Jean Dolbeault

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Keywords: Mathematics Subject Classification (2020). Primary: 46E35, 35K55, 35B06. Secondary: 26D10, 49J40, 35B40, 49K20, 49K30, 35J60, 35J20, 53C21 Interpolation, Gagliardo-Nirenberg inequality, Caffarelli-Kohn-Nirenberg inequality, stability, entropy methods, entropy-entropy production inequality, carré du champ, fast diffusion equation, Harnack Principle, asymptotic behaviour, Hardy-Poincaré inequalities, spectral gap, intermediate asymptotics, self-similar Barenblatt solutions, rates of convergence, symmetry, symmetry breaking, bifurcation

Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view.

This paper aims at illustrating some of these recent aspect of entropyentropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

The two main results of Sections 2 and 3 are based on recent papers. Stability results on Gagliardo-Nirenberg-Sobolev inequalities have been obtained in collaboration with M. Bonforte, B. Nazaret, and N. Simonov and are collected in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]. Symmetry results in Caffarelli-Kohn-Nirenberg inequalities rely on a series of papers with M.J. Esteban, M. Loss and M. Muratori, see [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF][START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]. The guideline of this paper is to clarify the interplay of nonlinear interpolation inequalities with their linearized counterparts using entropy methods and fast diffusion flows.

A brief historical perspective

The goal of this section is to sketch important steps in the development of the theory of some important functional inequalities, in the perspective of entropy methods. It is neither a complete review nor a detailed historical account, but more a personal selection of some results which are highlighted as representative of the evolution of the ideas. Several results have been discovered at least twice, in completely independent papers. Whenever possible, this has been clarified, but such issues are always delicate and the overall information is probably still incomplete.

Sobolev and some other interpolation inequalities

Unless it is specified, we consider functions on R d . For any q ∈ [1, ∞), for any smooth and compactly supported function, where 2 * = 2 d /(d -2) is the critical exponent and the optimal constant is

f q := R d | f | q d x 1/q , whilst f ∞ denotes the L ∞ (R d ) norm.
S d = π d (d -2) Γ(d /2) Γ(d ) 2/d = 1 4 d (d -2) 2 2 d π 1+ 1 d Γ (d + 1)/2 2/d = 1 4 d (d -2) S d 2 d .
Inequality (1.1) is easily extended to the Beppo-Levi space using standard completion results with respect to the norm

f → ∇ f 2 2 + f 2 2 *
1/2 , see for instance [START_REF] Lions | Les espaces du type de Beppo Levi[END_REF]. Inequality (1.1) appears in [START_REF] Sobolev | Sur un théorème d'analyse fonctionnelle[END_REF] and is extremely well understood:

see [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF]Elliptic equations and rearrangements[END_REF][START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF]. The question of the optimality of the constant S d has anyway a much longer history. Among radial functions and at least in dimension d = 3 for the critical exponent 2 * = 6, the solutions of the Euler-Lagrange equation enter in the study of the polytropic and isothermal gas spheres, which are models of gaseous stars in astrophysics and can be traced back to the work of H. Lane in [START_REF] Lane | On the theoretical temperature of the sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment[END_REF]. A detailed account can be found in [START_REF] Chandrasekhar | An introduction to the study of stellar structure[END_REF]Chapter IV]. The explicit computation of the optimal radial functions and the corresponding value of S d appears in [START_REF] Bliss | An integral inequality[END_REF]. Further considerations on the linear stability of the radial optimal functions can be found in [START_REF] Rosen | Minimum value for c in the Sobolev inequality φ 3 ≤ c ∇φ 3[END_REF].

The classical papers for the optimal Sobolev inequality are those of T. Aubin and G. Talenti [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF], with an earlier contribution by E. Rodemich in [START_REF] Rodemich | The Sobolev inequalities with best possible constants[END_REF] which is quoted for instance in [66, p. 158]. These papers rely on spherically symmetric rearrangements and particularly on the Pólya-Szegő principle: we refer to [START_REF] Kawohl | Rearrangements and convexity of level sets in PDE[END_REF][START_REF] Kesavan | Symmetrization & applications[END_REF][START_REF]Inequalities in rearrangement invariant function spaces[END_REF] and references therein for details. An alternative proof based on Riesz' lemma and duality can be found in [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF][START_REF] Lieb | Analysis[END_REF]. Many other proofs have been established using various techniques like, e.g., mass transport in [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities[END_REF]. Some results based on nonlinear flows and entropy methods, which completely avoid spherically symmetric rearrangements, will be exposed in Section 2.

Optimal functions in (1.1) are known as the Aubin-Talenti functions. The set M of these functions is a (d +2)-dimensional manifold generated by dilations, translations and multiplications by a constant of

g(x) := 1 + |x| 2 -(d -2)/2 ∀ x ∈ R d .
Stability is a delicate issue. As an answer to some of the questions raised in [START_REF] Lieb | Sobolev inequalities with remainder terms[END_REF], G. Bianchi and H. Egnell proved in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] the existence of C > 0 such that

∇ f 2 2 -S d f 2 2 * ≥ C inf g ∈M ∇ f -∇g 2 2 , (1.2)
The existence of C is obtained by concentration-compactness methods and by contradiction, hence comes with no estimate. Various refinements starting with [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF] and the more recent results of [START_REF] Figalli | Gradient stability for the Sobolev inequality: the case p ≥ 2[END_REF][START_REF] Neumayer | A note on strong-form stability for the Sobolev inequality[END_REF][START_REF] Zhang | Sharp gradient stability for the Sobolev inequality[END_REF] have been obtained, although without explicit and constructive estimates on C . By duality methods inspired by [START_REF]Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] and flows, quantitative and constructive results were obtained in weak norms in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF][START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF]. In [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF], a result of stability has been obtained, with a measure of the distance to M by a relative Fisher information, a strong norm: this result will be presented in Section 2.

Gagliardo-Nirenberg inequalities.

In two papers, [START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF][START_REF] Nirenberg | On elliptic partial differential equations[END_REF], E. Gagliardo and L. Nirenberg established a number of interpolation inequalities which are now famous. Some historical details can be found in [START_REF]Foreword: issue dedicated to professor Emilio Gagliardo[END_REF]. Here we are interested in the sub-family ∇ f θ 2 f 1-θ p+1 ≥ C GNS (p) f 2p (1.3) where the exponent θ =

d (p-1) (d +2-p (d - 2
)) p is fixed by scaling considerations and functions f are taken in the space obtained as the completion of the space of infinitely differentiable functions on R d with compact support, with respect to the norm

f → (1 -θ) f p+1 + θ ∇ f 2 . We shall say that the exponent p is an admissible exponent if p ∈ (1, +∞) if d = 1 or 2 , p ∈ (1, p ] if d ≥ 3 with p := d d -2 .
In the limit case where p = p , d ≥ 3, for which θ = 1, we notice that 2 p = 2 * and recover (1.1) as a limit case of (1.3). Other interesting limits can be obtained: as p → 1, Inequality (1.3) degenerates according to [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] into the Euclidean logarithmic Sobolev inequality in scale invariant form, while in the limit as p → +∞ when d = 2 (see for instance [START_REF]The Euclidean Onofri inequality in higher dimensions[END_REF]), we recover the Euclidean Onofri inequality. See [20, Chapter 1] for more details. Inequality (1.3) was proved in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]. Actually, in [START_REF] Gunson | Inequalities in mathematical physics[END_REF], an earlier result was stated with a rather complete scheme of a proof, up to the delicate question of the uniqueness. None of the authors of [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] were aware of [START_REF] Gunson | Inequalities in mathematical physics[END_REF] but they had by the time the paper was written fine uniqueness results available from the mathematical literature. Entropy methods are now providing alternative tools which have been exploited in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]Chapter 1]. In any case, the key issue is that optimal functions in (1.3) can be identified. They are given up to dilations, translations and multiplications by a constant by ) was apparently introduced first by V.P. Il'in in [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF] but is more known in the literature as the critical Caffarelli-Kohn-Nirenberg inequalities, according to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]. An important step in the understanding of this inequality came with [START_REF] Wang | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] where the question of the symmetry breaking problem was raised: in (1.5), is optimality achieved among radially symmetric functions ? Notice that partial symmetry results were known before, see, e.g., [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF][START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF]: the point in [START_REF] Wang | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] (also see [START_REF] Willem | Partial symmetry and asymptotic behavior for some elliptic variational problems[END_REF]) was to provide a mechanism for proving that symmetry breaking also occurs. If symmetry occurs, it can be proved that the equality case is achieved by

g(x) := 1 + |x| 2 -1 p-1 ∀ x ∈ R d . ( 1
R d | f | p |x| b p d x 2/p ≤ C a,b R d |∇ f | 2 |x| 2 a d x (1.5) is valid for any function f ∈ f ∈ L p R d , |x| -b d x : |x| -a |∇ f | ∈ L 2 R d , d x under the conditions that a ≤ b ≤ a +1 if d ≥ 3, a < b ≤ a +1 if d = 2, a +1/2 < b ≤ a +1 if d = 1,
g(x) := 1 + |x| (p-2)(a c -a) -2 p-2 ∀ x ∈ R d (1.6)
and the optimal constant is then given by

C a,b = 2 p ((a c -a) - p+2 p   (p -2) Γ 3 p-2 2 (p-2) 2 π S d -1 Γ p p-2   p-2 p
.

