Samuel Lellouch

Lih-King Lim

Laurent Sanchez-Palencia

Propagation of collective pair excitations in disordered Bose superfluids

Keywords: numbers: 05.30.Jp, 05.70.Ln, 05.60.Gg, 67.85.-d

We study the effect of disorder on the propagation of collective excitations in a disordered Bose superfluid. We incorporate local density depletion induced by strong disorder at the meanfield level, and formulate the transport of the excitations in terms of a screened scattering problem. We show that the competition of disorder, screening, and density depletion induces a strongly non-monotonic energy dependence of the disorder parameter. In three dimensions, it results in a rich localization diagram with four different classes of mobility spectra, characterized by either no or up to three mobility edges. Implications on experiments with disordered ultracold atoms are discussed.

I. INTRODUCTION

The dynamics of correlated quantum systems attracts a growing attention sparked by the recent development of quantum devices with long coherence times and dynamical control of parameters, e.g. superconducting circuits and ultracold atoms [1]. An additional asset of the latter is that disorder may be introduced in a controlled way [2]. Disorder may strongly affect dynamical processes, mainly due to Anderson localization [3]. Understanding the interplay of disorder and interactions in dynamical quantum systems is thus of fundamental importance and localization in quantum systems is still the subject of active research [4][5][6][7][8][9][10][START_REF] Scarola | Transport of Hubbard-band quasiparticles in disordered optical lattices[END_REF]. This topic has also been addressed in the context of quasi-periodic systems [START_REF] Michal | [END_REF]13].

In correlated quantum systems, most basic dynamical processes are determined by the transport properties of their collective excitations [14]. An important starting point in the understanding of localization in correlated systems thus relies on classification according to the symmetries of their excitations [15]. For Fermi systems, it is mostly based on the three classes of random matrices [16] as well as chiral or particle-hole symmetries [17]. For Bose systems, a strong distinction arises between Goldstone and non-Golstone modes [18]. For instance, in a Bose superfluid, while localization is at its strongest at low energy for particle-like excitations, it is suppressed for phonon excitations [5,7,10]. This conclusion is based on a weak disorder analysis and holds in dimension d ≤ 2 where localization occurs for arbitrary weak disorder. It is, however, challenged in higher dimension where the onset of the Anderson transition requires sufficiently strong disorder, which may alter the very nature of the excitations.

In this work we study the transport of collective excitations in a disordered, weakly-interacting Bose superfluid in dimension higher than one. We show that the competition of disorder, screening, and density depletion yields a strongly non-monotonic and non-universal energy dependence of the disorder parameter, which controls the localization properties. In three dimensions (3D), our analysis indicates that the localization diagram exhibits r Amplitudes (in arb. units) V(r) n (r) c µ+∆ Figure 1. One-dimensional cut of the density profile of a Bose superfluid in a disordered potential: Exact numerical solution of the GPE (1) [shaded area, nc(r)] vs. self-consistent solution using Eq. (2) [dashed line], for a disordered potential [full line, V (r)] of amplitude VR = 0.87µ and correlation length σR = ξ. The density profile follows the modulations of the potential, smoothed at the length scale of ξ, and may be locally depleted around disorder maxima.

several classes of mobility spectra, characterized by either no or several mobility edges. We finally discuss implications of these localization properties on quantum-quench experiments with disordered ultracold atoms.

II. TRANSPORT THEORY OF COLLECTIVE EXCITATIONS

A. Meanfield scattering theory

To study the transport of collective excitations in the presence of disorder, it is worth devising a scattering problem. For weakly-interacting Bose superfluids, we may rely on meanfield theory [START_REF] Popov | Functional Integrals in Quantum Field Theory and Statistical Physics[END_REF]. The background density field n c (r) obeys the Gross-Pitaevskii equation (GPE),

-2 ∇ 2 /2m + V (r) -µ + gn c (r) n c (r) = 0, (1)
where m is the particle mass, µ is the chemical potential, and g > 0 is the coupling constant of short-range repulsive interactions. The disordered potential V (r) is chosen to be spatially homogeneous, isotropic, and of vanishing statistical average. For weak disorder, Eq. ( 1) can be solved perturbatively [START_REF] Sanchez-Palencia | [END_REF]21]. Below, we consider regimes of strong disorder and it is necessary to extend this approach, including possible local depletion of the density around the disorder maxima. To do so, we generically write the density field in the form

n c (r) = [µ -η(r) + ∆]/g, (2)
where the field η(r) describes the modulations of the density due to the disorder, and the quantity ∆ is a shift in the chemical potential. The latter allows us to impose the conventional condition that η(r) is of zero statistical average. Notice that since the density is positive everywhere, n c (r) ≥ 0, the field η(r) is bounded above, η(r) ≤ µ + ∆.

We then insert Eq. ( 2) into the Gross-Pitaevskii equation (1), and linearize it, which yields

η(r) = (µ + ∆) min ∆ + Ṽ (r) µ + 3∆/2 , 1 , (3)
where, in Fourier space,

Ṽ (q) = V (q) 1 + ξ 2 ∆ |q| 2 . ( 4
)
The quantity Ṽ (r) is a generalized smoothed potential [START_REF] Sanchez-Palencia | [END_REF], where the healing length is renormalized by the shift ∆ to the value ξ ∆ = / 4m(µ + 3∆/2). The zeroaverage condition, η(r) = 0, then yields

0 = ∆ + min{ Ṽ (r), µ + ∆/2} , (5)
where . . . denotes statistical averaging. Note that Eq. ( 5) ensures that µ + 3∆/2 ≥ 0, so that ξ ∆ is well defined. In general, the density field n c (r) is therefore found by solving self-consistently Eqs. ( 4) and ( 5) for ∆ and Ṽ (r), and Eq. (3) for η(r). This self-consistent solution is in good agreement with the exact numerical solution of the GPE (1) (see Fig. 1). As expected, the density modulations η(r) follow those of the disorder, smoothed at the length scale of the healing length [START_REF] Sanchez-Palencia | [END_REF]21]. However, both the amplitude of the smoothed potential and the healing length are renormalized by the energy shift ∆. Moreover, for strong disorder, the field η(r) locally saturates to the constant value µ + ∆ at positions where Ṽ (r) typically exceeds the chemical potential µ (more precisely, where Ṽ (r) ≥ µ + ∆/2). In those regions, which will be referred to as depleted regions, we thus have n c (r) ≈ 0 (see Fig. 1). In order to interpret the shift ∆, we may rewrite Eq. ( 5) in the form ∆ = depl. d Ṽ P ( Ṽ ) [ Ṽ -(µ + ∆/2)], where P ( Ṽ ) is the probability distribution of the smoothed potential and the integral is restricted to the depleted regions. The quantity ∆ can thus be assimilated to the weight of the part of the smoothed potential that is truncated in the depleted regions. In particular, in the case of weak disorder for which Ṽ (r) never exceeds µ, we find ∆ = 0 and we recover the solution given by usual perturbation theory [START_REF] Sanchez-Palencia | [END_REF]21]. For stronger disorder, ∆ is finite, and our approach accounts for the local depletion of the density around the disorder maxima.

Knowing the density field n c (r), we now treat the collective excitations. The phase and density fluctuations, θ and δn, are readily found by developing the many-body Hamiltonian up to order two in the operator B(r) ≡ δn(r)/2 n c (r) + i n c (r) θ(r). The resulting quadratic Hamiltonian is then diagonalized by the Bogoliubov transform B(r) = ε {u ε (r) bε +v * ε (r) b † ε }, where bε is the annihilation operator of an elementary pair excitation of energy ε. It yields the Bogoliubov-de Gennes equations [START_REF] Popov | Functional Integrals in Quantum Field Theory and Statistical Physics[END_REF][START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF]

L 0 u ε v ε + U(r) u ε v ε = ε u ε v ε , (6)
where

L 0 =    - 2 ∇ 2 2m + µ + 2∆ +µ + ∆ -µ -∆ + 2 ∇ 2 2m -µ -2∆    and U(r) = +V (r) -2η(r) -η(r) +η(r) -V (r) + 2η(r) .
In this form, Eq. ( 6) devises a well-defined two-wave scattering problem. The dynamics of a given excitation at energy ε is governed by the homogeneous propagator L 0 and scattering from a disordered medium defined by U(r). The latter combines the two random fields V (r) and η(r), which are strongly correlated [see Eqs.

(3) to (5)].

At this point, one could wonder whether the Bogoliubov approach is valid even in the presence of strong density depletion. The main approximation here is the truncation of the many-body Hamiltonian at second order in the Bogoliubov operator. Since the latter is equivalent to the linearization of the time-dependent Gross-Pitaevskii equation (tGPE) [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF], it can be tested by comparing the meanfield dynamics predicted by the exact tGPE on the one hand and by the linearized tGPE on the other hand. Our results show excellent agreement between the two in all regimes, namely the phonon, particle, intermediate regimes for weak to strong density depletion (for details, see appendix. A). It validates the use of the Bogoliubov approach used here.

B. One-parameter scaling theory

Universal transport properties can now be inferred using the one-parameter scaling (OPS) approach [START_REF] Abrahams | [END_REF], which can be extended to the case of excitations, as we outline here. It consists in developing a renormalizationgroup (RG) analysis of the size-dependent conductance.

The latter is identified to the Thouless number [25], which is the ratio of the energy scale associated to diffusion across a finite sample of size L, δε = D B /L 2 (with D B = w ε l B /d the classical diffusion constant, w ε = -1 |∂ε/∂k| the excitation velocity, and l B the Boltzmann transport mean free path), to the energy-level spacing, ∆ε = 1/N (ε)L d (with N (ε) the density of states per unit volume). In diffusive regimes, if k ε is the momentum associated to the energy ε, then

N (ε) ∝ k d-1 ε /|∂ε/∂k| = k d-1 ε / w ε , so that G(L) ∝ (k ε l B )(k ε L) d-

C. Disorder parameter

In order to estimate the IDP for the scattering problem (6), we follow the approach of Ref. [7,10] and extend it to strong disorder with possible local density depletion (see details in appendix B). In brief, we note that the homogeneous propagator L 0 does not support only plane-wave modes with momentum

k ε such that 2 k 2 ε /2m = ε 2 + (µ + ∆) 2 -(µ + 2∆), but also evanescent modes of penetration length γ -1 ε such that 2 γ 2 ε /2m = ε 2 + (µ + ∆) 2 + (µ + 2∆
). The latter ensures that for a scattering length larger than the penetration length, the excitation modes (u ε , v ε ) can be decomposed into two fields (g + ε , g - ε ), where the second one is enslaved by the first one. Retaining only the leading disorder terms, the behavior of the excitation is then entirely determined by the field g + ε , which fulfills the closed equation

2 k 2 ε 2m g + ε (r) = - 2 2m ∇ 2 g + ε (r) + V ε (r)g + ε (r) (7)
where

V ε (r) = V (r) -f (ε)η(r), (8)
with

f (ǫ) = 2 √ ε 2 +(µ+∆) 2 -(µ+∆) √ ε 2 +(µ+∆) 2
. The so-called screened potential V ε (r) results from the competition of the bare disorder V (r) and the meanfield repulsive interaction, determined by the field η(r). This competition is strongly energy dependent due to the factor f (ε). Equation ( 7) describes an equivalent scattering problem, which can now be solved by standard quantum transport theory [START_REF] Rammer | Quantum Transport Theory[END_REF]. In the on-shell approximation [START_REF] Kuhn | [END_REF], it yields

1 k ε l B (ε) ≃ 2πm 2 4 k 4-d ε dΩ d (2π) d (1-cos θ)C ε [2k ε sin(θ/2)], (9)
where dΩ d denotes the infinitesimal solid angle in d dimensions and C ε (q) ∝ |V ε (q)| 2 is the power spectrum of the screened potential. Notice that in 1D, the angular integral in Eq. ( 9) reduces to θ = π so that one recovers the result of Ref. [7,10] for the Lyapunov exponent.

