the controllability and the solitary waves defined on the real line. In [START_REF] Gagnon | Lagrangian controllability of the 1-dimensional korteweg-de vries equation[END_REF], the N -solitons solution was used by the author as a trajectory to achieve small-time Lagrangian controllability for the Korteweg-de Vries equation.

Other results are found in the literature on the controllability of the ground state when defined on bounded domains. Lange and Teismann [START_REF] Lange | Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state[END_REF], considered the 1-D nonlinear focusing Schrödinger equation on a bounded domain with homogeneous Dirichlet boundary conditions and with a distributed control. In [START_REF] Lange | Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state[END_REF], the ground state is defined as the unique positive solution in x ∈ (0, 1) of

ψ (x) + ψ 3 (x) = ψ(x), x ∈ (0, 1), ψ(0) = ψ(1) = 0. (6)
The local controllability around the ground state was obtained using the HUM method [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] and a fine spectral analysis of the underlying linearized equation. A similar question was addressed by Castelli and Teismann in [START_REF] Castelli | Rigorous numerics for nls: Bound states, spectra, and controllability[END_REF], but this time with the size L(t) of the 1-D domain having the role of the control.

We also find few results on the stabilization of the solitary waves. In [START_REF] Mirrahimi | Lyapunov control of a quantum particle in a decaying potential[END_REF], the approximate stability of the bilinear Schrödinger equation in R d , modelling the action of a laser on a quantum particle, around ground state solutions to

(∆ + V (x))ψ(x) = 0, x ∈ R d ,
was proven by Mirrahimi under spectral hypothesis of the operator ∆ + V . In the particular case of a dipolar function µ(x) = x and with a domain equal to the interval (-1/2, 1/2), a global practical stabilisation was obtained by Beauchard and Mirrahimi through a Lyapunov analysis in [START_REF] Beauchard | Practical stabilization of a quantum particle in a onedimensional infinite square potential well[END_REF].

We now turn to general controllability properties of the nonlinear Schrödinger equation. We first emphasize on the result of Rosier and Zhang [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF], as the extension of their results shall be needed in the present work. They have proved the exact local controllability around smooth trajectories to (1) of

iψ t + ∆ψ + |ψ| 2 ψ = 0, (x, t) ∈ Ω × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ Ω, (7)
for where ν denotes the outward normal vector to Ω. More precisely, Theorem 1 (Theorem 1.1, [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF]). Let T > 0 be given and let w ∈ C ∞ ([-, T + ]; S(R)) be a solution of the nonlinear Schrödinger equation, with λ ∈ C * ,

Ω ⊂ R d ,
iw t + ∆w + λ|w| 2 w = 0, for any (x, t) ∈ Ω 1 × (-, T + ) where > 0 and Ω 1 is a bounded domain in R d with Ω ⊂ Ω 1 . Assume s > d 2 , or 0 ≤ s < d 2 with 1 ≤ d ≤ 2 + 2s, or s = 0, 1 with d = 2.
Then there exists a δ > 0 such that for any ψ 0 , ψ 1 ∈ H s (Ω) satisfying

ψ 0 -w(., 0) H s (Ω) ≤ δ, ψ 1 -w(., T ) H s (Ω) ≤ δ,
one can find an appropriate boundary control function v(x, t) such that (7)-( 8) admits a solution

ψ ∈ C([0, T ]; H s (Ω)) such that ψ(x, 0) = ψ 0 (x), ψ(x, T ) = ψ 1 (x), in Ω.
The control strategy deployed in [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] consists to extend the initial data ψ 0 to R d and to consider the control problem in the whole space

iψ t + ∆ψ + |ψ| 2 ψ = ϕ(x)h(x, t), (x, t) ∈ R d × R + , ψ(x, 0) = ψ 0 (x), x ∈ R d , (9)
where

ϕ ∈ C ∞ (R d ; [0, 1]) is a cut-off function such that ϕ(x) = 1, if |x| ≥ R + 1 0, if |x| ≤ R (10)
with R > 0 sufficiently large so that Ω ⊂ B R (0). This strategy of extending the solution to the whole space to deduce boundary controllability property is attributed to Russell who first used it for the wave equation in [START_REF] Russell | Boundary value control of the higher-dimensional wave equation[END_REF][START_REF] Russell | Boundary value control theory of the higher-dimensional wave equation[END_REF]. Littman and Taylor later on gave a general principle for the boundary controllability of a linear partial differential equation, that is reversibility, smoothing properties and uniqueness usually leads to controllability [START_REF] Littman | Smoothing evolution equations and boundary control theory[END_REF]. In [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF], the smoothing properties come from the Strichartz estimates available in R d . The controllability of the linearized equation associated to [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF] around the smooth trajectory w is obtained in [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] via Carleman estimates for C ∞ ([-, T + ]; S(R)) potentials. The local exact controllability of (9) around smooth trajectories is deduced by a fixed point argument. Theorem 1 is obtained by constructing the control v in (8) as the trace of ( 9), and the controlled solution of ( 7) is obtained as the restriction of the solution of [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF] to Ω. Such solutions may be defined in a weak sense (see Section 2.4).

Other controllability results were obtained for the nonlinear Schrödinger equation. The controllability of the nonlinear defocusing Schrödinger equation on a compact manifold without boundary was proved by Dehman, Gérard and Lebeau [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF] using the exponential decay of the nonlinear problem and local controllability, assuming that the support of the internal control satisfies the Geometric Control Condition [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], where the time of controllability depends on the size of the initial data. This argument of first stabilizing the solution and then using local controllability around zero was also used by Laurent in [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF][START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF]. The Geometric Control Condition (GCC) was shown by Lebeau to be sufficient for the controllability of the linear Schrödinger equation [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF], but unlike the linear wave equation where GCC is necessary and sufficient (see [START_REF] Burq | Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes[END_REF] for a precise statement), GCC is not always necessary, as illustred by Jaffard [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] and Burq and Zworski [START_REF] Burq | Geometric control in the presence of a black box[END_REF]. It was more recently proved by Jin that a control supported in any nonempty open set yields the controllability on hyperbolic surfaces (see [START_REF] Jin | Control for Schrödinger equation on hyperbolic surfaces[END_REF] and reference therein). Extensive study was also done on the controllability of the linear and bilinear Schrödinger equation and with boundary or internal control. We refer the reader to the surveys [START_REF] Laurent | Internal control of the Schrödinger equation[END_REF][START_REF] Zuazua | Remarks on the controllability of the Schrödinger equation[END_REF] on the subject

Main results

Let T > 0 and Ω ⊂ R d , with d ≥ 1 be an open, connected and bounded domain with a boundary ∂Ω of class C 2 . In the spirit of [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF], we consider the boundary controllability of the focusing nonlinear Schrödinger equation

     iψ t + ∆ψ + ψ|ψ| p-1 = 0, (x, t) ∈ Ω × (0, T ), ψ(x, t) = v(x, t), (x, t) ∈ ∂Ω × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ Ω, (11)
where v ∈ L 2 ((0, T ); L

ψ 0 -e iγ Q c 1 H 1 (Ω) < , inf γ∈[0,2π] ψ 1 -e iγ Q c 2 H 1 (Ω) < , there exists a control v ∈ C([0, T ]; H 1/2 -(∂Ω)) such that (11) admits a solution ψ ∈ C([0, T ]; H 1 (Ω)) satisfying ψ(x, 0) = ψ 0 (x), ψ(x, T ) = ψ 1 (x), in Ω, for T T * 1 -T * 2 > 0, where log | log(T * 1 )| T * 1 = c 1 , log | log(T * 1 -T * 2 )| T * 1 -T * 2 = c 2 .
Remark. The minimal time of controllability T * 1 -T * 2 given by Theorem 2 is approximately the minimal time of controllability with the technique employed in this article. A precise time of controllability is given in the proof of Theorem 5, and is obtained by taking into account the phase shift of the blow-up profile as well as the initial and final data. We highlight that

T * 1 -T * 2
is the minimal time of controllability if the phase of ψ 0 and ψ T are those of the blow-up profile, hence the statement of Theorem 2.

The proof of Theorem 2 relies on two main arguments. One is to exploit the nonlinear nature of the equation to use the blow-up profile ψ b in the set B α * , and such that E(ψ 0 ) < 0, as a trajectory to go from the vicinity of

Q c 1 to the vicinity of Q c 2 . The second is to use the C ∞ (R; S(R d ))
regularity of the ground state to adapt the exact local controllability around smooth trajectories [37, Theorem 1.1] for [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF] around the ground state. Theorem 3. Let T > 0, p ∈ [1, p * ] and λ ≥ 0. Then there exists a δ > 0 such that for any ψ 0 , ψ T ∈ H 1 (Ω) satisfying

ψ 0 -Q λ (., 0) H 1 (Ω) ≤ δ, ψ T -Q λ (., T ) H 1 (Ω) ≤ δ, (12)
one can find an appropriate boundary control function v(x, t) such that (11) admits a solution

ψ ∈ C([0, T ]; H 1 (Ω)) such that ψ(x, 0) = ψ 0 (x), ψ(x, T ) = ψ T (x), in Ω.
The control strategy therefore consists to drive the initial data to the blow-up profile ψ b (., ), for > 0 arbitrarily small, and for α * sufficiently small. Then, the blow-up profile is used for ( , T -) to reach the vicinity of Q c 2 where the local exact controllability around Q c 2 is used again to reach ψ T at time t = T . The minimal time is therefore dependent on the finite speed of blow-up. This strategy of proof is similar to the result of Muñoz [START_REF] Muñoz | On approximate controllability of generalized KdV solitons[END_REF], as the control strategy is non destructive and as the controlled solution ψ remains close to the ground state ∀t ∈ [0, T ]. Note that the assumption c 1 < c 2 is not restrictive as the case c 2 < c 1 is easily obtained by the time reversibility of (1) through the change of variables ψ(x, t) → ψ(x, -t) (the case c 1 = c 2 is trivially deduced from Theorem 3). In particular, we deduce the null-controllability in large time in the neighborhood of the ground state.

ψ 0 ∈ H 1 (Ω) such that inf γ∈[0,2π] ψ 0 -e iγ Q c H 1 (Ω) < there exists a control v ∈ C([0, T ]; H 1/2 -(∂Ω)) such that (11) admits a solution ψ ∈ C([0, T ]; H 1 (Ω)) satisfying ψ(x, 0) = ψ 0 (x), ψ(x, T ) = 0, in Ω,
for T 1/δ > 0, with δ > 0 given by the smallness assumption (12) of Theorem 3.

Proof. Theorem 2 implies Theorem 4. Indeed, choose c 2 = c and, for any given δ > 0, choose 0 < c 1 sufficiently small so that Q c 1 H 1 (Ω) < δ. Such c 1 always exist since Ω is bounded, and therefore . L 2 (Ω) is not invariant with respect to the scaling ψ λ (x, t) = λ 2 p-1 ψ(λx, λ 2 t). For instance, if Ω ⊂ B 1 (0), the ball of radius 1 and centered at the origin, then

Q c 1 L 2 (Ω) ≤ Q c 1 L 2 (B 1 (0)) = Q L 2 (B √ c 1 (0)) . Since Q is bounded on B √ c 1 (0), we have Q B √ c 1 (0) → 0 as c 1 → 0. The same argument is used for Q c 1 Ḣ1 (Ω)
. Then, the time reversibility of (11) yields a trajectory starting from ψ 0 to Q c 1 . We use Theorem 3 to drive the solution near Q c 1 to 0.