Assuming that this was the case, V. Felli and M. Schneider systematized the strategy of [START_REF] Wang | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF]: by linearizing the functional made of the difference of the two sides in (1.5), they found a negative eigenvalue if

b < b FS (a) with b FS (a) := d (a c -a) 2 (a c -a) 2 + d -1 + a -a c ,
thus proving the linear instability of g, a contradiction. As a consequence, symmetry breaking occurs if b < b FS (a). Reciprocally, one of the major achievements of entropy methods is the following symmetry result.

Theorem 1.1 (Symmetry vs. symmetry breaking, critical case, [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]). Assume that d ≥ 2, a < a c , and b FS (a) ≤ b ≤ a + 1 if a < 0. Then equality in (1.5) is achieved if and only if, up to a scaling and a multiplication by a constant, f = g.

The result in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] is actually stronger: it is proved that there is no other critical point of the Euler-Lagrange equation, which is a rigidity result. The method is not limited to critical inequalities and there is also a family of subcritical Caffarelli-Kohn-Nirenberg inequalities, exactly as in the case without weights. Similar results have been obtained in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF]Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods[END_REF][START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF] for the inequality

R d | f | 2 p |x| γ d x 1/p ≤ C β,γ,p R d |∇ f | 2 |x| β d x θ R d | f | p+1 |x| γ d x 1-θ (1.7)
with optimal constant C β,γ,p , parameters β, γ and p such that

γ -2 < β < d -2 d γ , γ ∈ (-∞, d ) , p ∈ 1, p with p := d -γ d -β -2 , (1.8)
and an exponent θ =

(d -γ) (p-1) p (d +β+2-2 γ-p (d -β-2))
which is determiend by scaling considerations. The admissible set of parameters (β, γ) is restricted by the condition p ≤ p , which means

β ≥ d -2 + γ -d p .
If symmetry holds, then the equality case in (1.7) is achieved by

g(x) := 1 + |x| 2+β-γ -1 p-1 ∀ x ∈ R d .
The following results are taken from [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Theorem 2] and in [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]Theorem 1.1].

Theorem 1.2 (Symmetry vs. symmetry breaking, critical case, [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]). Assume that d ≥ 2 and β, γ and p satisfy (1.8). Symmetry holds in (1.7) if and only if

γ < d , γ -2 < β < d -2 d γ and β ≤ β FS (γ) := d -2 -(γ -d ) 2 -4 (d -1) .

Branches of solutions

In 1983, H. Brézis and L. Nirenberg published a paper, [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], devoted to the positive solutions of the equation

-∆ f + λ f = f p-1 (1.9)
in bounded domains, with homogeneous Dirichlet boundary conditions. They were interested in the case of the critical exponent p = 2 * , d ≥ 3, and related issues of compactness in connection with the Yamabe problem. This paper had a considerable impact. With [START_REF] Floer | Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential[END_REF], it was a starting point for an amazing effort to construct various solutions by variational methods or using Lyapunov-Schmidt reduction techniques, not only on the Euclidean space but also on Riemannian manifolds (where ∆ has to be understood as the Laplace-Beltrami operator). One of the typical issues is to understand how branches of solutions depend on the parameter λ and at which values bifurcations may eventually occur, as part of the larger question of the multiplicity of the solutions and their classification. Nonexistence (typically based on Pohožaev's identity) and uniqueness results are another related issue, with a series of prominent papers, [START_REF] Coffman | Uniqueness of the ground state solution for ∆uu + u 3 = 0 and a variational characterization of other solutions[END_REF][START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF][START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF], that can now be reinterpreted using entropy methods. In [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] for instance, it is proved that (1.9) written on the sphere S d has a unique positive solution, f ≡ λ -1/(p-2) , if and only if λ ≤ d /(p -2). As a consequence, according to [14, Corollary 6.1] (also see [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF]), we obtain the inequality

∇ f 2 L 2 (S d ) ≥ d p -2 f 2 L p (S d ) -f 2 L 2 (S d ) ∀ f ∈ H 1 (S d , d µ) (1.10)
where d µ = S d -1 d v g denotes the uniform probability measure on the sphere and p ∈ (2, 2 * ), with the convention that 2 * = +∞ if d = 1 or 2. Inequality (1.10) has been studied in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] by rigidity methods, in [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF] by techniques of harmonic analysis, and using the carré du champ method in [START_REF] Bentaleb | Inégalité de Sobolev pour l'opérateur ultrasphérique[END_REF][START_REF] Ledoux | Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator[END_REF][START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF], for any p > 2, with improvements and extensions in [START_REF] Dolbeault | Improved interpolation inequalities on the sphere, Discrete and Continuous Dynamical Systems Series S[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF]. We refer to [START_REF]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF] for a review and [START_REF]Improved interpolation inequalities and stability[END_REF] for most recent results. If d ≥ 3, the case p = 2 * is also covered: Inequality (1.10) is the Sobolev inequality on the sphere and it is equivalent to (1.1), up to a stereographic projection. Taking into account the normalization of d µ, it is easy to recover the expression of S d . Another limit case is the logarithmic Sobolev inequality on S d ,

∇ f 2 L 2 (S d ) ≥ d 2 S d | f | 2 log | f | 2 f 2 L 2 (S d ) d µ ∀ f ∈ H 1 (S d , d µ) \ {0} , (1.11)
which corresponds to the limit as p → 2 + . Remarkably, (1.10) is also valid for any p ∈ [1, 2) where the case p = 1 corresponds simply to the Poincaré inequality on the sphere. In this range, and actually for any p

∈ [1, 2) ∪ (2, +∞) if d = 1 or p ∈ [1, 2) ∪ (2, 2 # ] if d ≥ 2, where 2 # := 2 d 2 + 1 (d -1
) 2 is the Bakry-Emery exponent, Inequalities (1.10) and (1.11) were proved earlier by the carré du champ method of D. Bakry and M. Emery in [START_REF] Émery | Hypercontractivité de semi-groupes de diffusion[END_REF].

From the point of view of elliptic PDEs, the carré du champ method amounts to test (1.9) by ∆ f + (p -1) |∇ f | 2 / f . After using the Bochner-Lichnerowicz-Weitzenböck formula

1 2 ∆ (|∇ f | 2 ) = |Hess f | 2 + ∇ • (∆ f ) • ∇ f + Ric(∇ f , ∇ f ) , (1.12) the fact that Ric(∇ f , ∇ f ) = (d -1) |∇ f | 2 on S d
, and a few integrations by parts, we obtain the identity

S d d d -1 |Q f | 2 + (d -λ) |∇ f | 2 + γ(p) |∇ f | 4 f 2 d µ = 0 with Q f := Hess f - 1 d ∆ f Id - d -1 d + 2 (p -1) ∇ f ⊗ ∇ f f - 1 d |∇ f | 2 f Id (1.13) and γ(p) := d -1 d + 2 2 (p -1) (2 # -p) . (1.14)
Here we use the notation

|A| 2 = d i , j =1 A 2 i j if A = (A i j ) d i , j =1 is a d × d matrix.
It is therefore clear that the unique positive solution of (1.9) is a constant if λ ≤ d . This result has an interesting consequence in terms of branches of solutions. On H 1 (S d , d µ), the functional

f → ∇ f 2 L 2 (S d ) - λ p -2 f 2 L p (S d ) -f 2 L 2 (S d )
always admits f ≡ 1 as a critical point, there is no other positive solution of (1.9) if λ ≤ d and the functional is nonnegative. For any λ > d , a simple linearization around the constant solution shows that there is a minimizer which makes the functional negative and λ = d , which corresponds to the first positive eigenvalue of the Laplace-Beltrami operator, is a bifurcation point. Let us define

µ(λ) := inf u∈H 1 (S d )\{0} (p -2) ∇ f 2 L 2 (S d ) + λ f 2 L 2 (S d ) f 2 L p (S d )
.