It is worth pointing out that the previous approach describes only excitations of energy ε > ε c ≡ 2∆(µ + 3∆/2). Otherwise, we have k 2 ε < 0 and all modes of L 0 are evanescent. In the following, we will thus disregard the case of excitations ε < ε c , which in most cases reduces to a very narrow energy range at the bottom of the spectrum, since ∆ ≪ µ. It should as well be pointed out that since it retains only leading terms in disorder, our theory is not expected to be quantitatively exact, but rather to provide a qualitative description of the relevant physics. Possible extensions of the approach are discussed in the conclusion.

III. LOCALIZATION OF BOGOLIUBOV QUASIPARTICLES IN AN IMPURITY MODEL A. The impurity model

We can now discuss the behavior of the disorder parameter of the excitations. For concreteness, let us consider a generic impurity model described by the potential

V (r) = j V j h(r -r j ) -V . ( 10
)
The impurities are independent Gaussian-shaped potentials of width σ R , h(r) = exp(-r 2 /2σ 2 R ), randomly distributed in space with density ρ, and with amplitudes V j that take the values +V + 0 > 0 or -V - 0 < 0 with equal probability (see Fig. 2). The constant term

V ≡ ρ( √ 2πσ R ) d (V + 0 -V - 0 )/2 ensures that the potential V (r) is of vanishing statistical average. The square disorder amplitude is V 2 R ≡ V 2 = ρ( √ πσ R ) d [(V + 0 ) 2 + (V - 0 ) 2 ]/2.
This two-impurity model generalizes the one-impurity model, which is widely used in studies of Anderson localization in non-interacting systems [START_REF] Lifshits | Introduction to the Theory of Disordered Systems[END_REF][START_REF] Abrahams | 50 years of Anderson Localization[END_REF]. Here, we introduce two types of impurities, namely repulsive (V j = +V + 0 > 0) and attractive (V j = -V - 0 < 0) ones. In contrast to non-interacting systems, it is crucial to distinguish repulsive and attractive impurities in the present work because they have radically different effects on the density background. For instance, only repulsive impurities can induce local density depletion. The two-impurity model is generic in the sense that it is the simplest one to describe a disordered system where different kinds of impurities are present. In addition, controlled impurity models can be realized in ultracold-atom systems where the impurities are made of individual atoms trapped at some random sites of an optical lattice [START_REF] Gavish | [END_REF]. So far, only one-impurity models have been realized [31] but they can be extended to models with different kinds of impurities using different atomic species.

B. Behavior of the disordered parameter

To compute the IDP for the previous impurity model, we first numerically determine the density background η(r) and ∆ following the previous self-consistent procedure. This permits to compute the screened potential Eq. ( 8) and its power spectrum, from which the IDP is inferred using Eq. ( 9). Figure 3 shows the energy dependence of the IDP for the 3D balanced impurity case (V + 0 = V - 0 ), plotted as a function of k ε ξ. Qualitatively similar curves are found for lower dimensions and for imbalanced impurity cases (V + 0 = V - 0 ). Depending on the disorder strength, the IDP exhibits three generic behaviors. Notice that k ε ξ → 0 corresponds to ε → ε c .

For weak disorder (case A in Fig. 3), the IDP shows a non-monotonic energy dependence, which can be understood as follows. At high energy, the excitations are insensitive to the density background and behave as particles in the bare disorder potential. Conversely, at low energy, the excitations are strongly affected by the density background, which screens the disorder and suppresses scattering. More precisely, this holds when the chemical potential exceeds the maximum of the smoothed potential, i.e. for V + 0 h(0) -V < µ where h(r) is the smoothed impurity. Then, the density background as no depleted region. The power spectrum of the screened potential, C ε (q), can be computed explicitly as a function of that of the bare disorder, C(q), and of the excitation energy ε using Eq. ( 8). It yields

C ε (q) = 1 - 1 + 4k 2 ε ξ 2 1 + 2k 2 ε ξ 2 1 1 + q 2 ξ 2 2 C(q). (11)
k ε l B k ε ξ (A) V 0 =0.32µ (B) V 0 =1.59µ (C) V 0 =6.35µ
≡ V + 0 = V - 0 .
Shown are the results of Eq. ( 9), where the screened power spectrum Cε(q) is calculated with the full screened disorder of Eq. ( 8) (dots) or with Eq. ( 11) (solid line), as well as the contributions of the bulk (dashed lines) and depleted (dotted lines) regions.

Inserting this Eq. ( 11) into Eq. ( 9), we find the solid line in Fig. 3, which reproduces very well the data in the full energy range for case A. Notice that the same formula as found from Eqs. ( 9) and ( 11) was inferred from calculations of the scattering mean free path using a different approach in Ref. [32]. Equation [START_REF] Scarola | Transport of Hubbard-band quasiparticles in disordered optical lattices[END_REF] generalizes the 1D case [7,10]. It defines a screening function [prefactor in the rhs of Eq. ( 11)], which can also be identified in single-scattering processes [21] and renormalizes the disorder by the interactions. The behavior of the IDP can now be found by inspection of Eqs. ( 9) and [START_REF] Scarola | Transport of Hubbard-band quasiparticles in disordered optical lattices[END_REF]. For

k ε ≫ ξ -1 , σ -1
R , the screening is irrelevant and we can replace C ε (q) by C(q) in Eq. (9). We then recover the free-particle behavior [START_REF] Kuhn | [END_REF]33],

k ε l B ∼ µ 2 σ R V 2 R ξ (k ε ξ) 5 , k ε ≫ ξ -1 , σ -1 R . (12)
Conversely, for k ε ≪ ξ -1 , σ -1 R , the screening strongly enhances k ε l B compared to the free-particle case and we find

k ε l B ∼ µ 2 ξ d V 2 R σ d R (k ε ξ) -d , k ε ≪ ξ -1 , σ -1 R . ( 13
)
This result is in agreement with the universal behavior expected for Goldstone modes [18]. Both low-energy and high-energy scalings reproduce the behavior of case A and locate the minimum of the IDP at k min ε ∼ min(1/ξ, 1/σ R ). Note that a non-monotonic behavior of the IDP is also found in the propagation of other kinds of waves, such as photonic [34] and acoustic [35] ones.

For intermediate to strong disorder (cases B and C in Fig. 3), the energy dependence of the IDP found from the solution of the full scattering problem [Eqs. (8) and ( 9)] strongly differs from the weak disorder case at low energy, where k ε l B now increases with the energy. To understand this, it should be noticed that, for V + 0 h(0) -V > µ, the background density is now locally depleted around the positive impurities. Hence, during its propagation, an excitation goes through two types of regions, namely the density depleted region, and the rest, which constitutes the density bulk. In the bulk, the field η(r) may be approximated by Ṽ (r), provided we neglect the quantity ∆, which is valid for low impurity density, ρσ d R ≪ 1. It yields a non-monotonic contribution to k ε l B similar to case A, with a smaller overall magnitude due to the truncation around the positive impurities (dashed lines in Fig. 3). Conversely, in the depleted regions, the field η(r) saturates to the value µ + ∆. The bare disorder in those regions is thus protected against screening and Eq. ( 8) may be replaced by

V ε (r) ≃ V (r) -(µ + ∆)f (ε).
In this field, the excitations behave as (non-Goldstone) free particles, yielding a monotonic contribution to k ε l B (dotted lines). In the white-noise limit (k ε σ R ≪ 1), this contribution is

k ε l B ∼ (k ε ξ) 4-d , k ε ≪ σ -1 R . (14)
The various behaviors of the IDP observed in Fig. 3 can then be interpreted as follows. Neglecting the correlations between the contributions of the bulk and depleted regions, the disorder parameter (k ε l B ) -1 is approximately the sum of these two contributions. Its inverse (the IDP k ε l B , which is plotted in Fig. 3), is thus dominated by the smallest corresponding contribution. At low energy, because of the screening in the bulk, the contribution of the depleted region always dominates if it exists, and captures the free-particle-like behavior of k ε l B . At intermediate energy the behavior of k ε l B crucially depends on the relative magnitude of the two contributions. When V + 0 h(0) -V µ (case B), only the upper fraction of the positive impurities is truncated and the bulk starts to dominate at moderate energy. It results in a turning point k max ε where bulk and depleted regions equally contribute to the IDP, yielding there a local maximum. When V + 0 increases, the depleted region dominates in a wider energy range and k max ε moves to higher values. When V + 0 h(0) -V ≫ µ (case C), the positive impurities are almost entirely truncated, and k max ε eventually merges with the local minimum k min ε , which does not significantly depend on V + 0 . The curve then becomes monotonic.

C. Localization diagram

We now turn to the localization properties of the collective excitations, focusing on the 3D case where mobility edges appear at the localization threshold k ε l B ∼ 1. We determine the latter from IDP curves as those of Fig. 3 for the general model with different amplitudes of the positive (V + 0 ) and negative (V - 0 ) impurities. In all cases, they are of type A, B, or C with an overall magnitude and positions of the local maximum and minimum that depend on The localization diagram of Fig. 4 is expected to be generic. In particular, the competition of disorder, screening, and density depletion determine the diversity of mobility spectra. Yet, a given model of disorder does not necessary display all cases and the imbalanced impurity model with a finite correlation length is the simplest we found that does. For instance, for only positive impurities or in the balanced case, the only possibilities are (0) or (I) because the minimum of the IDP cannot be controlled independently of the density depletion. Conversely, for only negative impurities, the depleted region is absent and the only possibilities are (0) or (II). The case of white-noise disorder is also limited because the smoothed impurity potential diverges in the center, h(r) = e -r/ξ /4πξ 2 r, so that depleted regions strictly appear as soon as V + 0 = 0, and the only possibilities left are (I) and (III).