A natural question following Theorem 2 and Theorem 4 is to understand whether this type of controllability results hold for the ground state in the mass subcritical regime. Indeed, the ground state is known to be orbitally stable in this regime, and one could assert that this stability is sufficient to disrupt the control strategy employed for Theorem 2. By proving that the blow-up profile of Bahri, Martel and Raphaël [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF] holds in the mass slightly subcritical regime, we prove the exact controllability between the vicinity of two different ground states for the mass slightly subcritical case.

∂Ω of class C 2 . Let c 1 , c 2 ∈ R + with c 2 > c 1 .
Then, there exists 1 < p < p * such that for every p ∈ (p, p * ), there exists > 0 such that, for every

ψ 0 , ψ 1 ∈ H 1 (Ω) such that inf γ∈[0,2π] ψ 0 -e iγ Q c 1 H 1 (Ω) < , inf γ∈[0,2π] ψ 1 -e iγ Q c 2 H 1 (Ω) < , there exists a control v ∈ C([0, T ]; H 1/2 -(∂Ω)) such that (11) admits a solution ψ ∈ C([0, T ]; H 1 (Ω)) satisfying ψ(x, 0) = ψ 0 (x), ψ(x, T ) = ψ 1 (x), in Ω, in time T T c 1 ,c 2 > 0, where T c 1 ,c 2 := C c 2 -c 1 c 1 c 2 , with C(p) > 0 independant of c 1 , c 2 .
As for Theorem 2, we deduce that Theorem 5 holds for c 2 < c 1 as well as the null controllability.

Theorem 6. Let d ≥ 1 and Ω ⊂ R d be an open, connected and bounded domain with a boundary ∂Ω of class C 2 . Let c ∈ R + . Then, there exists 1 < p < p * such that for every p ∈ (p, p * ), there exists > 0 such that, for every

ψ 0 ∈ H 1 (Ω) such that inf γ∈[0,2π] ψ 0 -e iγ Q c H 1 (Ω) < , there exists a control v ∈ C([0, T ]; H 1/2 -(∂Ω)) such that (11) admits a solution ψ ∈ C([0, T ]; H 1 (Ω)) satisfying ψ(x, 0) = ψ 0 (x), ψ(x, T ) = 0, in Ω,
in time T 1/δ > 0, with δ > 0 given by the smallness assumption (12) of Theorem 3.

We underline that Theorem 2, Theorem 4, Theorem 5 and Theorem 6 are closely related to the open question of global exact controllability of [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF]. Not only the solitary waves are special solutions to (1), but they are in fact conjectured to be generic in the decomposition of the solutions to [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]. Indeed, the soliton resolution, conjectured to hold in the energy subcritical regime [START_REF] Soffer | Soliton dynamics and scattering[END_REF], states that the solutions of (1) evolve as a finite number of solitons, and a radiation behaving as a solution to the linear Schrödinger equation. Therefore, any control strategy of [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF] relying on the extension on the Euclidean space either needs to rule out altogether the existence of solitary waves, or to understand, in some extent, their global controllability properties. In this sense, we view the results above as a step in this direction.

Structure of the article

In Section 2, we recall the well-posedness results for the focusing nonlinear Schrödinger equation. In Section 3, we extend the results of Rosier and Zhang [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] to deduce the exact local controllability around the ground state for the mass critical and subcritical regime. Section 4 is dedicated to the various results on the blow-up profiles that we shall need for the proof of the main results. Finally, Section 5 is reserved for the proof of Theorem 2 and Theorem 5.

Well-posedness

Functional framework

We consider, for Ω an open subset of R d , the space L p (Ω) of measurable complex-valued functions

u : Ω → C such that u L p (Ω) < ∞ with the norm u L p (Ω) = Ω |u(x)| p dx 1/p , if p < ∞, ess sup x∈Ω |u(x)|, if p = ∞.
The space L 2 (Ω) is a real Hilbert space equipped with the scalar product

(u, v) L 2 (Ω) = Re Ω u(x)v(x)dx , ∀u, v ∈ L 2 (Ω).
Likewise, H m (Ω), m ∈ N is a real Hilbert space with the scalar product

(u, v) H m (Ω) = 0≤|α|≤m Re Ω D α u(x)D α v(x)dx , ∀u, v ∈ H m (Ω), where α ∈ N d is a multi-index and D α = D (α 1 ,...,α d ) = ∂ α 1 x 1 • • • ∂ α d x d .

Strichartz estimates

Let us now introduce the definition of admissible pair for the Strichartz estimates. Let 1 ≤ q ≤ ∞ and r ∈ R + . We say that the pair (q, r) is admissible if and only if

2 q = d 1 2 - 1 r , (13)
and if 2 ≤ r ≤ ∞ for d = 1, 2 ≤ r < ∞ for d = 2 and 2 ≤ r ≤ 2d d -2 , if d ≥ 3.
We denote q the conjugate of q such that 1 q + 1 q = 1. Let us recall the Strichartz estimates that we shall use in this paper.

Lemma 1 (Strichartz's estimates, [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). For any s ∈ R, the following holds • If (q, r) is an admissible pair, then there exists a constant C > 0 such that, for every

ψ ∈ H s (R d ), S(t)ψ L q (R;W s,r (R d )) ≤ C ψ H s (R d ) ,
• Let I ⊂ R, be an interval, bounded or not , J = I satisfying 0 ∈ J. If (γ, ρ) and (q, r) are two admissible pairs, then there exists a constant C > 0 such that for every

f ∈ L γ (I, W s,ρ (R d )) t 0 S(t -τ )f (τ )dτ L q (I;W s,r (R d )) ≤ C f L γ (I,W s,ρ (R d )) .

Linear Schrödinger equation

We begin with the well-posedness of the linear Schrödinger equation. For any s ∈ R, denote

Aψ = i∆ψ, ∀ψ ∈ D(A) with A : D(A) = H s+2 (R) → H s (R)
the infinitesimal generator of the group of isometry S(t) on H s (R). The solution of

iψ t + ∆ψ = 0, (x, t) ∈ R d × R, ψ(x, 0) = ψ 0 (x), x ∈ R d , for ψ 0 ∈ H s (R)
, is given by ψ(., t) = S(t)ψ 0 and satisfies ψ(., t) ∈ C(R; H s (R d )). We now turn to the inhomogeneous problem. Consider I ⊂ R an interval such that 0 ∈ I, and consider

iψ t + ∆ψ = f, (x, t) ∈ R d × I, ψ(x, 0) = ψ 0 (x), x ∈ R d . ( 14
)
We follow [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for the definition of the well-posedness. We say that ψ ∈ C(I; H 1 (R d )) is a H 1 strong solution to [START_REF] Godet | Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces[END_REF] if and only if ψ satisfies

ψ(t) = S(t)ψ 0 -i t 0 S(t -τ )f (τ )dτ, ∀t ∈ I,
and the well-posedness is defined as follow.

Definition 1. We say that ( 14) is locally well-posed in H 1 if :

• There is uniqueness in H 1 of the solution ψ;

• For every

ψ 0 ∈ H 1 (R d ), there exists a strong H 1 (R d ) solution which is defined on a maximal interval (-T min , T max ) with T min = T min (ψ 0 ) and T max = T max (ψ 0 ) such that T min , T max ∈ (0, ∞].
• There is a blow-up alternative : if

T max < ∞, then lim t→T - max ψ(., t) H 1 (R d ) = ∞ (respec- tively T min < ∞, then lim t→-T + max ψ(., t) H 1 (R d ) = ∞).
• The solution depends continuously on the initial data : if ψ 0,n → ψ 0 in H 1 (R d ), and if I ⊂ (-T min (ψ 0 ), T max (ψ 0 )) is a closed interval, then the maximal solution ψ n of (14) with the initial data ψ n (0) = ψ 0,n is defined on I for n large enough and satisfies

ψ n → ψ, n → ∞ in C(I; H 1 (R d )).
Then, the Strichartz estimates and a classical semigroup result [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] yields the local wellposedness of [START_REF] Godet | Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces[END_REF]. 14) is locally well-posedness.

Proposition 1. Let I ⊂ R an interval such that 0 ∈ I. Then, for ψ 0 ∈ H 1 (R d ) and f ∈ L 1 (I; H s (R d )), (
The above Strichartz estimates allows us to deduce the local well-posedness of the linearized equation of ( 9) around smooth trajectories of (1). Indeed, consider

iψ t + ∆ψ + a(x, t)ψ + b(x, t)ψ = f (x, t), (x, t) ∈ R d × I, ψ(x, 0) = ψ 0 (x), x ∈ R d , (15)
with

ψ 0 ∈ H 1 (R d ), f ∈ L 1 (I; H s (R d )) and a, b complex-valued functions belonging to C ∞ (R; S(R d )).
The solution to ( 15) is given by the Duhamel formula,

ψ(t) = S(t)ψ 0 -i t 0 S(t -τ )(aψ + bψ + f )(τ )dτ, in H s (R d ).
The local well-posedness of (15) in C(I; H s (R d )) was proved [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] by proving that the Strichartz estimates of Lemma 1 also holds for the flow map S L (t) associated to

Lψ := iψ t + ∆ψ + a(x, t)ψ + b(x, t)ψ.
Proposition 2 (Proposition 2.4, [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF]). For any s ∈ R and any an admissible pair (q, r) such that q > 2, then the following holds:

• There exists a constant C > 0 such that, for every ψ ∈ H s (R d ),

S L (t)ψ L q (R;W s,r (R d )) ≤ C ψ H s (R d ) ,
• Let I ⊂ R, be an interval, bounded or not, J = I satisfying 0 ∈ J. Let (γ, ρ) be another admissible pair, then there exists a constant C > 0 such that for every f ∈ L γ (I, W s,ρ (R d )),

t 0 S L (t -τ )f (τ )dτ L q (I;W s,r (R d )) ≤ C f L γ (I,W s,ρ (R d )) .
These estimates together with standard semigroup arguments [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] allows to prove the wellposedness of

iψ t + ∆ψ + a(x, t)ψ + b(x, t)ψ = f (x, t), (x, t) ∈ R d × R + , ψ(x, 0) = ψ 0 (x), x ∈ R d . ( 16
)
Proposition 3. Let I ⊂ R such that 0 ∈ I. Let ψ 0 ∈ H 1 (R d ), f ∈ L 1 (I; H 1 (R d ))
. Then equation ( 16) is locally well-posed with ψ ∈ C(I; H 1 (R d )).

Well-posedness of the nonlinear equation on R d

We now turn to the well-posedness of the nonlinear equation

iψ t + ∆ψ + |ψ| p-1 ψ = 0, (x, t) ∈ R d × I, ψ(x, 0) = ψ 0 (x), x ∈ R d , (17)
where

ψ 0 ∈ H 1 (R d ).
The local well-posedness holds using Kato's argument [START_REF] Kato | On nonlinear Schrödinger equations[END_REF] (see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Section 4.4]).