The uniqueness result in the range λ ∈ [0, d ] and the bifurcation result at λ = d have the following consequences. The carré du champ method is limited to the range p ≤ 2 # , but the cases p ∈ (2 # , +∞) if d = 1 or 2, or p ∈ (2 # , 2 * ] if d ≥ 3 can be dealt with using another test function, as we shall see below: see [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF][START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF] and references therein for more details. The method admits various extensions, for instance to smooth compact connected Riemannian manifolds, or improvements which can be used in order to obtain stability estimates. For instance, on S d , the inequality

If λ ≤ d , then µ(λ) = λ, while µ(λ) < λ if λ > d . See Fig. 1.
∇ f 2 L 2 (S d ) ≥ d 2 -p -γ f 2 L 2 (S d ) -f 2- 2 γ 2-p L p (S d ) f 2 γ 2-p L 2 (S d ) ∀ f ∈ H 1 (S d ) (1.15) is established in [48] for d ≥ 1, p ∈ [1, 2) ∪ (2, 2 # ) with the convention 2 # = +∞ if d = 1 and γ = γ(p) given by (1.14) if d ≥ 3 and γ(p) = (p -1)/3 if d = 1, under the condition that γ(p) = 2 -p. If γ(p) = 2 -p,
there is a logarithmic version of the inequality. Inequality (1.15) can be used to obtain estimates on the branches of solutions as in [START_REF]Improved interpolation inequalities and stability[END_REF]Theorem 2]. A major difficulty in the elliptic point of view of [START_REF] Coffman | Uniqueness of the ground state solution for ∆uu + u 3 = 0 and a variational characterization of other solutions[END_REF][START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] is the construction of the appropriate test functions. Reinterpreted in terms of adapted entropy methods and gradient flow estimates, building the estimates becomes a much simpler task. This is what we will explain next.

Entropies and carré du champ methods

Carré du champ methods were introduced in [START_REF]Diffusions hypercontractives[END_REF] by D. Bakry and M. Emery and have been successfully applied to a wide range of questions: a general overview can be found in [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. Here we give two simple examples, first in the classical setting of linear diffusions with a confinement potential on R d , and then we revisit the inequalities (1.10) of Section 1.2 dealing with interpolation on S d .

Inequalities, rates of convergence and entropy methods for linear diffu-

sions. On R + × R d , let us consider the heat equation ∂u ∂t = ∆u , u(t = 0, x) = u 0 (x) (1.16)
and assume that u 0 ∈ L 1 ∩ L 2 (R d ) is nonnegative. Among various classical results, it is well known that as t → +∞, the solution behaves like the Green function

G(t , x) = (4 π t ) -d /2 exp(-|x| 2 /(4 t )).
As a consequence, we know that u(t , •) 2 2 = O(t -d /2 ) as t → +∞, which is easily recovered using Nash's inequality:

u 2+ 4 d 2 ≤ C Nash u 4 d 1 ∇u 2 2 (1.17)
For a solution of (1.16), we know that u(t , •) 1 = u 0 1 =: M for any t ≥ 0 and can indeed write that

d d t u(t , •) 2 2 = -2 ∇u(t , •) 2 2 ≤ - 2 C Nash u(t , •) 2+4/d 2 M 4/d
and the decay estimate easily follows after integrating with respect to t . A slightly more delicate issue is to obtain an intermediate asymptotics estimate, for instance an estimate of

u(t , •)-M G(t , •) in L 1 (R d ).
This can be done with entropy methods as follows. Using the change of variables [START_REF] Bliss | An integral inequality[END_REF]) is equivalent to solve the Fokker-Planck equation

u(t , x) = R(t ) -d v log R(t ), x R(t ) with R(t ) = 1 + 2 t ∀ (t , x) ∈ R + × R d , solving (1.
∂v ∂t = ∆v + ∇ • (x v) , v(t = 0, x) = u 0 (x) . (1.18)
A remarkable fact is that (1.18) admits a stationary solution M µ with µ(x) := G(1/2, x). Up to an additional change of unknown function, the function w(t , x) = v(t , x)/µ(x) solves the Ornstein-Uhlenbeck equation

∂w ∂t = ∆w + x • ∇w , w(t = 0, x) = u 0 (x) µ(x) . (1.19)
With the Gaussian measure d µ = µ(x) d x, an elementary computations shows that

d d t R d |w -M | 2 d µ = -2 R d |∇w| 2 d µ ≤ -2 R d |w -M | 2 d µ
where the last inequality is a simple consequence of the Gaussian Poincaré inequality

R d f -f 2 d µ ≤ R d |∇ f | 2 d µ where f := R d f d µ , ∀ f ∈ H 1 (R d ) .
It seems that this inequality was explicitly written in that form in [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF], in the very same paper in which J. Nash proved (1.17). With this observation and a Grönwall estimate, we obtain the exponential decay of

R d |w -M | 2 d µ ≥ R d |w -M | d µ 2 , which controls u(t , •) -M G(t , •) 1 .
Alternatively, one can directly consider the entropy and compute

d d t R d w log(w/M ) d µ = - R d w |∇ log w| 2 d µ ≤ -2 R d w log(w/M ) d µ
where the last inequality is a simple consequence of the Gaussian logarithmic Sobolev inequality

R d f 2 log( f 2 /M ) d µ ≤ 2 R d |∇ f | 2 d µ where M = R d f 2 d µ , ∀ f ∈ H 1 (R d ) .
Using again a Grönwall estimate and the Pinsker-Csiszár-Kullback inequality

R d w log(w/M ) d µ ≥ 1 4 M R d |w -M | d µ 2 ,
we can conclude as before. It turns out that the heat equation can be seen as the gradient flow of the entropy R d u log u d x with respect to Wasserstein's distance as shown in [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF] and the entropy formulation turns out to be better adapted when linear diffusion are replaced by nonlinear diffusions. For more details on Nash's inequality, consequences for the heat equation, the comparison of the decay estimates and more detailed references, we refer to [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF][START_REF]A variational proof of Nash's inequality[END_REF].

A natural extension of the above heat related estimates is to consider linear diffusions with confinement due to a given potential φ(x), which amounts to study the Ornstein-Uhlenbeck equation with potential φ, that is,

∂w ∂t = ∆w -∇φ • ∇w . (1.20) This equation is reduced to (1.19) if φ(x) = |x| 2 /2. The invariant measure is now d µ = Z -1 e -φ d x with Z = R d e -φ d x and the Ornstein-Uhlenbeck operator L := ∆ -∇φ • ∇ is self-adjoint on L 2 (R d , d µ) in the sense that R d w 1 L w 2 d µ = - R d ∇w 1 • ∇w 2 d µ
As seen above, obtaining convergence rates heavily relies on functional inequalities. One of the key features of entropy methods is that the flow of (1.20) can be used to establish these inequalities by the carré du champ method, under the assumption that Hess φ ≥ κ Id a.e. (1.21) for some κ > 0. Under this assumption, we claim that

R d |∇ f | 2 d µ ≥ κ p -2 R d | f | p d µ 2/p - R d | f | 2 d µ ∀ f ∈ H 1 (R d , d µ) (1.22)
for any p ∈ [1, 2), and also, by taking the limit as p → 2 -and with the notation

M = R d | f | 2 d µ, R d |∇ f | 2 d µ ≥ κ 2 R d | f | 2 log | f | 2 /M d µ ∀ f ∈ H 1 (R d , d µ) .
If w solves (1.20) with initial datum w 0 = | f | p , the strategy of the carré du champ method is to notice that d d t R d w d µ = 0 and that w 2 converges to the constant

M = R d | f | p d µ while d d t R d ∇w 1/p 2 d µ + κ p -2 R d w 2/p d µ ≤ 0 , so that R d |∇ f | 2 d µ - κ p -2 R d | f | p d µ 2/p - R d | f | 2 d µ ≥ lim t →+∞ R d |∇w(t , x) 1/p | 2 d µ - κ p -2 R d w(t , x) 2/p d µ 2/p -M 2/p = 0 .
The proof relies on a commutation of the derivatives which is is similar to the Bochner-Lichnerowicz-Weitzenböck formula (1.12). Performing a few integrations by parts, we compute for some function

h ∈ H 2 (R d , d µ) the integral R d (L h) 2 d µ = - R d ∇h • ∇(L h) d µ = - R d ∇h • (L ∇h) d µ + R d ∇h • [L , ∇] h d µ = R d |Hessh| 2 d µ + R d ∇h • [L , ∇] h d µ . Since ∇h • [L , ∇] h = Hess φ : ∇h ⊗ ∇h ≥ κ |∇h| 2 according to (1.21), this proves R d (L h) 2 d µ ≥ R d |Hessh| 2 d µ + κ R d |∇h| 2 d µ . (1.23)
Another elementary identity is given by an integration by parts:

R d (L h) |∇h| 2 h d µ = R d |∇h| 4 h 2 d µ -2 R d Hessh : ∇h ⊗ ∇h h d µ . (1.