IV. CONCLUSIONS

The physics we have discussed here is particularly relevant to ultracold-atom experiments. In these systems, out-of-equilibrium dynamics can be generated by a local quench, which produces collective excitations [36]. In the presence of disorder, their transport properties and ability to mediate long-range energy transfer are determined by the four classes of mobility spectra of the localization diagram. In case (0), all excitations are protected against localization and propagate diffusively, i.e. r 2 = 2D B t. In all other cases, energy can only be partially transferred since some excitation modes are localized. Energyresolved quenches may provide experimental evidence of such mobility spectra in ultracold gases. Moreover, these systems offer a wide range of models of disorder, e.g. impurities [START_REF] Gavish | [END_REF]31] and speckle potentials [2]. The statistical properties of the latter may be tailored, which may lead to even richer localization diagrams [33,37,38].

The observation of the localization effects we have discussed here requires that the lifetime τ of the Bogoliubov quasiparticles exceeds the transport meanfree time τ B ≡ l B /w ε (with w ε the excitation group velocity). On the one hand, for low temperature, the decay of Bogoliubov excitations is dominated by Beliaev processes [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF]39]. To estimate the corresponding decay rate Γ = 1/τ , we resort to local density approximation. The depleted regions, where the excitations behave as free particles with infinite lifetime, very weakly contribute to Γ. A good estimate of Γ is thus given by the bulk contribution. For a typical excitation ε ∼ µ, it yields τ ≃ m/10 n 3/2 a 5/2 sc , where a sc = mg/4π 2 is the scattering length [39]. On the other hand, w ε ≃ gn/m in the phonon regime and l B ∼ 1/k ε in the region of interest, which yields τ B ∼ m/4π na sc . Therefore, the validity condition of our approach reads τ B /τ ∼ √ na sc ≪ 1, which is the validity condition of the Bogoliubov approach, very verified in dilute-gas Bose-Einstein condensates [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF]. For instance, using the parameters of Ref. [40], we find find τ ∼ 6s and τ B ∼ 5ms, making experimental observation of our predictions possible. The approach used in this work provides a intuitive understanding of the physics at stake as well as generic qualitative predictions for the localization behavior of collective excitations. However, since it relies on lowestorder perturbation theory, it is not expected to be quantitatively accurate. In particular, it does not take into account the disorder-induced shift of the dispersion relation, which is known, in the non-interacting case, to result in a possibly significant shift on the position of the mobility edge [33,41]. Moreover, numerical calculations in the non-interacting case show that the mobility edge significantly depends on the model of disorder [42,[START_REF] Fratini | Anderson localization of matter waves in quantum-chaos theory[END_REF]. Therefore, the determination of the precise localization diagram for collective pair excitations in disordered Bose superfluids, as well as the identification of the various classes of mobility spectra predicted in the present work, require a full numerical resolution of the localization problem in the two-impurity model as well as in other models of disorder.

This research was supported by the European Research Council (FP7/2007-2013 Grant Agreement No. 256294), the Ministère de l'Enseignement Supérieur et de la Recherche, and the Institut Francilien de Recherche sur les Atomes Froids (IFRAF). We acknowledge the use of the computing facility cluster GMPCS of the LUMAT federation (FR LUMAT 2764).

In this appendix, we show that the Bogoliubov approach used in this paper, i.e. a development of the many-body Hamiltonian around the inhomogeneous density background up to second order in fluctuations terms, is valid even in the presence of strong local density depletion. We perform this check by comparing exact calculations and linearized theory at the meanfield level. We consider the time-dependent Gross-Pitaevskii equation (tGPE),

i ∂ t ψ = - 2 ∇ 2 2m ψ + V (r) + g|ψ| 2 ψ, (A1)
which governs the time-evolution of a condensate wavefunction ψ(r, t). In the linearized approach, one writes ψ(r, t) = n c (r) + δψ(r, t), where n c (r) is the density background found from the solution of the stationary GPE (1) and δψ(r, t) is a small perturbation. At lowest order, it yields the linearized equation

i ∂ t δψ δψ * = L GP δψ δψ * , (A2)
where the matrix

L GP =    - 2 ∇ 2 2m + V + gn c -µ gn c -gn c + 2 ∇ 2 2m -V -gn c + µ   
is exactly the one appearing in the Bogoliubov-de-Gennes equations [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF]. This linearization procedure thus turns out to be equivalent to the Bogoliubov development of the many-body Hamiltonian to second order. Therefore, to check the validity of the latter in the presence of density depletion, one can compare the results of the timeevolution of an excitation δψ(r, t) on top of a depleted condensate n c (r), using either the exact tGPE (A1) or its linearized version, Eq. (A2).

We have performed this test in one dimension, which is the most unfavorable dimension due to large fluctuations. For the sake of simplicity, the density depletion is induced by a single strong potential barrier V (x) = V 0 Θ(a -|x -x * |), of height V 0 = 30µ, width 2a = 0.05L, and centered on some position x * . The quantity Θ denotes the Heaviside function. In such a configuration, we first determine the density background n c (x) solving the stationary GPE Eq. ( 1) by imaginary time propagation. The latter is strongly depleted under the barrier (see Fig. 5). We then add a small excitation on top on this background, δψ 0 (x) = e ikx e -(x-x0) 2 /2σ 2 . The latter describes a plane wave of momentum k inside a Gaussian envelop of width σ. In practice, we choose kσ ≫ 1 so that it is sufficiently narrow around k in momentum space. We then compute the time evolution of this initial excitation either solving Eq. (A1) for ψ(x, t)

-L 0 L Ψ(x,t)
x t=0 t=100, from (tGPE) t=100, from linearized (tGPE) kξ=1 ξ/L=0.005

-L 0 L Ψ(x,t)
x t=0 t=5.9, from (tGPE) t=5.9, from linearized (tGPE) kξ=5 ξ/L=0.025 Time evolution of an excitation δψ0(x) = e ikx e -(x-x 0 ) 2 /2σ 2 on top of a depleted condensate in the presence of a strong potential barrier V (x) = V0Θ(a -|xx * |), for V0 = 30µ, 2a = 0.05L, and σ = 0.05L. Three different regimes of the excitation spectrum are considered: (i) kξ = 1 with almost total reflection (top), (ii) kξ = 5 with reflection and transmission of the excitation of the same order to magnitude (central), (iii) kξ = 10 with almost total transmission (bottom). Note that for clarity purposes, ξ is varied from a panel to another. Shown are the initial wavefunction ψ(x, 0) = nc(x) + δψ0(x) (solid red line) as well as the final wavefunction as given by the full time-dependent tGPE (solid orange line) and by the linearized tGPE (blue dotted line). The agreement between the two calculations is excellent in all cases.

-L 0 L Ψ(x,t)
with the initial condition ψ(x, 0) = n c (x) + δψ 0 (x), or solving Eq. (A2) for δψ(x, t) with the initial condition δψ(x, 0) = δψ 0 (x). We have performed this comparison in a wide range of parameters, from non-depleted to strongly depleted cases, and from the phonon (kξ ≪ 1) to the free-particle (kξ ≫ 1) regimes. As shown on Fig. 5 in the case of strong depletion, we found an excellent agreement in all cases, irrespective to the values of k and to the relative strength of reflection and transmission by the barrier. This validates the use of the linearized equation, and thus of the Bogoliubov approach, to study the dynamics of the collective excitations even in the presence of strong modulations of the potential.

where

-2 k 2 ε /2m ≡ -ε 2 + (µ + ∆) 2 + (µ + 2∆) and 2 γ 2 ε /2m ≡ ε 2 + (µ + ∆) 2 + (µ + 2∆
) are the eigenvalues of the homogeneous matrix M . Without any approximation at this stage, the BdGEs in the (g + ε , g - ε ) basis then read

2 k 2 ε 2m g + ε (r) = - 2 2m ∇ 2 g + ε (r) + V (r) -f -(ǫ)η(r) g + ε (r) +Φ + (ǫ)η(r)g - ε (r) (B4) - 2 β 2 ε 2m g - ε (r) = - 2 2m ∇ 2 g + ε (r) + V (r) -f + (ǫ)η(r) g - ε (r) +Φ -(ǫ)η(r)g + ε (r), (B5) with f ± (ǫ) = 2 √ ε 2 +(µ+∆) 2 ±(µ+∆) √ ε 2 +(µ+∆) 2 and Φ ± (ǫ) = √ ε 2 +(µ+∆) 2 ±(µ+∆) √ ε 2 +(µ+∆) 2
. In the absence of disorder, Eqs. (B4) and (B5) are now decoupled and are straightforward to solve. The g + ε modes are plane waves of momentum k ε , while the g - ε are evanescent waves of penetration length γ -1 ε . The latter vanish identically if the system is infinite or has periodic boundary conditions.

In the presence of disorder, we can therefore make the assumption |g - ε | ≪ |g + ε | since g - ε is at least one order of magnitude smaller that g + ε in V R /µ and ∆/µ. Keeping only the leading-order terms in Eq. (B5), we then find

g - ε ≃ - 2m 2 γ 2 ε Φ -(ε) dr ′ G 1/γε (r -r ′ )η(r ′ )g + ε (r ′ ), (B6)
where G 1/γε is the Green function associated to the differential operator -∇ 2 /γ 2 ε + 1. In Fourier space, it reads G 1/βε (q) = (2π) -d/2 /[1 + q 2 /β 2 ε ] and in real space it is a positive function of integral unity, decaying exponentially on a length scale 1/γ ε . Then, since 2m/ 2 γ 2 ε < µ and |Φ -(ε)| < 1 for any energy ε, we find

|g - ε | G 1/γε (r -r ′ )|η(r ′ )||g + ε (r ′ )|dr ′ /µ (V R /µ)|g + ε |,
which is consistent with the working assumption |g - ε | ≪ |g + ε |. Therefore, the last term of Eq. (B4) can be ne-glected, which yields a closed equation for g + ε ,

2 k 2 ε 2m g + ε (r) = - 2 2m ∇ 2 g + ε (r) + V ε (r)g + ε (r), (B7)
where

V ε (r) = V (r) -f (ε)η(r) (B8)
and, for simplicity, we now denote by f (ǫ) the quantity

f -(ǫ) = 2 √ ε 2 +(µ+∆) 2 -(µ+∆) √ ε 2 +(µ+∆) 2
. The quantity V ε (r) defines a so-called screened potential of zero-average. It can be viewed as the screening of the bare potential V (r) by the density background encoded in η(r). It notably depends on the energy ε of the Bogoliubov excitation. Therefore Eq. (B7), together with Eq. (B8), contains the leading disorder terms. It features an effective single-wave scattering problem, which can now be solved by standard quantum transport theory [START_REF] Rammer | Quantum Transport Theory[END_REF]. Localization properties are then determined in a two-step process [START_REF] Vollhardt | [END_REF]45]. Firstly, the transport meanfree path is calculated in the semi-classical approach where interference of multiple-scattering paths is neglected. Within the on-shell approximation, which amounts to assimilate the spectral function to the disorder-free one, diagrammatic theory yields Eq. ( 9),

1 k ε l B (ε) ≃ 2πm 2 4 k 4-d ε dΩ d (2π) d (1-cos θ)C ε [2k ε sin(θ/2)],
(B9) for models of disorder with isotropic correlation functions [START_REF] Kuhn | [END_REF], as considered in this work. For extension to anisotropic correlation functions, see Ref. [33]. Secondly, localization is found using either the one-parameter scaling theory [START_REF] Abrahams | [END_REF] or the self-consistent approach [START_REF] Vollhardt | [END_REF]45]. The one-parameter scaling theory is used and discussed in the main text. The self-consistent theory incorporates interference corrections on the top of diffusive dynamics, which yields a self-consistent equation for the diffusive constant or the localization length. Both approaches give the approximate localization threshold k ε l B ∼ 1 used in the present paper.