Theorem 7 ([18]

). Let j ∈ C(C; C) such that j(0) = 0,

|j(u) -j(v)| ≤ L(K)|u -v|, for all u, v ∈ C such that |u|, |v| ≤ K for K > 0 with    L(t) ∈ C([0, ∞)), d = 1, L(t) ≤ C(1 + t α ), with 0 ≤ α < 4 d -2 , if d ≥ 2.
Set g(u)(x) = j(u(x)) for all measurable u : R d → C almost everywhere in R d . Assume j, considered as a function of

R 2 → R 2 , is of class C 1 . Then, ( 17
) is locally well-posed in H 1 (R d ).
In turn,

iψ t + ∆ψ + |ψ| p-1 ψ = f (x, t), (x, t) ∈ R d × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ R d , (18)
is also locally well-posed with f ∈ L 1 (I; H 1 (R d )). In fact, one deduces the local well-posedness of ( 18) around any smooth trajectory w of (1) from [7, Theorem 4.4.6],

iψ t + ∆ψ + |w + ψ| p-1 (w + ψ) -|w| p-1 w = f (x, t), (x, t) ∈ R d × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ R d . ( 19
) Proposition 4. Let ψ 0 ∈ H 1 (R d ), f ∈ L 1 (I; H 1 (R d )), w ∈ C ∞ (I; S(R d )) solution to (1). Then, ( 19
) is locally well-posed in H 1 (R d ).
Proof. Indeed, [7, Theorem 4.4.6] is stated as follow for general nonlinear equation of the form

iψ t + ∆ψ + g(ψ) = 0, (x, t) ∈ R d × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ R d , (20)
with the following assumption on the non-linearity g ∈ C(H

1 (R d ); H -1 (R d )) : suppose there exist 2 ≤ r, ρ < 2d d-2 (2 ≤ r, ρ < ∞ if d = 1, 2) such that g(u) -g(v) L ρ (R d ) ≤ C(δ) u -v L r (R d ) , (21)
for all u, v ∈ H 1 (R d ) such that u H 1 (R d ) < δ, v H 1 (R d ) < δ and g(u) W 1,ρ (R d ) ≤ C(δ)(1 + u W 1,r (R d ) ), (22)
for all u ∈ H 1 (R d ) ∩ W 1,r (R d ), such that u H 1 (R d ) < δ.
Then, (we refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for a more precise statement)

Theorem 8 (Theorem 4.4.6 [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). Let g = g 1 + g 2 + . . . + g k such that each g j , j = 1, . . . k satisfies (21) and ( 22) for some exponent r j , ρ j . Then, for every ψ 0 ∈ H 1 (R d ), there exists a unique strong H 1 solution of (20) defined on a maximal time interval (0, T ψ 0 ).

Then, Proposition 4 follows easily. Indeed, define

g(ψ) = |w + ψ| p-1 (w + ψ) -|w| p-1 w, θ(z) = 1 if |z| ≤ 1 and g 1 (ψ) = θ(ψ)g(ψ), g 2 (ψ) = (1 -θ(ψ))g(ψ).
Then g 1 , g 2 satisfies ( 21) and [START_REF] Laurent | Internal control of the Schrödinger equation[END_REF] (see the proof of Proposition 6). Hence the local well-posedness follows for f = 0, and the case with f = 0 with f L 1 ((0,T );H 1 (R d )) < δ is dealt classically by including f in the Duhamel formula and using the Strichartz estimates.

Let us finally give a meaning to a solution of [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF], following for instance [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]. We underline that this definition is weaker than Definition 1, as nothing is stated about the uniqueness, the blow-up alternative or the continuous dependence on the initial data. Definition 2. We say that ψ is a solution to [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF]

if ∀τ ∈ [0, T ), i Ω ψ(x, τ )φ(x, τ ) -ψ 0 (x)φ(x, 0)dx - τ 0 Ω
∇ψ(x, t)∇φ(x, t)dxdt

+ τ 0 Ω ψ|ψ| p-1 φdxdt = τ 0 ∂Ω v(x, t)∂ ν φ(x, t)dxdt, ( 23
)
for any φ ∈ C([0, T ]; H 1 (Ω)) ∩ {∂ ν φ ∈ C([0, T ]; L 2 (∂Ω))}.
We underline that ( 23) make sense since, as p -

1 < 4/(d -2), |ψ| p-1 ψ ∈ C([0, T ); H -1 (R d )). Therefore, denoting φ the extension of φ by 0 outside Ω, τ 0 Ω ψ|ψ| p-1 φdxdt = τ 0 R d ψ|ψ| p-1 φdxdt ≤ Cτ |ψ| p-1 ψ L ∞ ((0,τ );H -1 (R d )) φ L ∞ ((0,τ );H 1 (Ω)) .

Local exact controllability around the ground state

The goal of this section is to extend the local exact controllability around smooth trajectories of (1) obtained by Rosier and Zhang [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] for p = 3 and d ≥ 1 to the local exact controllability around the ground state for the mass critical and mass subcritical regime 0 < p -1 ≤ 4/d with d ≥ 1, as given by Theorem 3. We begin by recalling their exact controllability result for the linear equation.

Controllability of the linearized equation

Let T > 0 and consider

iψ t + ∆ψ + a(x, t)ψ + b(x, t)ψ = ϕ(x)h(x, t), (x, t) ∈ R d × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ R d , (24)
with

ψ 0 ∈ H s (R d ), a, b ∈ C ∞ ((0, T ); S(R d )), h ∈ L 2 ((0, T ); H s (R d )) and ϕ ∈ C ∞ (R d ; [0, 1])
defined by [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF].

Theorem 9 ([37, Theorem 3.1]). Let T > 0 and s ≥ 0 be given and assume a, b ∈ C ∞ ((0, T ); S(R d )).

There exists a bounded linear operator

G : H s (R d ) × H s (R d ) → L 2 ((0, T ); H s (R d )),
such that for any ψ 0 , ψ T ∈ H s (R d ), if one chooses h = G(ψ 0 , ψ T ) as a control input, then the system (24) admits a solution ψ ∈ C([0, T ];

H s (R d )) satisfying ψ(., T ) = ψ T , in H s (R d ).

Local exact controllability around smooth trajectories

We now turn to the local exact controllability of,

iψ t + ∆ψ + |ψ| p-1 ψ = ϕ(x)h(x, t), (x, t) ∈ R d × (0, T ), ψ(x, 0) = ψ 0 (x), x ∈ R d , (25)
around ground state solitary wave ψ λ (x, t) = e iλt Q λ (x). We recall that, in this case, ψ λ ∈ C ∞ (R; S(R d )) and that Q λ is positive. We first write solution to [START_REF] Littman | Smoothing evolution equations and boundary control theory[END_REF] under the form ψ = ψ λ +y, where y solves,

   iy t + ∆y + p + 1 2 Q p-1 λ y + p -1 2 e i2λt Q p-1 λ y + g(y) = ϕ(x)h(x, t), (x, t) ∈ R d × (0, T ), y(x, 0) = y 0 (x), x ∈ R d , (26)
where,

g(y) = |e iλt Q λ + y| p-1 (e iλt Q λ + y) -e iλt Q p λ - p + 1 2 Q p-1 λ y - p -1 2 e i2λt Q p-1 λ y. (27)
Denote S L (t) the flow map associated to [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] with

a(x, t) = p + 1 2 Q p-1 λ (x), b(x, t) = p -1 2 e 2iλt Q p-1 λ (x). (28)
The following result give the sufficient functional framework to obtain the local exact bility of (26).

Proposition 5. Let s ≥ 0, T > 0 and g defined by [START_REF] Merle | Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation[END_REF]. If the following holds in a Banach space X s,T ⊂ C([0, T ];

H s (R d )) : there exists C > 0 such that φ ∈ H s (R d ), S L (t)φ X s,T ≤ C φ H s (R d ) , for every f ∈ L 2 ((0, T ); H s (R d )), t 0 S L (t -τ )f (., τ ) dτ X s,T ≤ C f L 2 ((0,T );H s (R d )) , for every z 1 , z 2 ∈ X s,T , t 0 S L (t -τ )g(z 1 )(τ ) dτ X s,T ≤ C(1 + z 1 X s,T ) z 1 p-1 X s,T , and
t 0 S L (t -τ )(g(z 1 ) -g(z 2 ))(τ ) dτ X s,T ≤ C( z 1 X s,T + z 2 X s,T + z 1 p-1 X s,T + z 2 p-1 X s,T ) z 1 -z 2 X s,T , then (26
) is locally exactly controllable in H s (R d ), that is, there exists δ > 0 such that for every

y 0 , y 1 ∈ H s (R d ) such that, y 0 H s (R d ) < δ, y 1 H s (R d ) < δ, then one can find a control h ∈ L 2 ((0, T ); H s (R d )) such that the solution y ∈ C([0, T ]; H s (R d )) of (26) satisfies, y(., T ) = y 1 , in H s (R d ).
Proof. The proof is very similar to [37, Theorem 4.1]. First, notice that (26) linearized around zero is given by ( 24), with a and b are defined by [START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF]. Since a, b defined by [START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF] belong to C ∞ (R; S(R d )), the exact controllability given by Theorem 9 holds.

The solution to ( 26) is given by the Duhamel formula,

y(t) = S L (t)y 0 - t 0 S L (t -τ )g(y)(τ )dτ + t 0 S L (t -τ )ϕ(x)h(x, τ )dτ.
Using the time-reversibility, we can assume y 1 = 0. The proof of Proposition 5 is obtained by a classical Fixed Point Theorem. Indeed, let

h = G y 0 , t 0 S L (t -τ )g(u)(τ )dτ for every u ∈ C([0, T ]; H s (R)). Then, let Γ(u)(t) = W S L (t)y 0 - t 0 S L (t -τ )g(y)(τ )dτ + t 0 S L (t -τ )ϕ(x)h(x, τ ))dτ.
We easily see that

Γ(u)(0) = y 0 , Γ(u)(T ) = 0.
It therefore suffices to prove that Γ(u)(t) is a contraction in X s,T thanks to the various estimates in hypothesis of Proposition 5. From now on, the proof is the same as in [37, Theorem 4.1] and we therefore omit it here.

It remains to define the proper space X s,T and to prove the estimates of Proposition 5, which yields Theorem 3.