24)

Hence with h = w 1/p and w solving (1.20), so that h solves

∂h ∂t = L h + (p -1) |∇h| 2 h with initial datum h(t = 0, •) = | f |, we find that d d t R d w 2/p d µ = d d t R d h 2 d µ = -2 (2 -p) R d |∇h| 2 d µ , d d t R d ∇w 1/p 2 d µ = d d t R d |∇h| 2 d µ = -2 R d (L h) L h + (p -1) |∇h| 2 h d µ
Altogether,

d d t R d ∇w 1/p 2 d µ + κ p -2 R d w 2/p d µ = -2 R d Hessh -(p -1) ∇h ⊗ ∇h h 2 d µ -2 (p -1) (2 -p) R d |∇h| 4 h 2 d µ ≤ 0 ,
which completes the proof of (1.22). We refer to [START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-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 | Φ-Entropies: convexity, coercivity and hypocoercivity for Fokker-Planck and kinetic Fokker-Planck equations[END_REF] for further details. Condition (1.21) is the standard criterion in the carré du champ method of D. Bakry and M. Emery, but it is a pointwise condition which is by far too strong, as we use it only after an integration. See [START_REF] Dolbeault | On the Bakry-Emery criterion for linear diffusions and weighted porous media equations[END_REF] for some considerations on nonlocal, spectral conditions. There is actually a deeper property, for the following reason. Let us assume that κ 1 > 0 is the optimal constant in

R d |∇ f | 2 d µ ≥ κ 1 p -2 R d | f | p d µ 2/p - R d | f | 2 d µ ∀ f ∈ H 1 (R d , d µ) (1.25) if p ∈ [1, 2)
, and in

R d |∇ f | 2 d µ ≥ κ 1 2 R d | f | 2 log | f | 2 /M d µ ∀ f ∈ H 1 (R d , d µ) (1.26) if p = 2, with M = R d | f | 2 d µ.
Now, let us assume that κ 2 > 0 is the optimal constant in

R d Hess f -(p -1) ∇ f ⊗ ∇ f f 2 d µ + (p -1) (2 -p) R d |∇ f | 4 f 2 d µ + R d Hess φ : ∇ f ⊗ ∇ f d µ ≥ κ 2 R d |∇ f | 2 d µ ∀ f ∈ H 2 (R d , d µ) . (1.27)
Finally, let us consider the Poincaré inequality

R d |∇ f | 2 d µ ≥ κ 0 R d | f | 2 d µ - R d f d µ 2 ∀ f ∈ H 1 (R d , d µ) (1.28)
and assume that κ 0 > 0 is the optimal constant.

Proposition 1.3. Let d ≥ 1, p ∈ [1, 2), e -φ ∈ L 1 (R d ),
and assume that the probability measure d µ = Z -1 e -φ d x with Z = R d e -φ d x is such that (1.28) holds for some κ 0 > 0. With the above notations, we have

κ 1 > 0 and κ 2 ≤ κ 1 ≤ κ 0 . Moreover, if κ 2 = κ 1 , then κ 2 = κ 1 = κ 0 .
Proof. By Hölder's inequality, we know that

R d | f | d µ 2 ≤ R d | f | p d µ 2/p , so that κ 1 ≥ κ 0 (2 -p) > 0.
The fact that κ 2 ≤ κ 1 is part of the standard carré du champ method and has already been observed above: we know that (1.25) and (1.26) are consequences of (1.27). Inequality (1.25) or (1.26) applied to the function h = 1 + ε f gives, at leading order in the limit as ε → 0, the Poincaré inequality

R d |∇ f | 2 d µ ≥ κ 1 R d | f | 2 d µ - R d f d µ 2 ∀ f ∈ H 1 (R d , d µ) ,
where the constant κ 1 is not necessarily optimal: κ 1 ≤ κ 0 .

If p = 1, we have κ 2 = κ 1 = κ 0 which easily follows from an expansion of the square: let f ∈ H 1 (d µ) be such that R d f d µ = 0. Using the fact that

R d (L f + κ 0 f ) 2 d µ ≥ 0, we find that R d Hess f 2 d µ + R d Hess φ : ∇ f ⊗ ∇ f d µ -κ 0 R d |∇ f | 2 d µ ≥ κ 0 R d |∇ f | 2 d µ -κ 0 R d | f | 2 d µ ≥ 0 ,
which proves that κ 2 ≥ κ 0 . In the remainder of the proof, we assume that p ∈ (1, 2). In the equality case κ 2 = κ 1 , we read from the previous computations that there is no non-constant optimal function for (1.25). This fact is easily obtained as follows: if f is an optimal function for (1.25), let us take w 0 = | f | p and consider the solution w of (1.20) with initial datum w 0 . At t = 0, we obtain

R d f -2 |∇ f | 4 d µ = 0, a contradiction. Hence, if we consider a minimizing sequence ( f n ) n∈N for κ 1 = inf Q p [ f ] where Q p [ f ] := (2 -p) R d |∇ f | 2 d µ R d | f | 2 d µ -R d | f | p d µ 2/p ,
we can write that f n = 1 + ε n h n for any n ∈ N, with lim n→+∞ ε n = 0 and (h n ) n∈N uniformly bounded in H 1 (R d d µ), so that

κ 0 ≥ κ 1 = lim n→+∞ Q p [1 + ε n h n ] = lim n→+∞ Q 1 [h n ] ≥ κ 0 .
For further details in a similar problem, see [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF].

It is possible to prove various other estimates. For instance, the function

p → 2 p 2 -p R d | f | 2 d µ - R d | f | p d µ 2/p
is non-decreasing according to [75, Lemma 1], so that we have the bounds

(2 -p) κ 0 ≤ κ 1 ≤ p κ 0 .
If the logarithmic Sobolev inequality (1.26) holds for some κ 1 > 0, then the Poincaré inequality (1.28) holds for some κ 0 ≥ κ 1 and (1.25) holds for some constant which can be estimated in terms of κ 0 and κ 1 . Additionally, if κ 0 = κ 1 , then κ 0 is also the optimal constant in (1.25) and does not depend on p ∈ [1, 2]: see [START_REF] Latała | Between Sobolev and Poincaré[END_REF][START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF]. This occurs if φ(x) = |x| 2 /2: in that case, we have

κ 2 = κ 1 = κ 0 , for any p ∈ [1, 2].
The carré du champ method gives only a sufficient condition to establish (1.25) or (1.26), and the constant κ in (1.21) is then a lower estimate for κ 2 .

In some cases, it can be proved by a direct computation that κ = κ 2 and then all optimal constants in (1.25)-(1.28) are known: this is the case for the harmonic potential and, as we shall see next, we have a similar picture in all cases of interest considered in this paper. For this reason, the method provides sharp results. From the point of view of the flow, if w solves (1.20) with initial datum w 0 = | f | p , for some non-constant function f ∈ H 1 (R d , d µ), then we have

Q p [ f ] ≥ lim t →+∞ Q p w(t , •) 1/p = lim t →+∞ Q 1 w(t , •) 1/p ≥ κ 0 . Then κ 1 = inf Q p [ f ] = κ 0 is recovered using f = 1 + ε h
as a test function and taking the limit as ε → 0, where h is optimal for (1.28).

1.3.2.