  2 , and β ≡ d log G/d log L ∼ d -2. In localized regimes, the conductance is exponentially small, G(L) ∼ exp(-L/L loc ) with L loc the localization length, and β ∼ log G. For d ≤ 2, β(G) is strictly negative and G(L) always flows down to the localized regime. Then, all states are localized, with the localization length L loc ∝ l B in 1D and log(L loc /l B ) ∝ k ε l B in 2D. Conversely, for d > 2, the RG flow has an unstable fixed point at k ε l B ∼ 1, known as the mobility edge or the Anderson localization transition [24]. Since the above scaling laws are independent of the dispersion relation, these features are all universal, except the transition point, which is determined by the value of the inverse disorder parameter (IDP) k ε l B .

Figure 2 .

 2 Figure 2. Schematic view of the two-impurity model defined by Eq.(10). It is made of two types of impurities with, respectively, positive (+V + 0 ) and negative (-V - 0 ) amplitudes. The impurities are randomly and independently spread over in space. Each impurity is Gaussian shaped with a width σR.

Figure 3 .

 3 Figure 3.Inverse disorder parameter (IDP) versus pairexcitation momentum for the 3D balanced impurity model, with σR = ξ, ρσ 3 R = 2 × 10 -4 , and for various values of the disorder amplitude V0 ≡ V + 0 = V - 0 . Shown are the results of Eq. (9), where the screened power spectrum Cε(q) is calculated with the full screened disorder of Eq. (8) (dots) or with Eq. (11) (solid line), as well as the contributions of the bulk (dashed lines) and depleted (dotted lines) regions.

Figure 4 .

 4 Figure 4. Localization diagram of pair excitations in the 3D impurity model, plotted as a function of the amplitudes of positive (V + 0 ) and negative (V - 0 ) impurities, for σR = ξ and ρσ 3 R = 2 × 10 -4 . It exhibits four classes of mobility spectra, characterized by zero (0), one (I), two (II), or three (III) mobility edges. Note the different scales on the two axes.

  Figure 5.Time evolution of an excitation δψ0(x) = e ikx e -(x-x 0 ) 2 /2σ 2 on top of a depleted condensate in the presence of a strong potential barrier V (x) = V0Θ(a -|xx * |), for V0 = 30µ, 2a = 0.05L, and σ = 0.05L. Three different regimes of the excitation spectrum are considered: (i) kξ = 1 with almost total reflection (top), (ii) kξ = 5 with reflection and transmission of the excitation of the same order to magnitude (central), (iii) kξ = 10 with almost total transmission (bottom). Note that for clarity purposes, ξ is varied from a panel to another. Shown are the initial wavefunction ψ(x, 0) = nc(x) + δψ0(x) (solid red line) as well as the final wavefunction as given by the full time-dependent tGPE (solid orange line) and by the linearized tGPE (blue dotted line). The agreement between the two calculations is excellent in all cases.

Appendix B: Derivation of the inverse disorder parameter

The background density field n c (r) being given by the solution of Eqs. ( 2) to ( 5), the collective excitations (u ε , v ε ) can now be determined by solving the Bogoliubov-de Gennes equations (6),

where

The dynamics of a given excitation at energy ε is thus governed by the homogeneous propagator L 0 and scattering from the disordered medium defined by U(r).

In order to solve the Bogoliubov-de Gennes equations (BdGEs), we generalize the approach of Ref. [7] to the strong disorder case where ∆ = 0. We first rewrite the BdGEs (B1) in the form

A suitable basis to perform diagrammatic expansion in leading disorder terms is found by applying the linear transform (u ε , v ε ) → (g + ε , g - ε ) that diagonalizes the homogeneous term in Eq. (B2), i.e. the matrix M ≡


Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Giuseppe Toscani
email: giuseppe.toscani@unipv.it.

Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities

Keywords: Sobolev inequality, logarithmic Sobolev inequality, Gaussian isoperimetric inequality, generalized Poincaré inequalities, Gagliardo-Nirenberg inequalities, interpolation, entropy -entropy production inequalities, extremal functions, optimal constants, relative entropy, generalized Fisher information, entropy power, stability, improved functional inequalities, fast diffusion equation, Ornstein-Uhlenbeck equation, porous medium equation, rates of convergence Mathematics Subject Classification (2010): 26D10, 46E35, 58E35

This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincaré inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.

Introduction

Several papers have recently been devoted to improvements of the logarithmic Sobolev inequality. [START_REF] Ledoux | Stein's method, logarithmic Sobolev and transport inequalities[END_REF] use the Stein discrepancy. Closer to our approach is [START_REF] Bobkov | Bounds on the deficit in the logarithmic sobolev inequality[END_REF], [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF], who exploit the difference between the inequality of [Stam, 1959, Inequality (2.3)] and the logarithmic Sobolev inequality to get a correction term in terms of the Fisher information functional. What we do here first is to emphasize the role of scalings and prefer to rely on [START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF] for a scale invariant form of the logarithmic Sobolev inequality on the Euclidean space. We also make the choice to get a remainder term that involves the entropy functional and is very appropriate for stability issues. This allows us to deduce striking results in terms of rates of convergence for the Ornstein-Uhlenbeck equation. Writing the improvement in terms of the entropy has several advantages: contraints on the second moment are made clear, improvements can be extended to all generalized Poincaré inequalities for Gaussian measures, which interpolate between the Poincaré inequality and the logarithmic Sobolev inequality, and stability results with fully explicit constants can be stated: see for instance Corollary 3, with an explicit bound of the distance to the manifold of all Gaussian functions given in terms of the so-called deficit functional. This is, for the logarithmic Sobolev inequality, the exact analogue of the result of [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] for Sobolev's inequality.

However, putting the emphasis on scalings has other advantages, as the method easily extends to a nonlinear setting. We are henceforth in a position to get improved entropy -entropy production inequalities associated with fast diffusion flows based on the scale invariant forms of the associated Gagliardo-Nirenberg inequalities, which cover a well-known family of inequalities that contain the logarithmic Sobolev inequality, and Sobolev's inequality as an endpoint. This is not a complete surprise because such improvements were known from [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] using detailed properties of the fast diffusion equation. By writing the entropy -entropy production inequality in terms of the relative entropy functional and a generalized Fisher information, we deduce from the scaling properties of Gagliardo-Nirenberg inequalities a correction term involving the square of the relative entropy, and this is much simpler than using the properties of the nonlinear flow. The method also works in the porous medium case, which is new, provides clear evidences on the role of the second moment, and finally explains the fast rates of convergence in relative entropy that can be observed in the initial regime, away from Barenblatt equilibrium or self-similar states.

The reader interested in further considerations on improvements of the logarithmic Sobolev inequality is invited to refer to [START_REF] Ledoux | Stein's method, logarithmic Sobolev and transport inequalities[END_REF] and [START_REF] Bobkov | Bounds on the deficit in the logarithmic sobolev inequality[END_REF], [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF] for probabilistic point of view and a measure of the defect in terms of Wasserstein's distance, and to [START_REF] Ledoux | The concentration of measure phenomenon[END_REF] for earlier results. Much more can also be found in [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. Not all Gagliardo-Nirenberg-Sobolev inequalities are covered by our remarks and we shall refer to [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a Schrödinger operator[END_REF] and references therein for the spectral point of view and its applications to the Schrödinger operator. The logarithmic Sobolev inequality in scale invariant form is equivalent to the Gaussian isoperimetric inequality: a study of the corresponding deficit can be found in [START_REF] Mossel | Robust dimension free isoperimetry in gaussian space[END_REF]. In the perspective of information theory, we refer to [START_REF] Toscani | An information-theoretic proof of Nash's inequality[END_REF]Toscani [ , 2014a] ] for a recent account on a concavity property of entropy powers that involves the isoperimetric inequality. It is not possible to quote all earlier related contributions but at least let us point two of them: the correction to the logarithmic Sobolev inequality by an entropy term involving the Wiener transform in [Carlen, 1991, Theorem 6], and the HWI inequality by [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF].

Gagliardo-Nirenberg inequalities (see [START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF], [START_REF] Nirenberg | On elliptic partial differential equations[END_REF]) have been related with fast diffusion or porous media equations in the framework of the so-called entropy methods by Del Pino and Dolbeault [2002]. Also see the papers by [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to selfsimilarity[END_REF], [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF], [START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF] for closely related issues. The message is simple: optimal rates of convergence measured in relative entropy are equivalent to best constant in the inequalities written in entropy -entropy production form. Later improvements have been obtained on asymptotic rates of convergence by [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF], [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF], [START_REF] Denzler | Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach[END_REF]. A key observation of [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF] is the fact that optimizing a relative entropy with respect to scales determines the second moment. This observation was then exploited by [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] to get a first explicit improvement in the framework of Gagliardo-Nirenberg inequalities. Notice that many papers on improved interpolation inequalities use the estimate of [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] with the major drawback that the value of the constant is not known. As a consequence the improved inequality, faster convergence rates for the solution to the fast diffusion equation were obtained and a new phenomenon, a delay, was shown by Dolbeault and Toscani [2015a]. Inspired by [START_REF] Villani | A short proof of the "concavity of entropy power[END_REF], [START_REF] Savaré | The concavity of Rényi entropy power[END_REF] studied the p-th Rényi entropy and observed that the corresponding isoperimetric inequality is a Gagliardo-Nirenberg inequality in scale invariant form. Various consequences for the solutions to the evolution equations have been drawn in [START_REF] Carrillo | Rényi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations[END_REF] and [START_REF] Dolbeault | Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays[END_REF], which are strongly related with the present paper but can all be summarized in a simple sentence: scales are important and a better adjustment than the one given by the asymptotic regime gives sharper estimates. The counterpart in the present paper is that taking into account the scale invariant form of the inequality automatically improves on the inequality obtained by a simple entropy -entropy production method. Let us give some explanations.