Proposition 6. Let T > 0, d ≥ 1, s = 1 and r = p + 1 for p ∈ (1, p * ]. Then the estimates of Proposition 5 hold in X T := C([0, T ]; H 1 (R d )) ∩ L q ((0, T ); W 1,r (R d )) endowed with the norm . X s,T := . L ∞ ((0,T );H 1 (R d )) + . L q ((0,T );W 1,r (R d )) .
Proof. Let r := p + 1 and q defined by ( 13). The pair (q, r) is admissible pair since r ≥ 2, r is always finite and satisfies

2 ≤ r ≤ 2d d -2 , if d ≥ 3.
Following [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF], let ξ ∈ C ∞ 0 (C) be such that ξ(z) = 1 for |z| ≤ 1 and set

g 1 (z) = ξ(z)g(z), g 2 (v) = (1 -ξ(z))g(z)
Since p > 1, we have the following for C i > 0,

g 1 (z) L 2 (R d ) ≤ C z L 2 (R d ) + ξ(z) (1 + |ψ c | p-1 )(z + ψ c ) -e ict Q p c L 2 (R d ) ≤ C 1 (C 2 + z L 2 (R d ) ) ≤ C 3 z L 2 (R d )
and

g 2 (z) L r (R d ) ≤ C z L r (R d ) + ξ(z) (1 + |ψ c | p-1 )(z + ψ c ) -e ict Q p c L r (R d ) ≤ C 1 (C 2 + z p-1 L r (R d ) ) ≤ C 3 z p-1 L r (R d )
A straightforward estimation yields

g 1 (z 1 ) -g 1 (z 2 ) L 2 (R d ) ≤ C z 1 -z 2 L 2 (R d )
and using the inequality ||u| p-1 u -|v| p-1 v| ≤ C(|u| p-1 + |v| p-1 )|u -v| for p > 1 ( [START_REF] Tao | Nonlinear dispersive equations[END_REF]), Hölder and Minkowski inequalities, we deduce

g 2 (z 1 ) -g 2 (z 2 ) L r ≤ C Q p-1 c (z 1 -z 2 ) L r + C (|z 1 + ψ c | p-1 + |z 2 + ψ c | p-1 )|z 1 -z 2 | L r ≤ C z 1 -z 2 L r + C z 1 + ψ c p-1 L r + z 2 + ψ c p-1 L r z 1 -z 2 L r ≤ C z 1 p-1 L r + z 2 p-1 L r z 1 -z 2 L r .
where r denotes the conjugate of r. Moreover, using the diamagnetic inequality ( [START_REF] Tao | Nonlinear dispersive equations[END_REF]) : for all

f ∈ H 1 (R d ) ∇|f | ≤ |∇f |,
in the sense of distribution, we obtain

∇g 1 (z) L 2 (R d ) ≤ C ∇z L 2 (R d )
and

∇g 2 (z) L r (R d ) ≤ C z p-1 L r (R d ) ∇z L r (R d ) Since p ∈ (1, p * ],
we have q > 2 (recall ( 13)) and therefore from [37, Proposition 2.4] (or see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]), we have that for every admissible pair (q, r), the following holds

S L (t)ψ 0 X T ≤ C ψ 0 H 1 (R d ) , t 0 S L (t -τ )ϕ(.)h(., τ )dτ X s,T ≤ C ϕh L 1 ((0,T );H 1 (R d )) ≤ C ϕh L 2 ((0,T );H 1 (R d )) , t 0 S L (t -τ ) (g(z)) (τ )dτ X T ≤ C g 1 (z) L 1 ((0,T );H 1 (R d )) + g 2 (z) L q ((0,T );W 1,r (R d )) , t 0 S L (t -τ ) (g(v 1 ) -g(v 2 )) (τ )dτ X T ≤ C g 1 (z 1 ) -g 1 (z 2 ) L 1 ((0,T );H 1 (R d )) + g 2 (z 1 ) -g 2 (z 2 ) L q ((0,T );W 1,r (R d )) .
In a similar fashion as above, one deduces that for z 1 , z 2 ∈ X T , g 1 (z 1 ) ∈ L ∞ ((0, T ); H 1 (R d )) and g 2 (z 1 ) ∈ L q ((0, T ); W 1,r (R d )) with the following estimates

g 1 (z 1 ) L ∞ ((0,T );H 1 (R d )) ≤ C z 1 L ∞ ((0,T );H 1 (R d )) g 2 (z 1 ) L q ((0,T );W 1,r (R d )) ≤ C z 1 p-1 L ∞ ((0,T );L 1,r (R d )) z 1 L q ((0,T );W 1,r (R d )) g 1 (z 1 ) -g 1 (z 2 ) L ∞ (H 1 ) ≤ C z 1 L ∞ (H 1 ) + z 2 L ∞ (H 1 ) z 1 -z 2 L ∞ (H 1 ) g 2 (z 1 ) -g 2 (z 2 ) L q (W 1,r ) ≤ C z 1 p-1 L ∞ (L 1,r ) + z 2 p-1 L ∞ (L 1,r ) z 1 -z 2 L ∞ (W 1,r )
Consequently,

g 1 (z) L 1 ((0,T );H 1 (R d )) + g 2 (z) L q ((0,T );W 1,r (R d )) ≤ C(1 + z X T ) z p-1 X T and g 1 (z 1 ) -g 1 (z 2 ) L 1 ((0,T );H 1 (R d )) + g 2 (z 1 ) -g 2 (z 2 ) L q ((0,T );W 1,r (R d )) ≤ C z 1 X T + z 2 X T + z 1 p-1 X T + z 2 p-1 X T z 1 -z 2 X T ,
that is, the desired estimates.

We finally obtain the local exact controllability of [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF] around smooth trajectories of (1) stated in Theorem 3 by taking the trace of the solution of ( 25) as the control. The well-posedness and the properties of the control operator G ensures that the solution ψ defined this way belongs to C([0, T ]; H 1 (R d )).

Properties of the blow-up trajectory

We present here the properties of the blow-up trajectory in the mass critical and mass slightly subcritical regime that shall be needed for the proof of Theorem 5. We begin by recalling the blow-up profile close in Ḣ1 (R d ) to the ground state in the mass slightly supercritical regime obtained in [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]. Using this construction, we prove first in Theorem 11 that this construction yields a blow-up profile close in Ḣ1 (R d ) to the ground state in the slightly subcritical regime. The second part of the proof of Theorem 11 lies in the proof that the blow-up profile is close to the ground state in H 1 loc (R d ). This closeness to the ground state shall be used in the proof of Theorem 5 to drive the initial or final data to the blow-up trajectory using the local controllability around the ground state. We recall that one cannot expect the blow-up profile to be close to the ground state in H 1 (R d ) as the blow-up profile does not belong to L 2 (R d ) [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF].

Blow-up profile in the slightly mass supercritical regime

The blow-up profile constructed in [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF] is based on the ansatz

ψ(x, t) = 1 λ 2 p-1 (t) e iθ(t) ψ x λ(t) , ( 29
)
where ψ is assumed to be radially symmetric, ψ(x) = ψ(|x|) = ψ(r), r := |x| and where, for a given T ∈ R and b > 0, λ(t) and θ(t) are defined for t ∈ [0, T ) by

λ(t) := 2b(T -t), θ(t) := - log(T -t) 2b . (30)
The parameter b, that shall be small, is used as a bifurcation parameter. Indeed, for ψ defined by ( 29) to be a solution to [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF], ψ has to satisfy

∆ ψ -ψ + ib 2 p -1 ψ + x.∇ψ + | ψ| p-1 ψ = 0, x ∈ R d . (31)
Notice that taking b = 0 in ( 31) is equivalent to (3), which makes the bifurcation parameter b apparent. Bahri, Martel and Raphaël are able to give a precise description of the finite energy self-similar blow-up according to the ansatz [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF].

Theorem 10. [1, Theorem 1] Let d ≥ 1, 0 < s c
1 and p * = 1 + 4/d be the mass critical exponent. There exists > 0 such that for any p satisfying 0 < p -p * < , there exists b(p) > 0 and a non-zero radially symmetric solution ψ to [START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF]

such that ψ ∈ Ḣ1 (R d ) ∩ C 2 (R d ), E( ψ) = 0.
Moreover, as p → p * + , the following hold

s c = κ 2 N c b sc exp - π b sc , N c = ∞ 0 Q 2 (r)r d-1 dr, (32)
and κ is defined as in (4);

• Bifurcation from the soliton profile :

ψ -Q Ḣ1 (R d ) = o(1);
• Non-oscillatory behaviour for the outgoing wave:

lim r→∞ r 2 p-1 | ψ(r)| = ρ sc (1 + o(1)), lim sup r→∞ r p+1 p-1 | ψ (r)| < ∞,
where

ρ sc = 2N c s c .
Based on the construction of [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF], we shall prove that there exists a blow-up profile in the mass slightly subcritical case satisfying the following.

Theorem 11. Let d ≥ 1, -1
s c < 0 and p * = 1 + 4/d be the mass critical exponent. Then there exists > 0 such that for any p satisfying -< p -p * < 0 there exists b(p) > 0 and a non-zero radially symmetric solution ψ to [START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF]

such that ψ ∈ Ḣ1 (R d ) ∩ C 2 (R d ), E( ψ) = 0.
|s c | = κ 2 N c b sc exp - π b sc , N c = ∞ 0 Q 2 (r)r d-1 dr;
• Bifurcation from the soliton profile :

ψ -Q H 1 loc (R d ) = o (1)
; • Non oscillatory behaviour for the outgoing wave:

lim r→∞ r 2 p-1 | ψ(r)| = ρ sc (1 + o(1)), lim sup r→∞ r p+1 p-1 | ψ (r)| < ∞, where ρ sc = 2N c |s c |.
For the controllability purposes of Theorem 5, we shall prove that the blow-up profile satisfies

ψ -Q H 1 loc (R d ) = o(1)
. Compared to Theorem 10, this requires to prove the additional convergence

ψ -Q L 2 loc (R d ) = o(1)
. We therefore recall below the main results behind Theorem 10 to collect the estimates of the blow-up profile in the different regions.

Let |x| = r ∈ [0, ∞) and consider the change of variables

ψ(x) = exp -i br 2 4 P (r). ( 33
)
Then ψ is a radial symmetric solution to [START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF] if we are able to find P : [0, ∞) → C solution to

   P + d -1 r P + b 2 r 2 4 -1 -ibs c P + |P | p-1 P = 0, r > 0, P (0) = 0. (34)
The dependence between b and s c in ( 34) is relaxed by looking for a solution P σ of

   P + d -1 r P + b 2 r 2 4 -1 -ibσ P + |P | p-1 P = 0, r > 0, P (0) = 0, (35)
with σ > 0 small. Clearly, if P σ is solution to [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] for any σ > 0 small, then taking σ = s c yields P sc solution to [START_REF] Muñoz | On approximate controllability of generalized KdV solitons[END_REF] and therefore a radially symmetric profile ψ satisfying [START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF]. Additional parameters (σ, ρ, γ, θ) ∈ R + × R + × R × R are introduced in order to construct the profile P σ , with an a priori control with respect to σ:

b ∈ b σ - 1 2 b 13 6 σ , b σ + 1 2 b 13 6 σ , (36)
where b σ > 0 is defined by

σ = κ 2 N c b σ exp - π b σ , N c = ∞ 0 Q 2 (r)r d-1 dr, and
ρ ∈ 1 2 ρ σ , 3 2 ρ σ , γ ∈ - 1 2 γ σ , 1 2 γ σ , θ ∈ - 1 2 θ σ , 1 2 θ σ , (37)
where

ρ σ = √ 2N 1 2 c √ σ, γ σ = b 1 6 σ exp - 2 √ b σ , θ σ = b 1 6 σ exp - π b σ exp 2 √ b σ .
In particular, there exists C > 0,

exp π b - π b σ -1 ≤ C π b - π b σ ≤ Cb 1 6 σ , σ ≤ Cb -1 exp - π b . (38)
These parameters allows are used to define the solutions of (35) over three intervals,

K := [0, r K ], J = [r K , r I ] and I = [r I , ∞), where r K = b -1/2 and r I = b -2 .
The three intervals take into account three different dynamic of the solution near the ground state : in K the non-linearity plays an important role, while in I, the ground state is exponentially small and therefore the equation is essentially linear. The interval J includes the so-called turning point, where the real part of the zeroth order operator vanishes2 . In the proof of Theorem 10, the solution in I is extended to J, and is denoted P ext . It is then matched to the solution P int in K at r = r K to yield the profile P σ .