Flows and interpolation inequalities on the sphere. On the sphere S d , exactly the same method as for the Ornstein-Uhlenbeck equation (1.20) applies. If we consider the heat equation

∂w ∂t = ∆w , w(t = 0, •) = f p ,
where ∆ is the Laplace-Beltrami operator or, with h = w 1/p , the equation

∂h ∂t = ∆h + (p -1) |∇h| 2 h , h(t = 0, •) = f , then we find that d d t S d h p d µ = 0 , d d t S d h 2 d µ = -2 (2 -p) S d |∇h| 2 d µ , d d t S d |∇h| 2 d µ = -2 S d (∆h) ∆h + (p -1) |∇h| 2 h d µ .
The strategy of the computation is similar to the case of Equation (1.20). We replace (1.21) by the Bochner-Lichnerowicz-Weitzenböck formula (1.12) applied to h. After an integration with respect to d µ, this proves

S d (∆h) 2 d µ = S d |Hessh| 2 d µ + (d -1) S d |∇h| 2 d µ . (1.29)
After a number integrations by parts, we obtain

d d t ∇h 2 L 2 (S d ) - λ p -2 h 2 L p (S d ) -h 2 L 2 (S d ) = -2 S d d d -1 |Q f | 2 + (d -λ) |∇ f | 2 + γ(p) |∇ f | 4 f 2 d µ
with Q f and γ(p) respectively given by (1.13) and (1.14). Using the fact that h p → M as t → +∞, we find that ∇h 2

L 2 (S d ) -λ p-2 h 2 L p (S d ) -h 2 L 2 (S d )
written with λ = d is monotone non-increasing, with limit 0 as t → +∞, so that (1.10) holds for the initial datum h(t = 0, •) = f . This method can be found in [START_REF]Diffusions hypercontractives[END_REF] but we refer also to [START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF] for details on the computations using the ultraspherical operator and to [START_REF] Dolbeault | Improved interpolation inequalities on the sphere, Discrete and Continuous Dynamical Systems Series S[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF] for some improvements. It is however limited to p ≤ 2 # . It is possible to overcome the limitation p ≤ 2 # by considering the nonlinear diffusion ∂w ∂t = ∆w m , w(t = 0, •) = f p , (

for some appropriately chosen m. The idea was introduced in [START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] and systematized in [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF][START_REF] Dolbeault | Improved interpolation inequalities on the sphere, Discrete and Continuous Dynamical Systems Series S[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF]. Let us give some details on how to use (1.30). We assume that

p ∈ [1, 2) ∪ (2, 2 * ] if d ≥ 3 and p ∈ [1, 2) ∪ (2, +∞) if d = 1, 2 . (1.31)
a range which covers the case 2 # < p < 2 * and also p = 2 * if d ≥ 3. It is convenient to replace the solution w of (1.30) by u such that w = u β p with

β = 2 2 -p (1 -m)
so that u solves the nonlinear diffusion equation

∂u ∂t = u 2-2β ∆u + κ |∇u| 2 u , u(t = 0, •) = | f | 1/β , (1.32) with κ = β (p -2) + 1. Then h = w 1/p = u β is such that d d t ∇h 2 L 2 (S d ) - d p -2 h 2 L p (S d ) -h 2 L 2 (S d ) = -2 β 2 S d d d -1 |Q u | 2 + δ(β) |∇u| 4 u 2 d µ with Q u := Hessu -1 d ∆u Id -β (p -1) d -1 d +2 ∇u⊗∇u u -1 d |∇u| 2
u Id and

δ(β) := - d -1 d + 2 (κ + β -1) 2 + κ (β -1) + d d + 2 (κ + β -1) . (1.33)
As a consequence, we can write an improved version of (1.10), with an integral remainder term

∇ f 2 L 2 (S d ) - d p -2 f 2 L p (S d ) -f 2 L 2 (S d ) ≥ 2 β 2 +∞ 0 S d d d -1 |Q u | 2 + δ(β) |∇u| 4 u 2 d µ d t .
We recall that the optimal constant cannot be larger than d , as shown by a Taylor expansion around f = 1 of (1.10) and by the Poincaré inequality on S d with optimal constant, which corresponds to the case p = 1 in (1.10). This explains why, along the flow (1.30), optimality in (1.10) is achieved only in the limit as t → +∞.

The infimum of the quotient The computation is therefore optimal for the solutions of (1.32) in the asymptotic regime as t → +∞. In the nonlinear regime, the constant which appears in the carré du champ computation is also d , which guarantees that the method gives the optimal constant in (1.10). In the words of Section 1.3.1 and with a straightforward analogy with Proposition 1.3, we have κ 2 = κ 1 = κ 0 . This feature will again appear in Sections 2 and 3. With the carré du champ method and the flow (1.32), not only the results of [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] are recovered in a natural parabolic setting, but improvements like (1.15) and stability results, are also obtained: see [START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF][START_REF]Improved interpolation inequalities and stability[END_REF] for more considerations in this direction.

Q p [h] := (2 -p) S d |∇h| 2 d µ S d |h| 2 d µ -S d |h| p d µ 2/p is obtained by considering lim ε→0 Q p [1 + ε h] = Q 1 [h]. Also notice that S d d d -1 |Q u | 2 + δ(β)

Entropy methods and the fast diffusion equation on the Euclidean space

Most of this section is a summary of the results of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]. The optimal constant in the entropy-entropy production inequality is given by the spectral gap of the linearized functional inequality. An important difference with Section 1.3.2 is that the linearization has to be done around the Barenblatt profile and not around constant functions. However, from the heuristic point of view, the situation is similar and the strategy is, mutatis mutandis, to prove that κ 2 = κ 0 . This explains why optimal constants are obtained by generalized carré du champ methods and why stability results can be achieved for (1.3), which is the major result of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF].

Rényi entropy powers

On R d , d ≥ 1, let us consider the fast diffusion equation

∂u ∂t = ∆u m (2.1)
with m ∈ [m 1 , 1), m 1 := (d -1)/d , and an initial datum u(t = 0, x) = u 0 (x) ≥ 0, with u 0 ∈ L 1 R d , (1+|x| 2 ) d x . Equation (2.1) was interpreted as the gradient flow of the entropy

E := R d u m d x
with respect to Wasserstein's distance in [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF]. Equation (2.1) admits self-similar solutions, the so-called Barenblatt functions which, up to a scaling, a multiplication by a constant and a translation, take the form

B (t , x) = 1 κ R(t ) d B τ(t ), x κ R(t ) ∀ (t , x) ∈ R + × R d
for some numerical constant κ > 0, an unbounded increasing function t → R(t ), τ(t ) = 1 2 log R(t ) and a Barenblatt profile B that can be written as

B(x) = 1 + |x| 2 1 m-1 ∀ x ∈ R d .
As in Section 1.3.2, we prove (1.3) directly by the carré du champ method.

The key property, inspired by the Rényi entropy powers of [START_REF] Savaré | The concavity of Rényi entropy power[END_REF] Here I and P are known respectively as the generalized Fisher information and the pressure variable.

(ii) The t -derivative of the generalized Rényi entropy powers, that is,

G := IE 2 m-m 1 1-m , satisfies the identity - 1 2
d d t log G = R d u m D 2 P - 1 d ∆P Id 2 d x + (m -m 1 ) R d u m ∆P + I E 2 d x . (2.2)
Hence G is monotone with a limit given by a self-similar Barenblatt function B (t , x). It turns out that writing

G[u] ≥ G[B] is exactly (1.
3) written with the optimal constant, using the relations f = u m-1/2 , so that, with

p = 1 2 m -1 , we have R d u d x = R d f 2p d x, E = R d f p+1 d x and I = (p + 1) 2 R d |∇ f | 2 d x.
Inequality (1.3) is then easily recovered. This gives the growth rate of E. Let

C 0 := 4 (1-m) 3 (2 m-1) 2 C GNS (p) 2 d (d +2) m-d (1-m) (2 m-1) M (d +2) m-d d (1-m)
where M = u 0 1 , and m c := (d -2)/d . According to [20, Lemma 2.1], we have the following result.

Lemma 2.1 ([20]). Assume that d

≥ 1, m ∈ [m 1 , 1) with additionally m > 1/2 if d = 1 or d = 2
, and consider a solution of (2.1)

with initial datum u 0 ∈ L 1 + R d , ( 1
+ |x| 2 ) d x such that u m 0 ∈ L 1 (R d ). Then R d u m (t , x) d x ≥ R d u m 0 d x + (1-m)C 0 m-m c t 1-m m-mc ∀ t ≥ 0 . (2.3)
Knowing the optimal constant C 0 in (2.3) is equivalent to identifying the optimal constant in (1.3). This is a standard property of entropy methods that optimal decay rates in the evolution equation are equivalent to optimal constants in the corresponding functional inequalities. The threshold case m = m 1 in (2.1) corresponds to p = p in (1.3), i.e., to the Sobolev inequality.

Relative entropy and relative Fisher information

Proving (2.2) requires some integrations by parts which are delicate to justify in the class of solutions considered in Lemma 2.1. A rigorous proof goes through the time-dependent rescaling (2.4) so that (2.1) is changed, with the choice R(0) = 1, into the Fokker-Planck type equation ∂v ∂τ

u(t , x) = 1 κ d R d v τ, x κ R where d R d t = R -d (m-m c ) , τ(t ) := 1 2 log R(t ) ,
+ ∇ • v ∇v m-1 -2 x = 0 , (2.5)
with an initial datum v 0 := κ d u 0 (κ•) ≥ 0, for some numerical parameter κ which depends only on m. For simplicity, we shall assume from now on that

R d v 0 d x = R d B d x. For a given function v ∈ L 1 + R d , (1 + |x| 2 ) d x such that v m ∈ L 1 (R d ) and R d v d x = R d B d x,
let us define the relative entropy (or free energy) and the relative Fisher information respectively by

F [v]:= R d B m-1 (v -B) -1 m (v m -B m ) d x , I [v]:= R d v ∇v m-1 + 2 x 2 d x .
It is proved in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] that the entropy-entropy production inequality

I [v] ≥ 4 F [v] (2.6)
is exactly equivalent to (1.3) written with the optimal constant. This is not difficult to check using f = v m-1/2 and B = g 2p with p = 1/(2 m -1).