At a formal level, the strategy of our paper goes as follows. Let us consider a generalized entropy functional E, which is assumed to be nonnegative, and a generalized Fisher information functional I. We further assume that they are related by a functional inequality of the form

I -λ E ≥ 0 .
We denote by λ the optimal proportionality constant. If the inequality is not in scale invariant form, we will prove in various cases that there exists a convex function ϕ, leaving from ϕ(0) = 0 with ϕ ′ (0) = λ such that I ≥ ϕ(E). Hence we have found an improved functional inequality in the sense that

I -λ E ≥ ϕ(E) -λ E = ψ(E)
where ψ(E) is nonnegative and can be used to measure the distance to the optimal functions. This is a stability result. The left hand side, which is called the deficit functional in the literature, is now controlled from below by a nonlinear function of the entropy functional. A precise distance can be obtained by the Pinsker-Csiszár-Kullback inequality, which is no more than a Taylor expansion at order two, and some generalizations.The key observation is that the optimization under scaling (in the Euclidean space) amounts to adjust the second moment (in the Euclidean space but also in spaces with finite measure, like the Gaussian measure, after some changes of variables).

At this point it is worth to emphasize the difference in our approach compared to the one of [START_REF] Bobkov | Bounds on the deficit in the logarithmic sobolev inequality[END_REF] for the logarithmic Sobolev inequality. What the authors do is that they write the improved inequality as ϕ -1 (I) ≥ E and deduce that

I -λ E ≥ I -λ ϕ -1 (I)
where the right hand side is again nonnegative because ϕ -1 is concave and λ (ϕ -1 ) ′ (0) = 1. This is of course a stronger form of the inequality, as it controls the distance to the manifold of optimal functions in a stronger norm, for instance. However, it is to a large extend useless for the applications that are presented in this paper, as the estimate in terms of the entropy is what matters, for instance, for application in evolution equations.

We shall apply our strategy to the logarithmic Sobolev inequality in Section 2, to the generalized Poincaré inequalities for Gaussian measures in Section 3 and to some Gagliardo-Nirenberg inequalities in Section 4. Each of these inequalities can be established by the entropy -entropy production method. By considering the Ornstein-Uhlenbeck equation in the first two cases, and the fast diffusion / porous medium equation in the third case, it turns out that dE dt = -I and

- d dt (I -λ E) = R ≥ 0 .
Hence, if lim t→∞ (I -λ E) = 0, this shows with no additional assumption that

∞ 0 R[v(t,
•)] dt is a measure of the distance to the optimal functions. Improved functional inequalities follow by ODE techniques if one is able to relate R with E and I. This is the method which has been implemented for instance in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF], [START_REF] Dolbeault | On the Bakry-Emery criterion for linear diffusions and weighted porous media equations[END_REF], [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] and it is well adapted when the diffusion equation can be seen as the gradient flow of E with respect to a distance. Typical distances are the Wasserstein distance for the logarithmic Sobolev inequality or the Gagliardo-Nirenberg inequalities, and ad hoc distances in case of the generalized Poincaré inequalities. See [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF], [START_REF] Dolbeault | A new class of transport distances between measures[END_REF][START_REF] Dolbeault | From Poincaré to logarithmic Sobolev inequalities: a gradient flow approach[END_REF] for more details on gradient flow issues. Improvements can also be obtained when dE dt differs from -I: we refer to [START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF], Dolbeault et al. [2014] for interpolation inequalities on compact manifolds, or to [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF] for improvements of Sobolev's inequality based on the Hardy-Littlewood-Sobolev functional. This makes the link with the famous improvement obtained by [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], and also [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF], but so far no entropy -entropy production method has been able to provide an improvement in such a critical case. For completeness, let us mention that other methods can be used to obtain improved inequalities, which are based on variational methods like in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], on symmetrization techniques like in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF] or on spectral methods connected with heat flows like in [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF]. Here we shall simply rely on convexity estimates and the interplay of entropy -entropy production inequalities with their scale invariant counterparts.

A very interesting feature of improved functional inequalities in the framework of the entropy -entropy production method is that the entropy decays faster than expected by considering only the asymptotic regime. In that sense, the improved inequality capture an initial rate of convergence which is faster than the asymptotic one. This has already been observed for fast diffusion equations in [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] with a phenomenon of delay that has been studied in Dolbeault and Toscani [2015a] and by [START_REF] Carrillo | Rényi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations[END_REF], by resorting to the concept of Rényi entropy. A remarkable fact is that the inequality is improved by choosing a scale (in practice by imposing a constraint on the second moment) without requesting anything on the first moment, again something that clearly distinguishes the improvements obtained here from what can be guessed by looking at the asymptotic problem as t → ∞. Details and statements on these consequences for diffusion equations have been collected in Section 5.

Stability results for the logarithmic Sobolev inequality

Let dµ = µ dx be the normalized Gaussian measure, with µ(x) = (2 π) -d/2 e -|x| 2 /2 , on the Euclidean space R d with d ≥ 1. The Gaussian logarithmic Sobolev inequality reads

R d |∇u| 2 dµ ≥ 1 2 R d |u| 2 log |u| 2 dµ (1) for any function u ∈ H 1 (R d , dµ) such that R d |u| 2 dµ = 1. This inequality is equivalent to the Euclidean logarithmic Sobolev inequality in scale invariant form d 2 log 2 π d e R d |∇w| 2 dx ≥ R d |w| 2 log |w| 2 dx (2)
that can be found in [Weissler, 1978, Theorem 2] in the framework of scalings, but is also the one that can be found in [Stam, 1959, Inequality (2.3)] or in [Carlen, 1991, Inequality (26)]. See [START_REF] Bobkov | Bounds on the deficit in the logarithmic sobolev inequality[END_REF], [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF] and Toscani [2014b] for more comments. The equivalence of ( 1) and ( 2) is well known but involves some scalings and we will give a short proof below for completeness. Next, let us consider the function

ϕ(t) := d 4 exp 2 t d -1 - 2 t d ∀ t ∈ R . (3)
Our first result is an improvement of (1), based on the comparison of ( 1) with ( 2), which combines ideas of [START_REF] Bakry | A logarithmic Sobolev form of the Li-Yau parabolic inequality[END_REF] and [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF]. It goes as follows.

Proposition 1. With ϕ defined by (3), we have

R d |∇u| 2 dµ - 1 2 R d |u| 2 log |u| 2 dµ ≥ ϕ R d |u| 2 log |u| 2 dµ ∀ u ∈ H 1 (R d , dµ) such that R d |u| 2 dµ = 1 and R d |x| 2 |u| 2 dµ = d . (4)
Inequality ( 4) is an improvement of (1) because ϕ(t) ≥ t 2 2 d for any t ∈ R and, by the Pinsker-Csiszár-Kullback inequality,

R d |u| 2 log |u| 2 dµ ≥ 1 4 R d |u| 2 -1 dµ 2 ∀ u ∈ L 2 (R d , dµ) such that u L 2 (R d , dµ) = 1 .
See [START_REF] Pinsker | Information and information stability of random variables and processes[END_REF], [START_REF] Csiszár | Information-type measures of difference of probability distributions and indirect observations[END_REF], [START_REF] Kullback | On the convergence of discrimination information[END_REF] for a proof of this inequality.

Proof . To emphasize the role of scalings, let us give a proof of Proposition 1, which follows the strategy of [START_REF] Bakry | A logarithmic Sobolev form of the Li-Yau parabolic inequality[END_REF]

R d |∇v| 2 dx ≥ 1 2 R d |v| 2 log |v| 2 dx + d 4 log(2 π e 2 ) (5)
which is the standard Euclidean logarithmic Sobolev inequality established in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] (also see [START_REF] Federbush | Partially alternate derivation of a result of Nelson[END_REF] for an earlier related result). This inequality is not invariant under scaling. By considering w such that v(x) = λ d/2 w(λ x), we get that

λ 2 R d |∇w| 2 dx - d 2 log λ ≥ 1 2 R d |w| 2 log |w| 2 dx + d 4 log(2 π e 2 ) . holds for any v ∈ H 1 (R d , dµ) such that v L 2 (R d ) =
1. An optimization on the scaling parameter shows that 4 λ 2 R d |∇w| 2 dx = d and establishes the scale invariant form of the logarithmic Sobolev inequality,

d 2 log 2 π d e R d |∇w| 2 dx ≥ R d |w| 2 log |w| 2 dx ∀ w ∈ H 1 (R d , dx) such that w L 2 (R d ) = 1 , (6)
which is equivalent to (2). This inequality can also be written as

R d |∇w| 2 dx ≥ 1 2 π d e exp 2 d R d |w| 2 log |w| 2 dx . If we redefine u such that w = u √ µ and assume that w L 2 (R d ) = 1, R d |x| 2 w dx = d, we have shown that R d |∇u| 2 dµ ≥ d 4 exp 2 d R d |u| 2 log |u| 2 dµ -1 . (7)
Inequality (4) follows by substracting 1 2 R d |u| 2 log |u| 2 dµ from both sides of the inequality, which is more or less the idea that has been exploited by [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF].

Consider a nonnegative function f ∈ L 1 2 (R d ) := g ∈ L 1 (R d ) : R d |x| 2 g dx < ∞ and, assuming that R d f dx > 0, define M f := R d f dx , θ f := 1 d R d |x| 2 f dx M f . ( 8
)
Let us define the Gaussian function

µ f (x) := M f (2 π θ f ) d/2 e - |x| 2 2 θ f ∀ x ∈ R d .
We shall denote by L 1 2 (R d ) the space of integrable functions on R d with finite second moment.

Lemma 2. Assume that f is a nontrivial, nonnegative function in f ∈ L 1 2 (R d ) such that ∇ √ f ∈ L 2 (R d ).
With θ f , µ f and ϕ defined by ( 3) and ( 8), we have

θ f 2 R d |∇f | 2 f dx - R d f log f dx - d 2 log 2 π e 2 θ f R d f dx ≥ 2 ϕ R d f log f µ f dx . (9) Proof . Let v be such that λ d f (λ x) = |u(x)| 2 µ(x) with λ 2 = θ f , µ(x) = (2 π) -d/2 e -|x| 2 /2
, and apply Proposition 1.