4.1.1 Solution P int of (35) in K

In K, the solution to P int of (35) on K satisfies the following Proposition 7 (Proposition 4.1, [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]). For σ > 0 small enough and for any b, γ satisfying (36) and (37), there exists a solution 4 )), 4 )),

P int = P int [σ, b, γ] of (35) on K satisfying Re(P int (r K )) = κb d-1 4 exp - 1 √ b (1 + O(b 1 3 )) + κ A γb d-1 4 exp 1 √ b , Re(P int (r K )) = -κb d-1 4 exp - 1 √ b (1 + O(b 1 3 )) + κ A γb d-1 4 exp 1 √ b , Im(P int (r K )) = κ B σb d+3 4 exp 1 √ b (1 + O(b 1
Im(P int (r K )) = κ B σb d+3 4 exp 1 √ b (1 + O(b 1
and there exists C > 0 such that

P int -Q Ḣ1 (K) ≤ Cb 1 12 .
Moreover, the map (σ, b, γ)

→ (P int [σ, b, γ](r K ), P int [σ, b, γ](r K )) is continuous.
The solution P int is obtained from the decomposition

P = (Q + γA + φ + ) + i(bσB + φ -) (39)
by a fixed point argument on (φ + , φ -) ∈ E K where E K is the complete metric space

E K := {(φ + , φ -) : K → R 2 is continuous and satisfies (φ + , φ -) K ≤ 1} (40)
endowed with the norm

(φ + , φ -) K := max(b -1/3 N + (φ + ), b -5/4 σ -1 N -(φ -)),
where

N + (φ + ) = φ + /Q L ∞ (K) , N -(φ -) = Hφ -L ∞ (K) , H(r) = (1 + r) d-1 Q(r), r ≥ 0.
Moreover, the functions A and B are associated with the linearized operators around the ground state, defined by

L + = -∂ rr - d -1 r ∂ r + 1 -pQ p-1 , (41)
L -= -∂ rr - d -1 r ∂ r + 1 -Q p-1 , (42)
and satisfies, by collecting the results from Lemma 4.1 and Lemma 4.4 of [1],

Lemma 2 ([1]

). There exist

C 2 functions A : [0, ∞) → R and B : [0, ∞) → R solutions respectively of L + A = 0 and L -B = -Q on (0, ∞), such that A(0) = 1, A (0) = B(0) = B (0) = 0 and
A(r) = κ A r -d-1 2 e r (1 + O(r -1 )), r ∈ [1, ∞), A (r) = κ A r -d-1 2 e r (1 + O(r -1 )), r ∈ [1, ∞), B(r) = κ B r -d-1 2 e r (1 + O(r -1 )), r ∈ [1, ∞), B (r) = κ B r -d-1 2 e r (1 + O(r -1 )), r ∈ [1, ∞),
for a constant κ A = 0 and κ B = N c /(2κ) > 0.

Proposition 7 is obtained in part with the following estimates on the decomposition of P int (39) which relates to the closeness to the ground state with respect to the parameter b (see for instance eq (4.15) in [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]). We shall use these estimates for the closeness of the ground state to the blow-up profile in H 1 loc (R d ).

Lemma 3 ([1]

). There exists C > 0 such that |γA| ≤ Cb

1 6 Q, |φ + | ≤ Cb 1 3 Q, b|σB| + |φ -| ≤ C exp - π b + 2 √ b Q.
Proof. These estimates come from the a priori bounds on the parameters σ, b, γ, (4), ( 36), [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] and ( 38) that, A and B given by Lemma 2 and for every (φ + , φ -) ∈ E K , using the definition of the norm N ± . Indeed, one has for r ∈ K, since Q is positive,

φ + Q ≤ φ + Q L ∞ (K) ≤ b 1/3 ,
from the definition of N + . Moreover, from the definition A and the a priori bound on γ, we have

|γA| ≤ 1 2 γ σ |A| ≤ Cb 1/6 σ exp - 2 √ b σ r -d-1 2 e r ,
Using (4) we have κr -(d-1)/2 e -r ≤ Q(r) + Cr -(d+1)/2 e -r ,

|γA| ≤ Cb 1/6 σ exp - 2 √ b σ e 2r Q(r) + Cr -(d+1)/2 e -r ≤ Cb 1/6 σ exp 2 √ b - 2 √ b σ (Q(r) + C) ≤ Cb 1/6 σ Q(r) + 2 √ b - 2 √ b σ Q(r)
≤ Cb 1/6 σ Q(r). The bound on bσB and φ -are obtained similarly.

Solution of (35) in I ∪ J

The profile in P ext in I ∪ J is constructed in two main steps. First, the solution is defined in

I = [b -2 , ∞) by letting P (r) = r -d-1 2 U (r), (43)
where

U + b 2 r 2 4 -1 - (d -1)(d -3) 4r 2 -ibσ U + r -1 2 (d-1)(p-1) |U | p-1 U = 0 (44)
The equation ( 44) is essentially linear near the profile Q in the region I as profile Q is exponentially decreasing.

Proposition 8 (Proposition 2.1, [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]). For σ > 0 small enough and for any b, ρ satisfying (36), [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF], there exists a C 2 solution U of (44) on I satisfying

U (r) = ρr -1 2 +σ exp ib r 2 4 exp -i ln r b (1 + O(b -3 r -2 )) (45)
U (r) = i b 2 ρr 1 2 +σ exp ib r 2 4 exp -i ln r b (1 + O(b -3 r -2 )), (46)
and

U -i br 2 U = O(b -1 r -1 |U |) = O(ρb -1 r -3 2 +σ ).
Moreover, the map (σ, b, ρ)

→ (U [σ, b, ρ](b -2 ), U [σ, b, ρ](b -2 )) is continuous.
As pointed out in [1, Remark 2.3], the profile P given by [START_REF] Tao | Nonlinear dispersive equations[END_REF] in I belongs to Ḣ1 (I) but not in L 2 (I) due to the asymptotic given by ( 45), [START_REF] Zuazua | Remarks on the controllability of the Schrödinger equation[END_REF]. However, ( 45) is sufficient to recover L 2 loc (I) for the profile P given by [START_REF] Tao | Nonlinear dispersive equations[END_REF], which shall be used to deduce P -Q L 2 loc (I) = o(1). The solution U on I is then extended to I ∪ J, the region including the turning point, which complicates significantly the analysis. Proposition 9 (Proposition 3.1, [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]). For σ > 0 small enough and for any b, ρ, satisfying (36), [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF], the solution U of (44) on I constructed in Proposition 8 extends to a solution of (44) on J ∪ I. Moreover, there exists a real θ ext ∈ [0, 2π) such that the function P ext defined by

P ext (r) = e iθext r -d-1 2 U (r), r ∈ J ∪ I,
is a solution of (35) on J ∪ I and satisfies 4 )) 4 ))

Re(P ext (r K )) = ρb d+1 4 √ 2 exp π 2b - 1 √ b (1 + O(b 1
Re(P ext (r K )) = - ρb d+1 4 √ 2 exp π 2b - 1 √ b (1 + O(b 1
Im(P ext (r K )) = ρb d+1 4 2 √ 2 exp 1 √ b - π 2b (1 + O(b 1 4
))

Im(P ext (r K )) = ρb d+1 4 2 √ 2 exp 1 √ b - π 2b (1 + O(b 1 4
))

Moreover, the map (σ, b, ρ) → (P ext [σ, b, ρ](r K ), P ext [σ, b, ρ](r K )) is continuous

Matching asymptotic

The solution P int , defined over K, and P ext , defined over J ∪ I, are matched at r K using a fixed-point argument on the parameters.

Theorem 12 (Theorem 2, [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]). Let d ≥ 1 and p * < p. There exists σ 0 > 0 such that for any σ ∈ (0, σ 0 ), and p ∈ [p * , p], there exists b, ρ, γ, θ satisfying (36), [START_REF] Rosier | Exact boundary controllability of the nonlinear Schrödinger equation[END_REF] such that the solution P ext [σ, b, ρ] of (35) on I ∪ J given by Proposition 3.1 and P int [σ, b, γ] of (35) on K given by Proposition 4.1 satisfy the following matching conditions

P int (r K ) = P ext (r K ), P int (r K ) = P ext (r K ).
In particular, the function P defined on [0, ∞) by

P (r) = P int (r), r ∈ K, P int (r), r ∈ I ∪ J,
is a C 2 solution of (35) on [0, ∞) satisfying the asymptotic: for r large,

|P (r)| = |ρ|r -d 2 +σ (1 + O(b -3 r -2 )) P (r) -i br 2 P (r) = O(ρb -1 r -d 2 -1+σ )
Theorem 10 is obtained from Theorem 12 by choosing σ = s c = d 2 -2 p-1 .

Blow-up profile for the mass slightly subcritical case

We deduce Theorem 11 in the mass slightly subcritical regime -1 s c < 0.

Proof. Consider ψ according to the ansatz

ψ(x, t) = 1 λ 2 p-1 (t) e iθ(t) ψ x λ(t) , ( 47
)
where ψ is assumed radially symmetric and λ(t), θ(t) defined by [START_REF] Merle | Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation[END_REF]. Then, if ψ satisfies

∆ ψ -ψ + ib 2 p -1 ψ + x.∇ψ + | ψ| p-1 ψ = 0, x ∈ R d . (48)
Then ψ defined by (47) satisfies [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]. Now, let

ψ(x) = exp -i br 2 4 P (r).
If P is a solution to

   P + d -1 r P + b 2 r 2 4 -1 -ibs c P + |P | p-1 P = 0, r > 0, P (0) = 0. ( 49
)
then ψ is a solution to (48). The proof of Theorem 10 is obtained from Proposition 7, Proposition 8 and Proposition 9 by relaxing the dependence of (49) with respect to s c > 0 by considering the family of equations,

   P + d -1 r P + b 2 r 2 4 -1 -ibσ P + |P | p-1 P = 0, r > 0, P (0) = 0. (50)
with respect to the parameter σ > 0. To prove that the conclusion of Theorem 10 holds in the mass slightly subcritical case -1 s c < 0, we seek a profile P solution to

   P + d -1 r P + b 2 r 2 4 -1 + ibσ P + |P | p-1 P = 0, r > 0, P (0) = 0. (51)
where the sign of ibσP is chosen so that b > 0 and σ > 0 (recall that in the mass subcritical case, Im(-ibs c ) > 0). Therefore, if P denotes the solution of (50) in the mass supercritical case, then P is solution to (51). Indeed, taking the complex conjugate of (51) yields, [START_REF] Zuazua | Remarks on the controllability of the Schrödinger equation[END_REF] of Proposition 8, we have, using again (4)