If additionally v solves (2.5), it is almost straightforward to check that

d d t F [v(t , •)] = -I [v(t , •)] ≤ -4 F [v(t , •)] ,
(2.7) so that we have the estimate

F [v(t , •)] ≤ F [v 0 ] e -4t ∀ t ≥ 0 .
This can be used to obtain intermediate asymptotics for a solution of (2.1) using (2.4) and a generalized Pinsker-Csiszár-Kullback inequality. Even more interesting is the fact that the carré du champ method applies:

d d t I [v(t , •)] ≤ -4 I [v(t , •)] , (2.8)
with equality up to a term which is similar to the one in (2.2) after applying (2.4).

The justification of the integrations by parts is easier than in the Rényi entropy powers approach, but still requires some care: see [START_REF] Jüngel | Entropy methods for diffusive partial differential equations[END_REF][START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF] and references therein for a complete proof. This is also the right setting to prove the identities in the Rényi entropy framework of Section 2.1, up to the change of variables (2.4): all details can be found in [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF].

Let us consider the quotient

Q[v] := I [v] F [v] .
Based on (2.7) and (2.8), we obtain that

d d t Q[v(t , •)] = Q[v(t , •)] Q[v(t , •)] -4 if v solves (2.5
). This proves the following initial time layer property.

Lemma 2.2 ([20]). With the above assumptions

, if Q[v(T, •)] ≥ 4 + η for some η > 0 and T > 0, then Q[v(t , •)] ≥ 4 + 4 η e -4 T /(4 + η -η e -4 T ) for any t ∈ [0, T ].

Linearization and Hardy-Poincaré inequalities

For reasons which are similar to the ones in Section 1.3.2, it turns out that inf Q[v] = 4 is not achieved among admissible functions, in the sense that

lim n→+∞ F [v n ] = 0 if (v n ) n∈N is a minimizing sequence for Q.
In other words and under appropriate normalization conditions, this means that v n → B and suggests to consider the limiting problem inf

I[h] F[h] = lim ε→0 Q B 1 + ε B 1-m h
where the linearized free energy and the linearized Fisher information are defined respectively by

F[h] := m 2 R d |h| 2 B 2-m d x and I[h] := m (1 -m) R d |∇h| 2 B d x .
By the Hardy-Poincaré inequality if d ≥ 1 and m ∈ (m 1 , 1), and for any function

h ∈ L 2 (R d , B 2-m d x) such that ∇h ∈ L 2 (R d , B d x) and R d h B 2-m d x = 0,

we have the Hardy-Poincaré inequality

I[h] ≥ 4 F[h].
If additionally we assume that R d x h B 2-m d x = 0, then we have the improved Hardy-Poincaré inequality

I[h] ≥ 4 α F[h] (2.9)
where

α = 2 -d (1 -m) if m ∈ (m 1 , 1). To get an improved inequality if m = m 1 , one has to assume additionally that R d |x| 2 h B 2-m d x = 0.
As in [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] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF], the improved estimate of (2.9) can be reimported in the nonlinear functionals as follows.

Lemma 2.3 ([20]). Let m ∈

(m 1 , 1) if d ≥ 2, m ∈ (1/3, 1) if d = 1, η = 2 d (m -m 1 ).
There exists an explicit χ ∈ (0, 1) such that,if

R d v d x = M , R d x v d x = 0 and (1 -ε) B ≤ v ≤ (1 + ε) B (2.10)
for some ε ∈ (0, χ η), then

Q[v] ≥ 4 + η .
In other words, since Condition (2.10) is stable under the action of (2.5) according to [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF], we have the asymptotic time layer property Q[v(t , •)] ≥ 4 + η if the solution v(t , •) of (2.5) satisfies (2.10) for any t > T , for some threshold time T > 0.

The threshold time estimate

The existence of a threshold time is known for a given initial datum from [START_REF] Vázquez | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF], but has been quantified only in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF] as follows.

Theorem 2.4 ([20]). Let m

∈ m 1 , 1 if d ≥ 2, m > 1/3 if d = 1, A > 0,
and G > 0 be given. Let ε ∈ (0, min{χ η, ε 0 }), with η and χ as in Proposition 2.3, for some explicit ε 0 ∈ (0, 1/2), and define

T := 1 2 α log 1 + α c 1 + A 1-m +G α 2 ε a (2.11) with α = d (m -m c
), for some explicit, positive numerical constants c > 0 and a.

Then for any solution v to (2.5)

with nonnegative initial datum v 0 ∈ L 1 (R d ), R d v 0 d x = M , R d x v 0 d x = 0 which satisfies sup r >0 r d (m-mc ) (1-m) |x|>r u 0 d x ≤ A , F [v 0 ] ≤ G , (2.12)
we have that

(1 -ε) B(x) ≤ v(t , x) ≤ (1 + ε) B(x) ∀ (t , x) ∈ [T , +∞) × R d . ( 2.13)
Based on a global Harnack Principle, this result is at the core of the method of [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]. It relies on a quantitative version of the results of J. Moser in [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF][START_REF]On a pointwise estimate for parabolic differential equations[END_REF], and a constructive proof of [START_REF] Vázquez | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF] based on the improved results of [START_REF] Bonforte | Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity[END_REF]. As discussed in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]Chapter 7], the tail decay in (2.12) is not only sufficient but also necessary for obtaining (2.13).

A stability result for Gagliardo-Nirenberg-Sobolev inequalities

By collecting the initial time layer property of of Lemma 2.2, the asympotic time layer property of Lemma 2.3 and the threshold time estimate of Theorem 2.4, we are able to write an improved entropy-entropy production inequality which amounts to Q[v(t , •)] ≥ 4 + 4 η e -4 T /(4 + ηη e -4 T ) for any t ∈ R + . This is the desired stability result, which goes as follows. There is a unique

g f ∈ M such that R d (1, x, |x| 2 ) f 2p d x = R d (1, x, |x| 2 ) g 2p f d x .
By the entropy-entropy production inequality (2.6) written for v = f 2p , we know that

δ[ f ] := p + 1 p -1 R d (p -1) ∇ f + f p ∇g 1-p f 2 d x -4 E [ f ]
is nonnegative, where the relative entropy with respect to g f is defined by

E [ f ] := 2 p 1 -p R d f p+1 -g p+1 f - 1+p 2 p g 1-p f f 2p -g 2p f d x .
An expansion of δ shows that, for some explicit numerical constant K GNS ,

δ[ f ] = (p -1) 2 ∇ f 2 2 + 4 d -p (d -2) p -1 f p+1 p+1 -K GNS p + 1 p -1 f 2pγ 2p
and the inequality δ[ f ] ≥ 0 is actually equivalent to (1.3).

Theorem 2.5 (Stability in Gagliardo-Nirenberg inequalities, [20]). Let d ≥ 1 and p ∈ (1, p ).

There is an explicit C such that, for any nonnegative f

∈ L 2p R d , (1 + |x| 2 ) d x such that ∇ f ∈ L 2 (R d ) and A = sup r >0 r d -p (d -4) p-1 |x|>r f 2p d x < ∞ and E [ f ] = G < ∞ , then we have δ[ f ] ≥ C inf ϕ∈M R d (p -1) ∇ f + f p ∇ϕ 1-p 2 d x .
The main point is that C has an explicit expression in terms of A and G, which does not degenerate to zero if f ∈ M. The critical case p = p can also be covered up to an additional scaling: see [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]Chapter 6].