The Gaussian function µ f is the minimizer of the relative entropy

e[f |µ] := R d f log f µ -(f -µ) dx w.r.t. all Gaussian functions in M := µ(x) = M (2 π θ) d/2 : e -|x| 2 2 θ , M > 0 , θ > 0 ,
that is, we have the identity

R d f log f µ f dx = e[f |µ f ] = min {e[f |µ] : µ ∈ M} .
Also notice that µ f is the minimizer of the relative Fisher information w.r.t. all Gaussian functions of mass M f :

R d ∇ f /µ f 2 dx = min R d ∇ f /µ 2 dx : µ(x) = M f (2 π θ) d/2 e -|x| 2 2 θ , θ > 0 .
Recall that by the Pinsker-Csiszár-Kullback inequality, the r.h.s. in ( 9) provides an explicit stability result in L 1 + (R d , dx) that can be written as

e[f |µ f ] ≥ 1 4 M f f -µ f 2 L 1 (R d ) ∀ f ∈ L 1 + (R d , dx) .
Combined with the observation that ϕ is nondecreasing and ϕ(t) ≥ t 2 2 d for any t ∈ R, we have shown the following global stability result.

Corollary 3. Assume that f is a nontrivial, nonnegative function in f ∈ L 1 2 (R d ) such that ∇ √ f ∈ L 2 (R d
). With θ f , µ f and ϕ defined by ( 3) and (8), we have

θ f 2 R d |∇f | 2 f dx - R d f log f dx - d 2 log 2 π e 2 θ f R d f dx ≥ 2 min µ∈M ϕ e[f |µ] = 2 ϕ e[f |µ f ] ≥ f -µ f 4 L 1 (R d ) 16 M 2 f .
3 An improved version of the generalized Poincaré inequalities for Gaussian measures

We consider the inequalities introduced by W. Beckner in [Beckner, 1989, theorem

1]. If µ(x) = (2 π) -d/2 e -|x| 2 /2 , then for any p ∈ [1, 2) we have u 2 L 2 (R d , dµ) -u 2 L p (R d , dµ) ≤ (2 -p) ∇u 2 L 2 (R d , dµ) ∀ u ∈ H 1 (R d , dµ) . ( 10
)
These inequalities interpolate between the Poincaré inequality (p = 1 case) and the logarithmic Sobolev inequality, which is achieved by dividing both sides of the inequality by (2 -p) and passing to the limit as p → 2. Some improvements were obtained already obtained in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF], [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF], [START_REF] Bartier | Convex Sobolev inequalities and spectral gap[END_REF]. What we gain here is that the improvement takes place also in the limit case as p → 2 and is consistent with the results of Proposition 1. Let us define dµ) . With the above notation, for any p ∈ [1, 2) we have

ϕ p (x) := d 4 (1 -x) -2 p d (2-p) -1 ∀ x ∈ [0, 1] . Corollary 4. Assume that u ∈ H 1 (R d , dµ) is such that R d |u| 2 |x| 2 dµ = d u 2 L 2 (R d ,
R d |∇u| 2 dµ ≥ u 2 L 2 (R d , dµ) ϕ p u 2 L 2 (R d , dµ) -u 2 L p (R d , dµ) u 2 L 2 (R d , dµ) . ( 11
)
By homogeneity we can assume that u L 2 (R d , dµ) = 1. The reader is invited to check that

lim p→2 ϕ p 1 -u 2 L p (R d , dµ) = d 4 e 2 d E[u] -1 where E[u] := R d |u| 2 u 2 L 2 (R d , dµ) log |u| 2 u 2 L 2 (R d , dµ) dµ .
The proof of Corollary 4 is a straightforward consequence of (7) and of the following estimate.

Lemma 5. For any p ∈ [1, 2) and any function u ∈ L p ∩ L 2 (R d , dµ), we have

u 2 L 2 (R d , dµ) u 2 L p (R d , dµ) ≤ exp 2-p p E[u]
and, as a consequence, for any

u ∈ L p ∩ L 2 (R d , dµ) such that u L 2 (R d , dµ) = 1, we obtain u 2 L 2 (R d , dµ) -u 2 L p (R d , dµ) ≤ 2 -p p R d |u| 2 log |u| 2 dµ . ( 12
)
Proof . The proof relies on an idea that can be found in [START_REF] Lata | Between Sobolev and Poincaré[END_REF] and goes as follows.

Let us consider the function

k(s) := s log R d u 2 s dµ .
Derivatives are such that

1 2 k ′ (s) = log R d u 2 s dµ - 1 s R d u 2 s log u dµ R d u 2 s dµ , s 3 4 R d u 2 s dµ 2 k ′′ (s) = R d u 2 s dµ R d u 2 s | log u| 2 dµ - R d log u u 2 s dµ 2 ,
hence proving that k is convex by the Cauchy-Schwarz inequality. As a consequence we get that

k ′ (1) ≤ k(s) -k(1) s -1 ∀ s > 1 .
Applied with s = 2/p, this proves that

- R d |u| 2 log |u| 2 u 2 L 2 (R d , dµ) dµ ≤ p 2 -p u 2 L 2 (R d , dµ) log u 2 L p (R d , dµ) u 2 L 2 (R d , dµ)
, from which we deduce the second inequality in (12) after observing that -log x ≥ 1 -x.

The result of Corollary 4 deserves a comment. As x → 0, ϕ p (x) ∼ p 2 x 2-p , so that we do not recover the optimal constant in (10) in the asymptotic regime corresponding to u L p (R d , dµ) / u L 2 (R d , dµ) → 1, that is when u approaches a constant, because of the factor p 2 . On the other hand, ( 11) is a strict improvement compared to (10) as soon as u 2 L p (R d , dµ) / u 2 L 2 (R d , dµ) < x ⋆ (p) where x ⋆ (p) is the unique solution to ϕ p (x) = x 2-p in (0, 1). Let Φ p be the function defined by

Φ p (x) = ϕ p (x) if x ∈ (0, x ⋆ (p)) , Φ p (x) = x 2 -p if x ∈ [x ⋆ (p), 1] . ( 13
)
Collecting these estimates of ( 10) and ( 11), we can write that dµ) . This is an improvement with respect to (10) because Φ p (x) ≥ x 2-p , with a strict inequality if x < x ⋆ (p). The right hand side in (11) controls the distance to the constants. Indeed, using for instance Hölder's estimates, it is easy to check that

R d |∇u| 2 dµ ≥ u 2 L 2 (R d , dµ) Φ p u 2 L 2 (R d , dµ) -u 2 L p (R d , dµ) u 2 L 2 (R d , dµ) for any function u ∈ H 1 (R d , dµ) such that R d |u| 2 |x| 2 dµ = d u 2 L 2 (R d ,
u 2 L 2 (R d , dµ) -u 2 L p (R d , dµ) ≥ u 2 L 2 (R d , dµ) -u 2 L 1 (R d , dµ) = R d |u -u| 2 dµ with u = R d |u| dµ.
Sharper estimates based for instance on variants of the Pinsker-Csiszár-Kullback inequality can be found in [START_REF] Cáceres | Nonlinear stability in L p for a confined system of charged particles[END_REF], [START_REF] Bartier | A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients[END_REF].

4 Stability results for some Gagliardo-Nirenberg inequalities

4.1 A first case: q > 1
We study the case of Gagliardo-Nirenberg inequalities

∇w ϑ L 2 (R d ) w 1-ϑ L q+1 (R d ) ≥ C GN w L 2q (R d ) (14)
with ϑ = d q q-1 d+2-q (d-2) . The value of the optimal constant has been established in Del Pino and Dolbeault [2002] (also see [START_REF] Gunson | Inequalities[END_REF] for an earlier but partial contribution).

Let us start with some elementary observations on convexity. Consider two positive constants a and b. Let us define

ζ = b a + b , κ = a b ζ + b a 1-ζ = a + b a 1-ζ b ζ .
Next let us take three positive numbers, A, B, and C such that A ζ B 1-ζ ≥ C and consider the function

h(λ) = λ a A + λ -b B -κ C .
The function h reaches its minimum at λ = λ * := b B a A 1 a+b and it is straightforward to check that

h(1) ≥ inf λ>0 h(λ) = h(λ * ) = κ A ζ B 1-ζ -C .
This computation determines the choice of κ. Using the assumption A ζ B 1-ζ ≥ C, we get the estimate

A + B -κ C ≥ C 1 ζ B 1-1 ζ + B -κ C = ϕ(B * -B)
where

B * := C 1 -ζ ζ ζ and ϕ(s) := C 1 ζ (B * -s) 1-1 ζ -B 1-1 ζ * -s . (15)
Indeed, ζ = b a+b leads to the identity

κ C = B * + C 1 ζ B 1-1 ζ *
.

Note that ϕ is a nonnegative strictly convex function such that ϕ(0) = 0 and ϕ ′′ (s

) > ϕ ′′ (0) = 1-ζ ζ 2 C 1 ζ B -1-1 ζ *
for any s ∈ (0, B * ).

We apply these preliminary computations with

a = d q -(d -2) , b = d q -1 2 q A = 1 4 (q 2 -1) R d |∇w| 2 dx , B = β R d |w| q+1 dx , β = 2 q q -1 -d C = ( 1 4 (q 2 -1)) ζ β 1-ζ C GN w L 2q (R d ) α , α = q + 1 -ζ (q -1) for any q ∈ 1, d d-2 . With K := ( 1 4 (q 2 -1)) ζ β 1-ζ κ, the functional J[w] := 1 4 (q 2 -1) R d |∇w| 2 dx + β R d |w| q+1 dx -K C α GN R d |w| 2q dx α 2q
is nonnegative and achieves its minimum at w * (x) = (1 + |x| 2 ) 1 1-q . Hence we have that

J[w] ≥ J[w * ] = 0
and this inequality is equivalent to ( 14), after an optimization under scaling. Notice that ϑ = 2 ζ/α. Theorem 6. With the above notations and ϕ given by ( 15), we have

J[w] ≥ ϕ β R d |w * | q+1 dx - R d |w| q+1 dx (16) for any w ∈ L q+1 (R d ) such that R d |∇w| 2 dx < ∞ and R d |w| 2q |x| 2 dx = R d w 2q * |x| 2 dx.
Proof . The reader is invited to check that, with the above notations,

B * -B = β R d |w * | q+1 dx - R d |w| q+1 dx .
As a last remark in this section, let us observe that the logarithmic Sobolev inequality appears as a limit case of the entropy -entropy production inequality, and that (2) is also obtained by taking the limit as q → 1 in Gagliardo-Nirenberg inequalities in ( 14): see Del Pino and Dolbeault [2002] for details. Also, when d ≥ 2, the convexity of ϕ is lost as q → d d-2 , which corresponds to Sobolev's inequality. This shows the consistancy of our method.