   P + d -1 r P + b 2 r 2 4 -1 + ibσ P + |P | p-1 P = 0, r > 0, P (
-Q L 2 loc (R d ) = o(1). Let C be a compact subset of R d . Denote first C K = B r K (0) ∩ C, where B r K (0) is the ball of R d centered
ψ -Q 2 L 2 (C I∪J ) ≤ ψ -Q 2 L 2 (Br C (0)\Br K (0)) ≤ r K ≤r≤r C |P (r) -Q(r)| 2 r d-1 dr + r K ≤r≤r C 1 -exp -i br 2 4 Q(r) 2 r d-1 dr
Dealing with the second integral as for (52), we deduce

r K ≤r≤r C 1 -exp -i br 2 4 Q(r) 2 r d-1 dr ≤ Ce -b -1/2 .
We are left with the first integral. We use the exponential decay of the ground state given by ( 45) to obtain

r K ≤r≤r C |P (r) -Q(r)| 2 r d-1 dr ≤ r K ≤r≤r C r -(d-1)/2 U (r) -κr -(d-1)/2 e -r 2 r d-1 dr + C r K ≤r≤r C e -2r r -1 dr ≤ r K ≤r≤r C |U (r)| 2 dr + κ r K ≤r≤r C e -2r dr + C r K ≤r≤r C e -2r r -1 dr.
The last two integrals are bounded by Ce -2b -1/2 . It remains to bound the first integral. In order to do so, we use (52), and the bound on ρ with respect to b given by ( 37) and (38) to deduce

r K ≤r≤r C |U (r)| 2 dr ≤(r C -b -1/2 ) U 2 L ∞ (r K ≤r≤r C ) ≤|r C -b -1/2 |b -5 e -π/b
which allows to finally prove that ψ -

Q L 2 (C I∪J ) → 0 as b → 0. Combining this fact with ψ -Q Ḣ1 (R d ) → 0 as b → 0, we finally deduce ψ -Q H 1 loc (R d )
→ 0 as b → 0, which ends the proof.

Controllability to the blow-up trajectory

We are now in position to prove Theorem 2 and Theorem 5. We begin by the proof of Theorem 5.

Proof. Let d ≥ 1 and > 0. Consider for now p ∈ [1, p * ). Denote ψ b the blow-up profile

ψ b (x, t) = 1 λ(t) 2 p-1 e iθ(t) ψ x λ(t) .
where ψ is given by Theorem 11, where

λ(t) = 2b(T -t), θ(t) = - log(T -t) 2b , t < T,
and where T > 0 is fixed the following way,

T -= 1 2bc 1 , T -T c 1 ,c 2 + = 1 2bc 2 . (53)
Hence,

ψ b (x, ) = e iθ( ) c 1 p-1 1 ψ( √ c 1 x), ψ b (x, T c 1 ,c 2 -) = e iθ(Tc 1 ,c 2 -) c 1 p-1 2 ψ( √ c 1 x). (54)
For p ∈ (1, p * ], denote δ 1 (p), δ 2 (p) ∈ R + such that Theorem 3 apply in time T = for the smooth trajectory ψ c 1 (x, t) = e itc 1 Q c 1 (x) and ψ c 2 (x, t) = e itc 2 Q c 2 (x) respectively. Define δ := min p∈[1,p * ] {δ 1 (p), δ 2 (p)}. We easily see that δ > 0 since Theorem 3 holds for p ∈ [1, p * ]. We fix p ∈ (1, p * ) such that, ∀p ∈ (p, p * ),

Q c 1 -c 1 p-1 1 ψ( √ c 1 x) H 1 (Ω) < δ, Q c 2 -c 1 p-1 2 ψ( √ c 2 x) H 1 (Ω) < δ. ( 55
)
This is always possible since Q -ψ ψ T -e iγ Q c 2 H 1 (Ω) .

We are now in position to prove Theorem 2. First, we need to adjust the phase of the blow-up profile as well as the initial and final data. Using the phase invariance of the solutions to (1), we define the trajectory ψ traj (x, t) = e i(c 1 +γ 1 -θ( )) ψ b (x, t).

By construction, using (54), ( 55) and ( 57), e i(c 1 +γ 1 ) Q c 1 -ψ traj (., ) H 1 (Ω) < δ.

Therefore, we can apply Theorem 3 in time T = around the smooth trajectory e i(c 1 t+γ 1 ) Q c 1 . This implies that the solution to [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF], denoted ψ 1 , satisfies ψ 1 (., 0) = ψ 0 and ψ 1 (., ) = ψ traj (., ).

We denote the associated control v 1 .

During the time interval ( , T c 1 ,c 2 -) we define ψ 2 (., t) = ψ traj (., t). Then, using again (54), ( 55) and (57), we apply Theorem 3 to drive the solution to [START_REF] Fibich | Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation[END_REF], denoted ψ 3 , from ψ 3 (., T c 1 ,c 2 -) = ψ traj (., T c 1 ,c 2 -) to ψ 3 (., T c 1 ,c 2 ) = e i(c 1 +γ 1 +θ(Tc 1 ,c 2 -)-θ( )) Q c 2 . Now, define the smallest T γ 2 ≥ 0 such that e ic 2 Tγ 2 e i(c 1 +γ 1 +θ(Tc 1 ,c 2 -)-θ( )) = e iγ 2 , and define ψ 4 (., t) = e i(c 1 +γ 1 +θ(Tc 1 ,c 2 -)-θ( )) e ic 2 (t-Tc 1 ,c 2 ) Q c 2 . We therefore have ψ 3 (., T c 1 ,c 2 ) = ψ 4 (., T c 1 ,c 2 ) and ψ 4 (., T c 1 ,c 2 + T γ 2 ) = e iγ 2 Q c 2 . We can finally apply Theorem 3 one last time to deduce that there exists a solution ψ 5 to (11) such that ψ 5 (., T c 1 ,c 2 + T γ 2 ) = ψ 4 (., T c 1 ,c 2 + T γ 2 ) = e iγ 2 Q c 2 and ψ 5 (., T c 1 ,c 2 + T γ 2 + ) = ψ T .

Hence, the solution We now turn to the proof of Theorem 2.

ψ(x, t) =                ψ 1 (x,
Proof. The proof of Theorem 2 is very similar to the proof above. Indeed, the only modification needed in the proof is the profile to go from one scaled ground state from another. In order to do so, we use the simple remark that the solution to (1) with the initial data ψ 0 = (1 + )Q λ satisfies the requirement for > 0 sufficiently small. Indeed, we have ψ 0 ∈ B α * using that the L 2 (R d )-norm is invariant under the scaling in the mass critical case. Moreover, using Pohozaev's identity E(Q λ ) = λE(Q) = 0, < 0

E(ψ 0 ) = (1 + ) 2 2 |∇Q λ | 2 - ( 1
Therefore the solution ψ to (1) starting from ψ 0 = (1 + )Q λ blow-up in finite time and belongs to C([0, T ); H 1 (R d )). We use the exact local controllability to reach this blow-up profile from the initial and to the final data in arbitrarly small time.

Conclusion

In lights of the results presented here, we obtained the controllability of initial and final states close to ground state solitary waves with different scaling. In some sense, this strategy is close to the return method, as a trajectory with "good control properties" [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF] was used to connect two different states. However, the small-time global controllability of ( 11) remains an open question.

Moreover, using the results presented here, we are able to address the question of finite time blow-up in the mass subcritical case on bounded domain. Indeed, consider

     iψ t + ∆ψ + |ψ| p-1 ψ = 0, (x, t) ∈ Ω × R + ψ(x, t) = v(x, t), (x, t) ∈ ∂Ω × R + ψ(x, 0) = ψ 0 , x ∈ Ω, (58)
with ψ 0 ∈ H 1 (Ω) and v ∈ L 2 (R + ; H 1/2 -(∂Ω)). Then Theorem 13. Let Ω ⊂ R d be a bounded domain and p < p * for 0 < p * -p 1. Then there exists ψ 0 ∈ H 1 (Ω) close to Q in H 1 (Ω) and v ∈ L 2 (R + ; H 1/2 -(∂Ω)) such that the solution blow-up in finite time with

∇ψ L 2 (Ω) 1 b(T -t) , t → T -.
The framework given by (58) allows to properly define blow-up solutions in H 1 (Ω) in the mass subcritical regime, that is, without having solutions not belonging to L 2 (|x| > R) for R > 0 large. We highlight that Theorem 13 is close to the blow-up phenomenon exhibited in for (58) in the mass critical case with v ≡ 0 where the ground state was defined as the unique positive steady state.

Theorem 4 .

 4 Let d ≥ 1 and Ω ⊂ R d be an open, connected and bounded domain with a boundary ∂Ω of class C 2 . Let c ∈ R + . For d ≥ 5, assume that the spectral property ([29]) holds. Then there exists > 0 such that, for every

Theorem 5 .

 5 Let d ≥ 1 and Ω ⊂ R d be an open, connected and bounded domain with a boundary

•

  Law for the nonlinear eigenvalue : b = b sc (1 + o(1)) where 0 < b sc 1 is defined by

  Moreover, as p → p * -, the following hold • Law for the nonlinear eigenvalue : b = b sc (1 + o(1)) where 0 < b sc 1 is defined by

  0) = 0. with b > 0 and σ > 0. Hence, Proposition 7, Proposition 8 and Proposition 9 hold, and taking σ = s c = d 2 -2 p-1 , the conclusion of Theorem 10 holds in the mass slightly subcritical case. To conclude the proof of Theorem 11, it remains to prove ψ

H 1 loc

 1 (R d ) → 0 as b → 0. Now consider ψ 0 , ψ T ∈ H 1 (Ω) such that min γ∈[0,2π] ψ 0 -e iγ Q c 1 H 1 (Ω) < δ, min γ∈[0,2π] ψ T -e iγ Q c 2 H 1 (Ω) < δ,(56)and denoteγ 1 = arg min γ∈[0,2π] ψ 0 -e iγ Q c 1 H 1 (Ω) , γ 2 = arg min γ∈[0,2π]

  d ≥ 1 an open, connected and bounded domain with a boundary ∂Ω of class C 2 .

	The boundary control v is imposed either on the Dirichlet or Neumann boundary condition,
	ψ(x, t) = v(x, t), (x, t) ∈ ∂Ω × (0, T ), or	∂ψ ∂ν	(x, t) = v(x, t), (x, t) ∈ ∂Ω × (0, T ), (8)

  2 (∂Ω)) is the boundary control. We first state the exact local controllability around ground state with different scaling in the mass critical regime p -1 = 4/d. This result covers the physically relevant case p = 3 in dimension d = 2. Theorem 2. Let d ≥ 1 and Ω ⊂ R d be an open, connected and bounded with a boundary ∂Ω of class C 2 . Let c 1 , c 2 ∈ R + and c 1 < c 2 . For d ≥ 5, assume that the spectral property ([29]) holds.Then there exists > 0 such that, for every ψ 0 , ψ 1 ∈ H 1 (Ω) such that

	inf
	γ∈[0,2π]

  t), t ∈ (0, ),ψ 2 (x, t), t ∈ ( , T c 1 ,c 2 -), ψ 3 (x, t), t ∈ (T c 1 ,c 2 -, T c 1 ,c 2 ), ψ 4 (x, t), t ∈ (T c 1 ,c 2 , T c 1 ,c 2 + T γ 2 ), ψ 5 (x, t), t ∈ (T c 1 ,c 2 + T γ 2 , T c 1 ,c 2 + T γ 2 + ), belongs to C([0, T c 1 ,c 2 + T γ 2 + ]; H 1 (Ω)). By a classical trace theorem, the trace of ψ provides the definition of the control v ∈ C([0, T c 1 ,c 2 + T γ 2 + ]; H 1/2 -(∂Ω)).

Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France. E-mail : ludovick.gagnon@inria.fr.

it is in fact on[START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF] that this turning point is seen, and it depends on the dimension d (see[START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF][START_REF] Godet | Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces[END_REF])

Acknowledgements

The author would like to thanks Pierre Raphaël for the enlightening discussions and advices.

Well-posedness in the different regimes in R d

The ground state Q plays an important role in the description of the global well-posedness of [START_REF] Bahri | Self-similar blow-up profiles for slightly supercritical nonlinear schrödinger equations[END_REF]. The local well-posedness of (1) with initial data ψ 0 ∈ H 1 (R d ) is established by the Cauchy theory if the energy is subcritical for d ≥ 3, and without any restriction on p > 1 at the origin and of radius r K = b -1/2 . We have,

We divide the region of integration of the first integral,

Then, on one hand

On the other hand, using the exponential decay of the ground state (4)

(52)

To deal with the profile P , we use the profile decomposition of P in K, the bounds of Lemma 3. Indeed, there exists (φ + , φ -) ∈ E K (recall [START_REF] Soffer | Soliton dynamics and scattering[END_REF]) such that

and such that Lemma 3 holds. Then,

Hence, we deduce ψ -Q L 2 (C K ) → 0 as b → 0. Second, we deal with C I∪J := C \ C K . Denote r C > 0, the radius of B r C (0) such that C ⊂ B r C (0). Since the profile P int on I ∪ J satisfies (45),


M Frewer

published or not. The documents may come

Q&A, review comments and final remarks to "Symmetry induced turbulent scaling laws for arbitrary moments and their validation with DNS and experimental data: A fact-check"

Comments, explanations and remarks

In the following all text that appears in quoted italics is the feedback or comments I received. The text in normal font is my response.

1.1. "An interesting point you make is this: 'Infinite things which do not converge, irrespective of their representation, cannot be considered as closed.' Here, I am not sure. It depends. Consider a standard Taylor expansion, for example 1/(1 + x 2 ) = 1 -x 2 + x 4 -x 6 + • • • . It diverges for x > 1. However, if all the terms in the expansion are known, and the function is known to be analytic, then the function is known. In this particular example it can even be calculated using this series in combination with the accelerating convergence trick dating back to Euler. Of course, e -1/x 2 + 1/(1 + x 2 ) has the same asymptotic expansion, which does not represent the exact function however many terms are known. Coming back to the point you are making, it is not impossible that moment expansions encode the full information about the statistics of a turbulent flow in spite of being divergent."

When I prepared the Fact-Check, I was aware of the problem that my statement 'Infinite things which do not converge, irrespective of their representation, cannot be considered as closed' could be interpreted differently than what I actually want to say with it. I know that such a strong statement should be accompanied with additional explanations and demonstrating examples, but the problem I had on the other hand was that I wanted to keep the Fact-Check concise and not lose focus on the more obvious problems the Webinar has.

Here are my arguments why I currently think that my statement is valid and why this statement applies to turbulence in particular. Crucial hereby is the part 'irrespective of their representation', which I highlighted in the Fact-Check. If we take your example, 1-x 2 +x 4 -x 6 +• • • , it can be seen as one particular representation of an abstract mathematical object that we want to consider, let's call it F (x), while 1/(1+x 2 ) is another, different representation of that same object F (x). However, what the entire object F exactly is, I can not tell you or write it down in any closed form. At this stage, I only know two different representations of this F .

The first (Taylor) representation has the problem that it does not converge for any real numbers |x| ≥ 1. This problem is fixed through the second representation 1/(1 + x 2 ), which is an analytic continuation of the first representation on its ill-defined domain of real numbers, which finally zips all infinite terms into a single expression. As a result, we now gained a special type of 'convergence' on this domain, in that we can identify, for example, for the specification x = 2, the divergent series 1-

2 2 +2 4 -2 6 +• • • as the closed term 1/(1+2 2 ) = 1/5, which is F (2).
And this identification can now be done for all numbers on the real line. We therefore can now say which finite values F will take on the real number line. Alternatively, one can say that the lower-level Taylor representation already encodes this information of F through its divergent series. One just has to change the representation to a higher algebraic level in order to get a finite and therefore measurable value.

The question now is, do we fully know what F is in its entirety? I don't think so, because the second, higher-level representation of F , namely 1/(1 + x 2 ), has a divergent problem too, namely for x = i and x = -i. So, maybe there exists a third representation on an even higher algebraic level, which can fix this problem and offers us a finite value, e.g., for

F (x = i), which is encoded in the divergent series 1 -(i) 2 + (i) 4 -(i) 6 + • • • = 1 + 1 + 1 + 1 + • • • of the first representation.
So, what I want to say here is that everything depends on the representation, whether an infinite process can be identified as something finite, which in the end can then be measured with a physical apparatus.

In Quantum Field Theory, for example, we deal with an infinite series S = S 1 + S 2 + • • • of Feynman diagrams S i , to determine a probability amplitude S. Some of these diagrams, i.e., some of the terms in the infinite series itself are divergent and need to be renormalized, which constitutes a first change in representation. Then, when all problematic diagrams have been renormalized, it leads to a new representation of S, call it S = T 1 + T 2 + T 3 + • • • , which still is an infinite series, but now consisting of finite terms, the renormalized diagrams T i . But in order to get a finite value of the probability amplitude S, the problem is still not solved yet, because the infinite series with the finite terms T i still diverges. Hence, one needs a second representation of S to get a finite value for it, e.g., through the method of Padé approximants, e.g. as explained in Bender & Orszag's book 'Advanced Mathematical Methods' .

Back to my statement 'Infinite things which do not converge, irrespective of their representation, cannot be considered as closed': It's clear now that this statement should be understood in the above sense, namely that when no finite representation for an infinite process can be found, then one cannot consider it as closed. Now, the abstract mathematical object which is being considered in the Webinar, namely the statistical Navier-Stokes entity, which I will call N , is way more complicated than the F (your example) or the S (QFT) considered above. The big difference is that while each of the single terms in the lowest-level representation of F and S are already in a closed form themselves, this is not the case for the object N , because there the terms are not some 'simple algebraic' terms, but are partial differential equations, which are coupled from one order to the next.

Hence, this is uncomparably more difficult than the two examples F and S discussed before. Even if we would change the representation here, and go in N from the momentformulation to the higher-level PDF-formulation, the problem remains and nothing is gained.

Instead of partial differential equations, we then face an infinite hierarchy of coupled integrodifferential equations. Thus the infinity problem remains, even when changing the representation, and therefore such systems cannot be considered as closed.

Back to your statement that the 'moment expansions encode the full information about the statistics of a turbulent flow', I see it sceptical also from another viewpoint. It's known that from the knowledge of the moments alone, irrespective to which order but finite, it is a challenging inverse problem to construct the associated PDF, where in the end there is even no guarantee that if a particular PDF-solution is found that it will also be unique. Additional physical constraints, e.g. as the maximum entropy approach, has to be placed to satisfactorily solve such a problem.

The aim is clear here: To get the PDF is more essential than only to have the moments, if one pursues to extract more detailed statistical information from a dynamical system. In this sense, I do not concur that the moment expansions encode the full information about the statistics of a turbulent flow. Said differently, in the above context of representations: The moment expansion is only the lowest-level representation of the abstract 'statistical-Navier-Stokes-object'. To get more information one has to change the representation to a higher level, but since this change does not bring along any improvement to the infinity problem, I can only conclude the following:

While the infinite PDF hierarchy encodes more information about the statistics of a turbulent flow than the infinite moment hierarchy, they both are unclosed, despite their infinite length. A reason is that the change of representation from the lower-level momentformulation to the higher-level PDF-formulation does not fix or improve the infinity problem. Also when changing the representation to the Hopf or Liouville equation, which zips the infinite hierarchy of equations into one functional equation through a continuum limit, nothing is gained in my opinion. Because without some form of coarse-graining, the Hopf equation is not a true statistical equation. Only by coarse-graining, useful statistical information can be extracted from the Hopf equation. But when coarse-graining this equation, it will just throw us back again to the infinite PDF hierarchy where we started from. As Orszag correctly phrased the problem: "From a mathematical point of view, turbulence theory is embedded in the theory of approximation of functional-differential equations" [p.363 in 'Analytical theories of turbulence', S.A. Orszag, JFM 1970 ].

• "Overall, while I am not sure that the moment expansion does not really work, I think it is worth trying. As for it being a success, yes, I was not fully convinced, and your points clarify the situation. We will see."

"

To the point about dissipation anomaly: Of course, the 1/Re → 0 limit is singular, and this indeed can lead to the symmetries of the Euler equations being not applicable to finite Re even approximately for arbitrary large but finite Re. You have a point here, yes. Mathematically, what is shown in the Webinar is based on a critical assumption, which is not proven." 1.3. "I agree. Something is flawed with the derivation of the scaling law for H i {n} beyond first order n = 1. It has to be in the assumptions, having performed a symmetry analysis that most probably is not compatible to the shear flow conditions posed."

or: "The [scaling] constant implied by the i = 1 profiles does not match the constant implied by the i = 3 profiles, as they are of opposite sign (at least over the Re range covered by our experiments, up to Re τ = 14000). ... The proposed symmetry solution cannot describe both the i = 1 and i = 3 cases with the same constant for our Reynolds number range."

If the algebraic scaling law (L):

H + i {n} = c i {n} (x + 2 + A + ) α•n+β + b i {n} , n ≥ 2
, would have been declared as a phenomenological result for shear flows that can be well matched to data in the streamwise direction i = 1, there would be no objection.

The problem here, however, is that in the Webinar this scaling law (L) is claimed as a first-principle solution of the statistical Navier-Stokes equations, since the full infinite set of MPC equations was allegedly considered to derive it -see again the last point of the Webinar's conclusion at minute 46 , how the emphasise is put on the word 'solution'.

The problem is compounded further by the fact that this 'solution' was derived specifically for a shear-flow configuration to be valid for arbitrary velocity correlations, and not only for the streamwise direction i = 1. Despite this shear-flow setting, the symmetry-induced scaling law (L) nevertheless resulted in a scaling exponent that is isotropic, i.e., independent of the velocity direction i. The component dependence only enters in trivial normalization and shift constants (c and b), but not in the essential scaling exponent, which only depends (linearly) on n, the order of the correlation, but not on i, the direction of the correlation. It is more than clear that such a result (L) can not be a first-principle solution of the statistical Navier-Stokes equations for turbulent shear flows, as it falsely will predict that all directionally different velocity correlations, for a fixed correlation order n, will all universally scale the same (up to a normalization and shift constant). So, from a theoretical point of view something is seriously flawed here.

Here is the correct take of the scaling law (L): It is definitely not a first-principle solution of the Navier-Stokes equation, as falsely propagated. Although it was somehow 'analytically derived', it nevertheless is a result based on several severe assumptions, which unfortunately were not made explicit in the Webinar.