Theorem 2.6 (Stability in Sobolev inequalities, [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]). Let d ≥ 3 and A > 0.There is an explicit C such that, for any nonnegative function f

∈ L 2 * R d , (1 + |x| 2 ) d x such that ∇ f ∈ L 2 (R d ) and R d (1, x, |x| 2 ) f 2 * d x = R d (1, x, |x| 2 ) g d x and sup r >0 r d |x|>r f 2 * d x ≤ A , we have ∇ f 2 2 -S d f 2 2 * ≥ C R d 2 d -2 ∇ f + f d d -2 ∇g -2 d -2 2 d x . (2.14)
The stability constant is C in Theorem 2.6 depends only on A. In Theorems 2.5 and 2.6, the stability is measured by the relative Fisher information, that is, by

R d (p -1) ∇ f + f p ∇ϕ 1-p 2 d x .
Such a quantity differs from the distance to M in (1.2) and deserves some comments. A simple expansion of the square and an optimization under scalings proves the following nonlinear extension of the Heisenberg uncertainty principle,

d p + 1 R d f p+1 d x 2 ≤ R d |∇ f | 2 d x R d |x| 2 f 2p d x . (2.15)
As a consequence of (1.3), i.e., of δ[ f ] ≥ 0, we have that

R d (p -1) ∇ f + f p ∇ϕ 1-p 2 d x ≥ 4 p -1 p + 1 E [ f ] .
According to [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]Lemma 1.7], by the Pinsker-Csiszár-Kullback inequality, E [ f ] controls a more standard distance between f 2p and g 2p f , namely

E [ f ] ≥ p + 1 8 p R d g 3p-1 f d x -1 f 2p -g 2p f 2 1 .

Symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

This section is devoted to Theorem 1.1. The interpretation of (1.5) as an entropyentropy production inequality paves the way to a flow method which generalizes the computations done for the fast diffusion equation in Section 2. By the carré du champ method, we are able to reduce the symmetry issue to a simple dichotomy: either the entropy-entropy production inequality is linearly unstable around the Barenblatt type profiles and optimal functions are not radially symmetric, or the linearized entropy-entropy production inequality admits a spectral gap which also enters in the carré du champ estimates, in the nonlinear regime, as for the fast diffusion equation, and then the inequality holds with precisely the spectral gap as optimal constant. Using the analogy with Section 1.3.2, we are again in a case for which κ 2 = κ 0 . More details can be found in [51, Section 4.2].

Symmetry versus symmetry breaking

Let us consider the critical Caffarelli-Kohn-Nirenberg inequalities (1.5). In dimension d ≥ 2, it has been proved in [START_REF] Wang | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] that an optimal function f ≥ 0 exists, if b < a + 1, and also b > a if a < 0. With the appropriate normalization, f solves

-div |x| -2a ∇ f = |x| -b p f p-1 . ( 3.1)
The symmetry versus symmetry breaking issue is to decide whether or not f is radially symmetric. The answer is given by Theorem 1.1, but a slightly more general rigidity result is given in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]:

if b ≤ b FS (a) < b < a +1,

Strategy of the proof

We summarize here the strategy of the proof in [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF], which we decompose in four steps.

A modified

R d |D α F | 2 d µ ≥ C d a,b α 1-2 p R d |F | p d µ 2 p (3.2)
where

d µ = s n-1 d s d ω = |x| n-d d x, D α F := α ∂F ∂s , 1 s ∇ ω F so that |D α F | 2 = α 2 ∂F ∂s 2 + |∇ ω F | 2 s 2
, and α and n are two parameters defined by

α = (1 + a -b) (a c -a) a c -a + b , n = 2 p p -2 .
Here n plays the role of an artificial dimension while α reflects a discrepancy between derivatives with respect to the radial and angular variables. (3.4)

The flow. The fast diffusion flow

∂u ∂t = L u 1-1 n = (1 -n) D * α • u D α P (3.5)
is similar to the flow studied in Section 2.1 and admits self-similar solutions

B (t , x) = t -n c + |x| 2 2 (n -1) α 2 t 2 -n
where c > 0 can be adjusted so that R d B (t , x) d µ = for any given M > 0. Let us notice that M = R d u d µ is conserved by (3.5) and formally compute

d d t R d u |D α P| 2 d µ = -2 R d 1 2 L |D α P| 2 -D α P • D α L P - 1 n (L P) 2 d µ . Hence, if u is optimal for (3.3), it is clear that d d t R d u |D α P| 2 d µ = 0.

The carré du champ method.

Let us denote by ∇ the gradient with respect to the angular variable ω ∈ S d -1 and by ∆ the Laplace-Beltrami operator on S d -1 . The following algebraic computation is adapted from [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] and extends to (3.5) the carré du champ method. With n > d ≥ 3 and the notation = ∂ s , we have 1 2

L |D α P| 2 -D α P • D α L P - 1 n (L P) 2 = α 4 n -1 n P - P r - ∆P α 2 (n -1) r 2 2 + 2 α 2 r 2 ∇P - ∇P r 2 + 1 r 4 1 2 ∆|∇P| 2 -∇P • ∇∆P - 1 n -1 (∆P) 2 -(n -2) α 2 |∇P| 2 .
If P is a positive function in C 3 (S d -1 ) and d ≥ 3, the computation of Section 1.3.2 can be adapted to prove that

S d -1 1 2 ∆|∇P| 2 -∇P • ∇∆P - 1 n -1 (∆P) 2 -(n -2) α 2 |∇P| 2 P 1-n d ω = (n -2) (d -1) (n -1) (d -2) S d -1 |Q| 2 P 1-n d ω + n -d 2 (d + 1) n + 3 2 + 3 (n -1) (n + 1) (d -2) 2 (n -2) (d + 1) S d -1 |∇P| 4 P 2 P 1-n d ω + (n -2) α 2 FS -α 2 S d -1 |∇P| 2 P 1-n d ω where Q := (∇⊗∇) P-1 d -1 (∆P) Id-3 (n-1) (n-d ) 2 (n-2) (d +1) ∇P⊗∇P P -1 d -1 |∇P| 2
P Id . For the case d = 2 we refer the reader to [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF].

L |D α P| 2 -D α P • D α L P - 1 n (L P) 2 d µ = α 4 n -1 n R d P - P r - ∆P α 2 (n -1) r 2 2 d µ + 2 α 2 R d 1 r 2 ∇P - ∇P r 2 d µ + (n -2) (d -1) (n -1) (d -2) R d 1 r 4 |Q| 2 P 1-n d µ + n -d 2 (d + 1) n + 3 2 + 3 (n -1) (n + 1) (d -2) 2 (n -2) (d + 1) R d 1 r 4 |∇P| 4 P 2 P 1-n d µ + (n -2) α 2 FS -α 2 R d 1 r 4 |∇P| 2 P 1-n d µ .
Up to regularity issues and decay properties as |x| → +∞, we conclude that a solution of (3.5) such that d d t R d u |D α P| 2 d µ = 0 satisfies ∇P = 0 and P -P /r = 0 if α ≤ α FS , which means that it takes the form u -n = P = a + br 2 for some t -dependent coefficients a and b. Inserting this expression in (3.5), we obtain u = B , up to a scaling, that is, a translation in time in self-similar variables.

Regularity issues.

The above computation is so far formal, as integrations by parts are not justified. In [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF], the regularity and decay properties needed for the method are established for positive solutions of (3.4) by elliptic estimates. In [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF], partial results are obtained for the parabolic point of view using selfsimilar variables as in Section 2.2. A complete parabolic proof is however still missing.

Conclusion and open problems

Entropy methods provide a framework which is extremely convenient for solving some fundamental variational problems and identifying the optimal constant in the corresponding functional inequalities. In some cases, evolving the functional by a nonlinear diffusion amounts to test the equation by a complicated expression involving a second order differential operator, powers of the solution and eventually homogeneous weights, which gives rises to delicate and complex computations. The use of a flow and of the corresponding entropy functional provides a framework to reinterpret the various terms and order the computations, as in Section 3.2. Identifying the optimality cases becomes also much easier and the stationarity with respect to the flow even provides uniqueness results which would be otherwise difficult to guess or to reduce to more standard techniques. Parabolic equations come with some extra properties, like regularization effects, which are crucial to obtain the stability results of Section 2.5. However, its major advantage is to relate the nonlinear regime with the linearized one around selfsimilar or stationary solutions, using quantities which are meaningful in the two cases: from the gradient flow point of view in the nonlinear regime and in terms of orthogonality in the linear regime. It is a remarkable feature that the use of a parabolic flow allows to bypass symmetrization techniques. This opens new directions of research in non-real valued problems (complex valued functions in quantum mechanics, in presence of magnetic fields, see for instance [START_REF] Bonheure | Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields[END_REF][START_REF]Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3[END_REF], or systems of equations).