4.2 A second case: q < 1 Now we study the case of Gagliardo-Nirenberg inequalities

∇w ϑ L 2 (R d ) w 1-ϑ L 2q (R d ) ≥ C GN w L q+1 (R d ) (17) with q = 1 2 p-1 < 1, ϑ = d 1+q 1-q d-q (d-2) and we denote by w L 2q (R d ) the quantity R d |w| 2q dx 1 2q
for any q ∈ (0, 1), even for q < 1/2 (in that case, it is only a semi-norm).

Our elementary estimates have to be adapted. Consider two positive constants a and b, with a > b. Let us define

η = b a -b , κ = b a η - b a 1+η = a -b b -η a 1+η .
Next let us take three positive numbers, A, B, and C such that A -η B 1+η ≤ C and consider the function

h(λ) = λ a A -λ b B + κ C .

The function h reaches its minimum at

λ = λ * := b B a A 1 a-b and it is straightforward to check that h(λ) ≥ h(λ * ) = κ C -A -η B 1+η .
Using the assumption A -η B 1+η ≤ C, we get the estimate

A -B + κ C ≥ C -1 η B 1+ 1 η -B + κ C = ϕ(B -B * )
where

B * := C η 1 + η η and ϕ(s) = C -1 η (B * + s) 1+ 1 η -B 1+ 1 η * -s (18)
is a nonnegative strictly convex function such that ϕ(0) = 0 and ϕ ′′ (s) > ϕ ′′ (0) > 0 for any s > 0.

We apply these preliminary computations with

a = d q -(d -2) , b = d 1 -q 2 q A = 1 4 (q 2 -1) R d |∇w| 2 dx , B = β R d |w| q+1 dx , β = 2 q 1 -q + d C = ( 1 4 (q 2 -1)) -η β 1+η (C GN ) -(q+1)(1+η) w α L 2q (R d )
, α = q + 1 + η (q -1) for any q ∈ (0, 1). With K := ( 14 (q 2 -1)) -η β 1+η κ, the functional

J[w] := 1 4 (q 2 -1) R d |∇w| 2 dx -β R d |w| q+1 dx + K (C GN ) -(q+1)(1+η) R d |w| 2q dx α 2q
is nonnegative and achieves its minimum at w * (x) = (1 -|x| 2 )

1 q-1 + . Hence we have that J[w] ≥ J[w * ] = 0
and this inequality is equivalent to (17), after an optimization under scaling. Notice that ϑ = 2 η (q+1) (1+η) . Theorem 7. With the above notations and ϕ given by ( 18), we have

J[w] ≥ ϕ β R d |w| q+1 dx - R d |w * | q+1 dx ∀ w ∈ L q+1 (R d ) such that R d |∇w| 2 dx < ∞ . ( 19
)
Proof . The proof is similar to the proof of Theorem 6 except that the roles of

∂f ∂t = ∆f -x • ∇f (20) with initial datum f 0 ∈ L 1 + (R d , (1 + |x| 2
) dµ and define the entropy as

E[f ] := R d f log f dµ .
Using (1), a standard computation shows that a solution f = (t, •) to (20) satisfies

d dt E[f ] = -4 R d |∇ f | 2 dµ ≤ -2 E[f ] , thus proving that E[f (t, •)] ≤ E[f 0 ] e -2t ∀ t ≥ 0 . (21)
It is well known that M = R d f (t, •) dµ does not depend on t ≥ 0. Since the second moment evolves according to

d dt R d f |x| 2 dµ = 2 R d f (d -|x| 2 ) dµ , if we assume that R d f 0 |x| 2 dµ = d M , then we get also that R d f (t, •) |x| 2 dµ = d M for any t ≥ 0.
Theorem 8. Let d ≥ 1 and consider a nonnegative solution to (20) with initial datum

f 0 such that E[f 0 ] is finite and R d f 0 |x| 2 dµ = d R d f 0 dµ. Then we have E[f (t, •)] ≤ - d 2 log 1 -1 -e -2 d E[f0] e -2t ∀ t ≥ 0 . (22)
Proof . The proof relies on the estimate

d dt E[f ] = -d R d |∇ f | 2 dµ ≥ d 4 exp 2 d E[f ] -1 ,
according to (7).

Let us conclude this section on the Ornstein-Uhlenbeck equation with some remarks.

(i) The estimate ( 22) is better than ( 21): if we let x = e -2t and a = e -2 d E[f0] , then

- d 2 log 1 -1 -e -2 d E[f0] e -2t ≤ E[f 0 ] e -2t
for any t ≥ 0 is equivalent to prove that h

(x) = 1 -a x -(1 -a)
x is nonnegative for any a ≥ 0 and any x ∈ (0, 1). This is indeed the case because h(0) = h(1) = 0 and h ′′ (x) = -a x (log x) 2 < 0.

The improvement degenerates as E[f 0 ] → 0 + . Indeed, we may observe that, for a given t ≥ 0,

- d 2 log 1 -1 -e -2 d E0 e -2t ∼ E 0 e -2t as E 0 → 0 + .
(ii) Similar results can be obtained using the generalized Poincaré inequalities for Gaussian measures of Section 3. With q = 2/p, we get that

d dt R d f q -M q q -1 dµ = - 4 q R d |∇f q/2 | 2 dµ ≤ Ψ p R d f q -M q q -1 dµ with Ψ p (t) := max 2, 2 p 2-p ϕ p ( 2-p p t) = 2 p 2-p Φ p ( 2-p p t)
with Φ p as in (13), and thus get an improvement of the standard estimate

R d f q -M q q -1 dµ ≤ R d f q 0 -M q q -1 dµ e -2t ∀ t ≥ 0 if R d f q 0 -M q q-1 dµ < p 2-p x ⋆ (p).
(iii) None of the above improvements requires that R d f x dµ = 0. If this condition is added at t = 0, it is preserved by the flow corresponding to (20) and spectral methods allows to prove that

R d f q -M q q -1 dµ ≤ R d f q 0 -M q q -1 dµ e -2λt ∀ t ≥ 0
where λ is the best constant in the inequality

u 2 L 2 (R d , dµ) -u 2 L p (R d , dµ) ≤ 1 λ ∇u 2 L 2 (R d , dµ) for any u ∈ H 1 (R d , dµ) such that R d u (1, x, |x| 2 -d) dµ = (0, 0, 0).
According to [Arnold et al., 2007, Theorem 2.4], we know that λ ≥ λ3 1-(p-1) λ 3 /λ 1 where (λ i ) i≥0 are the eigenvalues of the Ornstein-Uhlenbeck operator -∆ + x • ∇. Standard results on the harmonic oscillator allow us to prove that λ i = i, where the eigenspaces associated with i = 0, i = 1 and i = 2 are generated respectively by the constants, x i with i = 1, 2,... d and |x| 2 -d.

Nonlinear case: the fast diffusion equation

The inequality J[w] ≥ 0 in Section 4.1 is also known as the entropy -entropy-production inequality for the fast diffusion equation, as was shown in Del Pino and Dolbeault [2002]. Also see [START_REF] Arnold | Entropies and equilibria of many-particle systems: an essay on recent research[END_REF] for a review on these methods. Here are some details. Let us consider the free energy

F [v] := 1 p -1 R d v p -B p -p B p-1 (v -B) dx
where the Barenblatt profile B is defined by

B(x) = 1 + |x| 2 1 p-1 ∀ x ∈ R d .
and has mass

M * := R d B dx = π d 2 Γ 1 1-p -d 2 Γ 1 1-p .
Next we can define the generalized Fisher information by

I[v] := p 1 -p R d v ∇v p-1 -2 x 2 dx
and consider the deficit functional

J [v] := I[v] -4 F [v] .
We may also define the temperature as

Θ[v] := 1 d R d |x| 2 v dx R d v dx . It turns out that J [v] = J[w] if v p-1 2 = w and q = 1 2 p-1 . Notice that q ∈ 1, d d-2 is equivalent to p ∈ (p 1 , 1) with p 1 := d-1 d . Recall that β = 2 q q-1 -d = 1 1-p -d. If Θ[v] = Θ[B], we observe that F [v] := 1 p-1 R d (v p -B p ) dx because R d B p-1 (v -B) dx = 0.
Theorem 6 can be rephrased in terms of v as follows.

Corollary 9. Let p ∈ (p 1 , 1). Assume that v is a nonnegative function in L 1 (R d ) is such that v p and v |x| 2 are both in L 1 (R d ), and ∇v is in L 2 (R d ). With the above notations and ϕ defined by (15) we have

J [v] ≥ ϕ β (1 -p) F [v] if R d v dx = M * and Θ[v] = Θ[B] .
Since ϕ ′′ (s) ≥ ϕ ′′ (0) for any admissible s ≥ 0, we have in particular that

J [v] ≥ κ F [v]
2 under the assumptions of Corollary 9, with κ = 1 2 β 2 (1 -p) 2 ϕ ′′ (0). Hence Corollary 9 allows to recover [Dolbeault and Toscani, 2013, Theorem 8] with a much simpler proof.

In a second step, we can get rid of the constraints on the mass and on the second moment. Let us define

Θ * := Θ[B] . If we write that u(x) = λ 1+α d σ -d 2 v λ α x/ √ σ with λ = M M *
for some α ∈ R to be determined later and choose σ > 0 such that

σ = σ[u] := λ -(1+2α) R d u |x| 2 dx R d B |x| 2 dx ,
then we can define the corresponding Barenblatt profile by

B M,σ (x) = λ 1+α d σ -d 2 B λ α x/ √ σ ∀ x ∈ R d ,
and the relative entropy and the relative Fisher information respectively by

F M,σ [u] := 1 p -1 R d u p -B p M,σ -p B p-1 M,σ (u -B M,σ ) dx and, with p c := d-2 d , I M,σ [u] := p 1 -p σ d 2 (p-pc) R d u ∇u p-1 -∇B p-1 M,σ 2 dx .
The Barenblatt profile B M,1 (x) takes the form (C M + |x| 2 ) 1/(p-1) for some positive constant

C M = λ 1+α d p-1 if and only if α = 1 -p 2 -d (1 -p) .
With this choice, the functionals F M,σ [u] and I M,σ [u] have the same scaling properties, so that we get

J M,σ [u] := I M,σ [u] -4 F M,σ [u] = λ 2 p-d (1-p) 2-d (1-p) σ d 2 (1-p) J [v] .
By expressing the values of λ and σ in terms of M and Θ[u], we have that

λ = M M * and σ = σ[u] = M * M 2 (2-p)-d (1-p) 2-d (1-p) Θ[u] Θ * , λ 2 p-d (1-p) 2-d (1-p) σ d 2 (1-p) = h(M, Θ[u]) with h(M, Θ) := M M * d 2 (1-p) 2 -2 d (p 2 -4 p+3)+4 p 2 [2-d (1-p)] Θ Θ * d 2 (1-p)
We can now rephrase Corollary 9 for a general function u as follows.