The scaling law (L) is as far from the Navier-Stokes equation as any other invariant scaling law obtained by the classical invariance method as first used by Kármán and Prandtl a century ago. That the scaling law (L) was derived by a Lie-group symmetry analysis is just eye-candy and of no theoretical use, because when applying the mathematical tool of Lie-groups in turbulence correctly and properly, any thinkable invariant scaling law can be derived. Simply said, the closure problem of turbulence cannot be bypassed when using this method. It only shifts the closure problem of equations to a closure problem of symmetries. Without prior modelling or physical assumptions in the analysis, the Lie-group method alone does not give any information as how turbulence should scale.

Briefly summarized, the bigger picture for this failure is that the Lie-group symmetry method in turbulence is not free of assumptions. It is an ad-hoc method too, not in the same but in a similar way as the classical self-similarity method as first used by Kármán and Prandtl: Instead of using an a priori set of scales, the Lie-group method has to make use of an a priori set of symmetries, namely to select the correct relevant symmetries from an infinite (unclosed) set. Even when including all unclosed higher order correlation equations, one still gets an infinite and therefore unclosed set of functionally independent invariances if the analysis is properly performed, and not only those few as always reported. In other words, the Lie-group method in turbulence is effectively no different to the classical invariance method of Kármán and Prandtl.

Hence, the scaling law (L) is definitely not a result that directly follows from theory and is therefore also nothing special. Instead it's based on trial-and-error assumptions in choosing the 'correct symmetries' from an infinite (unclosed) set of invariant possibilities that the infinite (unclosed) MPC hierarchy is admitting. Important to note here is the fact that due to the arbitrariness involved when making a particular choice from an infinite (unclosed) set of possible symmetries, there is a high chance that one will select a nonphysical symmetry which is not reflected by experiment or numerical simulation. This clearly is the case for (L), since it is based on two nonphysical symmetries, as revealed and explained in the Fact-Check.

And therefore, it's not surprising why the scaling law (L) is not working for different velocity components 'i': Simply the wrong symmetries have been chosen for shear flow. For the quest to get better symmetry-induced results, different and especially physical symmetries have to be found by trial and error from the infinite (unclosed) set of invariant possibilities.

Finally, to counter the argument that no one is claiming a result as 'first principle solutions' when applying the method of Lie-group symmetry analysis to turbulence, here a few links to corroborate my claim about misleading statements made so far, e.g., as claiming to have derived "first principle solutions", "solutions without any assumptions", or that "no intuition is needed to find self-similar solutions", or even that with Lie-groups "the closure problem is somewhat bypassed" , etc.: JFM, 2001 : "The theory is fully algorithmic and no intuition is needed to find a selfsimilar mean velocity profile." [p.321],

or: "In the case of the logarithmic law of the wall, the scaling with the distance from the wall arises as a result of the analysis and has not been assumed in the derivation." [p.299] JFM, 2014 : "It was shown that they are exact solutions of symmetry invariant type of the infinite-dimensional series of MPC equations. They [Oberlack et al.] have shown that turbulent scaling laws may be generated from first principle and that most of the classical and new symmetry invariant solutions are based on one or several of the newly discovered statistical symmetry groups." [p.102] Mech. Eng. Rev., 2015 : "In fact, it was demonstrated that it is exactly these symmetries which are essentially needed to validate certain classical scaling laws such as the log-law from first principles and also to derive a large set of new scaling laws." [p.68] JFM, 2020 : "As noted earlier, this is particularly important in fluid mechanics and turbulence studies, as we can construct invariant solutions from first principles through symmetry." [p.5] 1.4. Is it a surprising result, that for the full-field correlations H i {n} the power-law matching of (L) to channel and pipe flow data works so well in the streamwise direction (i = 1)?

No, it is not a surprising result. In fact, it is a trivial result, because it's a direct consequence of the fact that for channel and pipe flow we approximately have H 1 {n} := U n 1 ≈ U 1 n , due to a strong mean field that washes out all streamwise fluctuations. Once the lowest-order cases n = 1 and n = 2 are matched, all higher orders will follow suit, because in the end it's just the exponentiated mean velocity U 1 n that is being matched. Said differently, if the matching to n = {1, 2} turns out to be good, then, automatically, the matching for all higher orders will be good too (relative to the matching error of n ≤ 2). Therefore, it is no surprise that the power law slope is linearly proportional to 'n' as shown in the Webinar, for both channel and pipe flow, simply because by definition the exponent of the exponentiated mean velocity U 1 n is linear in 'n', too. The interesting and difficult physics lies in the fluctuation correlations and not in the full field correlations, which in strong channel or pipe flow are trivial to predict once the mean field is known or has been matched to data. Also, in my opinion, I do not see a physical motivation why to consider the full-field correlations in the first place. As a theorist, I'm only used to work with correlations where the background or the bulk motion has been subtracted, as is also usual practice in statistical physics in general. If possible, one should always co-move with the phenomenon being considered. Because in statistical physics the finding insight or the gain in knowledge lies in the fluctuation correlations.

1.5. "Regarding determining the fluctuation correlations (R) based on the fits to the full-field correlations (H): This process is essentially ill-conditioned. Any degree of uncertainty in the measured values of U will dramatically alter the resulting predicted values of R. With perfect data, it should of course be possible to compute R from the fits of H profiles, but this is well beyond the reach of our experimental data.

In the Fact-Check it was tested with 'perfect data', namely with DNS data, and it's not working! As explained therein, this is a clear indication that the symmetry-induced transition from H to R, as put forward in the Webinar, is nonphysical. The finding is that the higher the order, the greater the error, which in turn grows exponentially with the order.

However, the strongest argument of all that we are dealing with a nonphysical invariance transformation, is the fact that (i) the scaling error will be of the exact same (exponential) order of magnitude when matching the fluctuation correlations R also directly (!) to the data, and (ii) if we would remove all nonphysical invariances or put them to zero, then the scaling theory proposed in the Webinar just reduces back to the classical dimensional scaling method, with the result that the matching process for R then improves by several orders of magnitude again. These two facts alone are already clear proof that the symmetry-based scaling method as presented in the Webinar is flawed (for more details, see e.g. the extensive discussion in Sec.5 in arXiv:1412.3061 ).

1.6. The bottom line is that the Webinar makes use of two nonphysical invariants, which not only generate the scalings in the wrong way, but also give a false picture of the intermittency and the non-Gaussian behavior in turbulence.

Q&A

Also worth taking a look is at the Q&A round of the Webinar, where the participants asked some important questions. Here are the correct answers to their questions: Q1: How to translate from the H-moments to the physically interesting R-moments? A1: Although there exists an analytical bijective mapping between H and R, this mapping leads to poor results when constructing R from a best-fit of H, where the results get more worse (exponentially worse) the higher the order of the moments are. The reason for this failure is that a best-fit for H does not imply a best-fit for R when using this map.

But, as was explained and discussed in the Fact-Check, and again in Sec.1.5 above, the actual problem is bigger than that, because at a certain order, the R cannot even be matched directly to the data anymore. For ZPG-boundary-layer flow, for example, this failure already starts at the third moment -for details see Sec.5 in arXiv:1412.3061 ). This shows that the underlying invariances are indeed nonphysical. Therein we even show that if these nonphysical invariances are removed or put to zero, then the proposed scaling theory just reduces back to the classical dimensional scaling method, with the result that the matching process for R then improves by several orders of magnitude again. A clear proof that the proposed scaling method in the Webinar is flawed.

Q2: What about the scaling factor of the intermittency invariance, and the subsequent discussion on the intermittency invariance itself ? A2: First of all, this in the Webinar so-called 'intermittency invariance' is nonphysical in that it violates several physical principles -see the explanations to statements #9 and #8 in the Fact-Check. Even if we would assume for a minute that this invariance is physical, then such an invariance surely cannot and does not reflect intermittency in turbulence. Besides of not knowing which kind of intermittency this invariance actually should describe (transitional large-scale, or fully-developed small-scale), it is an isotropic, global and for all orders of the moments universal invariance, which surely is not a characteristic of intermittency, irrespective of whether large-scale or small-scale. Hence the intermittency interpretation of this invariance is highly misleading -see also the explanation to statement #10.

Also from a pure phenomenological viewpoint, it's abundantly clear that intermittency is a symmetry breaking phenomenon and not a symmetry-existing or symmetry-preserving one (Frisch, 1996 ). Even if we would wrongly assume this to be the case, intermittency is definitely not described or featured by any global scaling symmetry, particularly not by the one given in the Webinar, which, on top of that, is a symmetry that also violates the classical principle of cause and effect. In fact, spatial-global symmetries, i.e., symmetries with constant group parameters, as all those shown in the Webinar, are not able to describe intermittent phenomena; at most, these can only be described by local space-dependent symmetries whose group parameters depend on the space coordinates. In order to describe intermittent phenomena, regardless of whether they are on a small or large scale, multifractal scaling approaches are necessary (see e.g. Frisch, 1996 ). A global and universal scaling ansatz, as presented in the Webinar, is therefore definitely not sufficient.

Q3: How to connect the symmetry-induced scaling theory to turbulence modelling? A3: There is no doubt that in order to improve the predictive quality of a turbulence model for a particular flow configuration, it should reflect as many symmetries as the statistical Navier-Stokes equations for that flow configuration can physically admit. All nonphysical symmetries which potentially can also be admitted by this system should, of course, not be included. The presented paper in the Webinar (Phys. Fluids, 2020: "Symmetries and turbulence modeling" ), however, builds up a model that is based on two nonphysical symmetries. The consequence: The model is not self-consistent and leads to contradictive and conflicting results, as can be analytically shown, for example, in the proof-of-concept case of a turbulent planar jet flow. For all details, see arXiv:2008.08580 , or follow this link , which also provides a summarized short version of this discussion.

What every turbulence modeler also should keep in mind when making use of the symmetry method is that although a symmetry can be physical, it can nevertheless be broken in the process considered. Turbulence is the result of successive symmetry break-downs, dynamical as well as static ones, where in the end, in the fully developed regime, some may, but also some may not be restored statistically. An eminent example is, e.g., the anomalous scaling behaviour of a fully developed turbulent flow, which significantly deviates from the statistical scaling symmetry of the governing Navier-Stokes equation. In general, noncompact groups as that of scale invariance are more prone to be broken statistically than compact groups as that of rotation invariance. For example, in flows which were initially prepared and held under isotropic conditions, global statistical isotropy may also break, however then in a more weaker sense. It are these statistical symmetry break-downs what ultimately produces complexity and diversity in nature.

Q4: Has the proposed scaling method anything in common with the operator expansion method, e.g., as in conformal field theory? A4: No, there is no common ground at all and also no connection can be established. Regardless this fact, the central claim is rather made that with the tools of a plain Lie-group invariance analysis alone, it has been analytically proven that the 2D inviscid Navier-Stokes equations admit conformal invariance. But this claim we already clearly refuted three years ago in arXiv:1802.02490 , which recently was confirmed again by the responsible journal J. Phys. A.

Anonymity is preserved throughout, because it's only about the scientific content here, and not about who gave feedback or who made a comment.