Corollary 10. Let p ∈ (p 1 , 1). Assume that u is a nonnegative function in L 1 (R d ) is such that u p and u |x| 2 are both in L 1 (R d ), and ∇u is in L 2 (R d ). With the same notations as in Corollary 9 and ϕ defined by ( 15) we have

J [u] M,σ ≥ h(M, Θ[u]) ϕ β (1 -p) F M,σ [u] h(M, Θ[u]) with σ = σ[u] .
The choice σ = σ[u] is remarkable because

F M,σ[u] [u] = inf σ>0 F M,σ [u] ,
so that B M,σ[u] is the best matching Barenblatt profile, among all Barenblatt profiles with mass M and characteristic scale σ > 0, when measured in relative entropy. See Dolbeault andToscani [2011, 2013] for more details.

One of the interests of the statement of Corollary 10 is that the free energy F M,σ[u] [u] is an explicit distance to the manifold of the optimal functions for the Gagliardo-Nirenberg inequalities. We have indeed the following Csiszár-Kullback inequality.

Theorem 11. [Dolbeault and Toscani, 2013, Theorem 4

] Let d ≥ 1, p ∈ (d/(d + 2), 1) and assume that u is a non-negative function in L 1 (R d ) such that u p and x → |x| 2 u are both integrable on R d . If u L 1 (R d ) = M , then F M,σ[u] [u] ≥ p σ[u] d 2 (1-p) 8 R d B p M,1 dx C M u -B M,σ[u] L 1 (R d ) + 1 σ[u] R d |x| 2 |u -B M,σ[u] | dx 2 .
The fast diffusion equation written in self-similar variables is

∂u ∂t + ∇ v • σ d 2 (p-pc) ∇v p-1 -2 x = 0 ( 23
)
and it is equivalent to the fast diffusion equation ∂v ∂t = ∆v p up to a rescaling: see for instance Del Pino and Dolbeault [2002], [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF] when σ is taken constant and [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF][START_REF] Toscani | An information-theoretic proof of Nash's inequality[END_REF][START_REF] Dolbeault | Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays[END_REF] when σ = σ[u(t, •)] depends on t. By using Corollary 10, we get that

d dt F M,σ[u(t,•)] [u(t, •)] = -I M,σ[u(t,•)] [u(t, •)]
while a direct computation shows that

d dt σ[u(t, •)] = - 2 (1 -p) 2 p M Θ[B M,1 ] σ[u(t, •)] d 2 (p-pc) F M,σ[u(t,•)] [u(t, •)] ≤ 0 .
Altogether this establishes a faster convergence rate of the solutions towards the Barenblatt profiles than one would get using the entropy -entropy production inequality J M,σ[u] [u] ≥ 0, and even a better rate that the one found in Dolbeault andToscani [2013, 2015b] because ϕ ′′ (s) > ϕ ′′ (0). Details are out of the scope of this paper and a simplified method will appear in [START_REF] Dolbeault | Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays[END_REF].

Nonlinear case: the porous medium equation

Now we turn our attention to the inequality J[w] ≥ 0 in Section 4.2 which was also studied in Del Pino and Dolbeault [2002]. With p > 1, we may consider the free energy

F [v] := 1 p -1 R d v p -B p + p |x| 2 (v -B) dx for any nonnegative function v such that R d v dx = R d B dx, where the Barenblatt profile B is now defined by B(x) = 1 -|x| 2 1 p-1 + ∀ x ∈ R d .
and has mass

M * := π d 2 Γ p p-1 Γ p p-1 + d 2 .
Next we can define the generalized Fisher information by

I[v] := p p -1 R d v ∇v p-1 -2 x 2 dx
and consider the deficit functional

J [v] := I[v] -4 F [v] .
As before, we may also define the temperature as

Θ[v] := 1 d R d |x| 2 v dx R d v dx . It turns out that J [v] = J[w] if v p-1 2 = w and q = 1 2 p-1 . Notice that q ∈ (0, 1) is equivalent to p ∈ (1, ∞). Recall that β = 2 q 1-q + d = 1 p-1 + d. If Θ[v] = Θ[B], we observe that F [v] := 1 p-1 R d (v p -B p ) dx.
Theorem 7 can be rephrased in terms of v as follows.

Corollary 12. Let p ∈ (1, +∞). Assume that v is a nonnegative function in L 1 (R d ) is such that v p and v |x| 2 are both in L 1 (R d ), and ∇v is in L 2 (R d ). With the above notations and ϕ defined by (18) we have

J [v] ≥ ϕ β (p -1) F [v] if R d v dx = M * and Θ[v] = Θ[B] . Θ[u] Θ * , λ 2 p-d (1-p) 2-d (1-p) σ d 2 (1-p) = h(M, Θ[u]) with h(M, Θ) := M M * d 2 (1-p) 2 -2 d (p 2 -4 p+3)+4 p 2 [2-d (1-p)] Θ Θ * d 2 (1-p)
We can now rephrase Corollary 12 for a general function u as follows.

Corollary 13. Let p ∈ (1, +∞). Assume that u is a nonnegative function in L 1 (R d ) is such that u p and u |x|2 are both in L 1 (R d ), and ∇u is in L 2 (R d ). With the same notations as in Corollary 12 and ϕ defined by (18) we have

J [u] M,σ ≥ h(M, Θ[u]) ϕ β (p -1) F M,σ [u] h(M, Θ[u]) with σ = σ[u] .
For the same reason as in the fast diffusion case, the choice σ = σ[u] is remarkable because

F M,σ[u] [u] = inf σ>0 F M,σ [u] ,
so that B M,σ[u] is the best matching Barenblatt profile, among all Barenblatt profiles with mass M and characteristic scale σ > 0, when measured in relative entropy. One of the interests of the statement of Corollary 13 is that the free energy F M,σ[u] [u] is an explicit distance to the manifold of the optimal functions for the Gagliardo-Nirenberg inequalities. Variants of the Pinsker-Csiszár-Kullback inequality can be found in [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to selfsimilarity[END_REF], Del Pino and Dolbeault [2002] and allow to control u -B M,σ[u] L 1 (R d ) under additional assumptions. Here we shall give a simpler result which goes as follows.

Theorem 14. Let d ≥ 1, p ∈ (1, ∞) and assume that u is a non-negative function in L 1 (R d ) such that u p and x → |x| 2 u are both integrable on R d . With previous notations, we have that

F M,σ[u] [u] ≥ p p -1 min{1, p -1} u -B M,σ[u] p L p (R d ) .
Proof . Let us consider the function χ(t) := t p -1 -p (t -1) -c p |t -1| p with c p := min{1, p -1}. The reader is invited to check that for some t p ∈ (0, 1), χ ′′ (t) < 0 for any t ∈ (0, t p ), χ ′′ (t) > 0 for any t ∈ (t p , 1) ∪ (1, +∞). Since χ(0) = χ(1) = 0 and χ ′ (0) = 0, we get that χ(t) ≥ 0 for any t ≥ 0. As above, let us define v by the rescaling (24) and consider

F [v] = 1 p -1 |x|≤1 χ v B B p dx + 1 p -1 |x|>1 u p -p (1 -|x| 2 ) u dx .
It is straightforward to check that

F [v] = c p p -1 |x|≤1 |v -B| p dx + 1 p -1 |x|>1 u p dx ≥ c p v -B p L p (R d )
and we conclude using the identities

F M,σ[u] [u] = h(M, Θ[u]) F [v] and u -B M,σ[u] p L p (R d ) = h(M, Θ[u]) v -B p L p (R d ) .
The porous medium equation written in self-similar variables is

∂u ∂t = ∇ v • σ d 2 (p-pc) ∇v p-1 + 2 x = 0 ( 25
)
and it is equivalent to the porous medium equation ∂v ∂t = ∆v p up to a rescaling: as in [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF][START_REF] Toscani | An information-theoretic proof of Nash's inequality[END_REF][START_REF] Dolbeault | Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays[END_REF] we may choose σ = σ[u(t, •)] to depend on t. By using Corollary 13, we get that Altogether this establishes a faster convergence rate of the solutions towards the Barenblatt profiles than one would get using the entropy -entropy production inequality J M,σ[u] [u] ≥ 0. This is new. More details based on a simpler method will appear in [START_REF] Dolbeault | Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays[END_REF].

As a concluding remark, which is valid for the porous medium case, for the fast diffusion case, and also for the linear case of the Ornstein-Uhlenbeck equation, we may observe that two moments are involved while the center of mass R d x u dx is not supposed to be equal to 0. In that sense our measurement of the distance to the manifold of optimal functions by the deficit functional is an explicit but still a rough estimate, which is definitely not optimal at least in the perturbation regime, that is, close to the optimal functions, which is precisely the regime that was considered in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF].

5

  R d |w * | q+1 dx and R d |w * | 2q dx are exchanged. Some consequences for diffusion equations 5.1 Linear case: the Ornstein-Uhlenbeck equation Let us consider the Ornstein-Uhlenbeck equation (or backward Kolmogorov equation)

  σ[u(t,•)] [u(t, •)] = -I M,σ[u(t,•)] [u(t, •)] while a direct computation shows that d dt σ[u(t, •)] = -2 (p -1) 2 p M Θ[B M,1 ] σ[u(t, •)] d 2 (p-pc) F M,σ[u(t,•)] [u(t, •)] ≤ 0 .

  , Proposition 2, p. 694]. As a preliminary step, we recover the scale invariant, Euclidean, version of the logarithmic Sobolev inequality from (1). Let v := u √ µ. We observe that R d |v| 2 dx = 1 and R d |x| 2 |v| 2 dx = d. With one integration by parts, we get that

April 17, 2015 c 2014 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

(2-p)-d (1-p) 2-d (1-p)

Acknowlegments. This work has been partially supported by the projects STAB, NoNAP and Kibord of the French National Research Agency (ANR). J.D. thanks the Department of Mathematics of the University of Pavia for inviting him and G. Savaré for stimulating discussions. The authors thank F. Bolley for pointing them a missing reference. c 2014 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Since ϕ ′′ (s) ≥ ϕ ′′ (0) for any admissible s ≥ 0, we have in particular that

2 under the assumptions of Corollary 12, with κ = 1 2 β 2 (1 -p) 2 ϕ ′′ (0). The result of Corollary 12 is new. In a second step, we can get rid of the constraints on the mass and on the second moment. Let us define

If we write that

for some α ∈ R to be determined later and choose σ > 0 such that

then we can define the corresponding Barenblatt profile by

and the relative entropy and the relative Fisher information respectively by

where p c = d-2 d . The Barenblatt profile B M,1 (x) takes the form (C M -|x| 2 )

1/(p-1) +

for some positive constant

.

With this choice, the functionals F M,σ [u] and I M,σ [u] have the same scaling properties, so that we get

By expressing the values of λ and σ in terms of M and Θ[u], we have that