Entropy methods have been introduced in partial differential equations, see [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF], in order to handle systems of charged particles, with the aim of obtaining rates in kinetic and related equations. There is a whole area of research which has emerged during the last 20 years under the name of hypocoercivity, with H 1 methods (see [START_REF] Villani | Hypocoercivity[END_REF]) or L 2 methods (see [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF][START_REF] Arnold | Sharpening of decay rates in Fourier based hypocoercivity methods[END_REF]). In L 2 hypocoercive approaches, there is a simple strategy: if the kinetic equation admits a diffusion limit (under the appropriate parabolic scaling) whose asymptotic behaviour is governed by a functional inequality, then the corresponding rates of convergence or decay can be reimported in the kinetic equation: see for instance [START_REF] Bouin | Hypocoercivity and subexponential local equilibria[END_REF][START_REF] Bouin | Diffusion and kinetic transport with very weak confinement[END_REF][START_REF] Bouin | Hypocoercivity without confinement[END_REF]. This is even true for systems with a non-local Poisson coupling, as shown in [START_REF] Addala | L 2 -hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system[END_REF]. However, identifying sharp rates and relating nonlinear models with their linearized counterparts, as can be done in the above examples of linear and nonlinear parabolic equations, is not done yet, even in the simplest benchmark cases considered in [START_REF] Arnold | Sharpening of decay rates in Fourier based hypocoercivity methods[END_REF].

Let us conclude this paper by some open problems:

Unconfined or weakly confined diffusions. Diffusion equations without any external potential or with an external potential which is insufficient to balance the diffusion are now rather well understood and even the hypocoercive theory in corresponding kinetic equations is essentially under control, see [START_REF] Bouin | Hypocoercivity without confinement[END_REF][START_REF] Bouin | Diffusion and kinetic transport with very weak confinement[END_REF][START_REF] Bouin | Hypocoercivity and subexponential local equilibria[END_REF]. In [START_REF] Bouin | Hypocoercivity without confinement[END_REF]Section 6], improved decay rates are obtained by Fourier estimates when moments of low order are set to zero. A spectral interpretation of these moment conditions, in absence of a bounded stationary solution and the corresponding functional inequalities, written as entropy-entropy production inequalities, is however still missing. Reinterpreting these inequalities in terms of spectral gaps would also be very useful. In kinetic theory, recent results of [START_REF] Mourrat | Variational methods for the kinetic fokkerplanck equation[END_REF][START_REF] Brigati | Time averages for kinetic fokker-planck equations[END_REF] also point in this direction, with an explicit role played by a Poincaré-Lions type inequality.

Branches of solutions, optimal constants and stability for Sobolev's inequality: a conjecture on the sphere. Let us consider the inequality = µ(λ). Since the optimal constant µ(λ) is defined as an infimum (in u) of an affine function of λ, λ → µ(λ) is a concave function. The key estimate is the rigidity result: u ≡ λ 1/(p-2) is the unique solution of (4.2) if λ ≤ d , and as a consequence we know that µ(λ) = λ.

The first positive eigenvalue of the Laplace-Beltrami operator ∆ on S d is λ 1 = d . Let ϕ 1 be an associated eigenfunction. By taking as a test function u ε (z) = 1 + ε ϕ 1 (z) in the limit as ε → 0 + , it is easy to see that λ = λ 1 is a bifurcation point: there is a branch of non-constant solutions emerging from constant solutions at λ = d . Hence d can be characterized as the minimum of {λ > 0 : µ(λ) < λ}, and symmetry breaking, in the sense of non-constant optimal functions, occurs if and only if λ > d . See Fig. 2.

For p < 2 * , it is not very difficult to establish the existence of optimal functions with antipodal symmetry, that is, of solutions which satisfy the condition u(x) = (-x) ∀ x ∈ S d . .

This can be done using partial symmetries and a convenient reparametrization of the branch emerging from λ 2 . The branch is also shown on Fig. 2. The intersection with the line µ = λ is expected to determine the optimal constant κ p in the inequality

∇u 2 L 2 (S d ) ≥ κ p p -2 u 2 L p (S d ) -u 2 L 2 (S d ) ∀ u ∈ H 1 a (S d , d µ) (4.
3)

The curve p → κ p is shown on Fig. 3. The loss of compactness as p → 2 * corresponds to a concentration on two antipodal points. Based on these numerical computations, we expect that κ 2 = 2 1-2/2 * d = 2 2/d d . Because of the stereographic projection and after taking into account the normalization of the volume of the sphere (d µ is a probability measure), this supports the conjecture:

The optimal constant κ in (4.3) for p = 2 * , on S d , is κ 2 * = 2 2/d d . Such a conjecture is an analogue of the Bianchi-Egnell estimate (1.2) written on S d instead of R d , where S d is the optimal constant in (1.1). It is compatible with the behaviour of the branches originating from λ 2 , whose lower part converges to an horizontal line as p → 2 * . Such a phenomenon is similar to what has been observed in [START_REF]Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF][START_REF] Esteban | A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF] for Caffarelli-Kohn-Nirenberg inequalities.

1. 1 . 1 . 2 2 ≥ S d f 2 2 *

 1122 Sobolev inequality. On R d with d ≥ 3, the classical Sobolev inequality asserts that ∇ f

  and a < a c := (d -2)/2 where the exponent p = 2 d /(d -2+2 (b -a)) is again fixed by scaling considerations. Here C a,b denotes the optimal constant. Inequality (1.5

FIGURE 1 .

 1 FIGURE 1. Numerical computation of the bifurcation diagram for d = 3 and p = 3. The branch emerging at λ = d from the straight line µ = λ of the constant solutions is expected to realize the infimum λ → µ(λ).

  is based on: (i) The entropy growth estimate: E = (1m) I where I := R d u |∇P| 2 d x and P := m m -1 u m-1 .

  up to a scaling, g as defined in(1.6) is the unique positive solution of(3.1). This symmetry result cannot be handled so far with symmetrization methods: in the range of (a, b) for which around g, a local property, then g is a global minimizer and, as a consequence, C a,b = ∇g |x| -a 2 2 / g |x| -b 2 p .

  Sobolev inequality. Let us use spherical coordinates with r = |x| and ω = x/r for any x ∈ R d \ {0}. With the change of variables f (r, ω) = F (s, ω) with s = r α , Inequality (1.5) is transformed into a Sobolev type inequality

1 .

 1 The parametrization by a and b of (1.5) is now replaced by the parametrization by α > 0 and n > d . The Felli & Schneider curve b = b FS (a) becomes α = α FS in the new set of parameters α and n, with α FS := d -1 n -For α > α FS the minimizers are not radial. With u = |F | p = P -n , Inequality (3.2) becomes R d u |D α P| 2 d µ ≥ p = 2 n/(n -2) while (3.1) is transformed into -L F = F p-1 where L := -D * α • D α .

Lemma 3 . 1 .

 31 Assume that ≥ 3. If u is a smooth enough and sufficiently decaying function as |x| = r → +∞, then R d 1 2

  p (S d ) ∀ u ∈ H 1 (S d , d µ) (4.1)with optimal constant µ(λ). For an optimal function u, the corresponding Euler-Lagrange equation is-(p -2) ∆ u + λ u = u p-1 (4.2)under the normalization condition u p-2 L p (S d )

FIGURE 2 .

 2 FIGURE 2. Case of the sphere in dimension d = 5: in the subcritical case p < 2 * , the optimal constant of (4.1) is given by λ → µ(λ). The dashed line is the branch of constant solutions u ≡ 1. The first bifurcation point is at λ 1 = d . From the second bifurcation point, at λ 2 = 2 (d +1), emerges another branch with a turning point: they intersect at λ = κ p . The curve λ → µ a (λ) is obtained as the minimum of this branch and of the line µ = λ.

FIGURE 3 .

 3 FIGURE 3. A numerical computation of the optimal constant κ p for (4.3) when d = 5. Only a few points have been computed on the dashed curve. The value of κ 1 (at p = 1) is known and equal to λ 2 = 2 (d +1) = 12. At p = 2 * , we get that κ 10/3 ≈ 6.59754.

  .[START_REF] Arnold | Sharpening of decay rates in Fourier based hypocoercivity methods[END_REF] Hence they generalize the Aubin-Talenti functions. For this reason, we shall still use this denomination and denote again by M the corresponding manifold. Actually there are deeper reasons for that than the algebraic expression of g, that were clarified in [9, Section 6.10] (also see[START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF] Section 1.3.1]). Stability results for (1.3) is one of the main achievement of entropy methods based on nonlinear diffusions and will detailed in Section 2 (see Theorems 2.5 and 2.6).

1.1.3. Caffarelli-Kohn-Nirenberg inequalities. The critical inequality

Acknowledgments

This work was supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). © 2021 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes.


V Mazières

R Pascaud

L Stafford

P Bonnet

A Al

C Chauvière

L Liard

Space-Time Plasma Steering Source: Synthesis and Outlooks

The aim of this paper is to present the promising outlooks that lies ahead for the innovative plasma source we introduced recently. The idea of this source is to dynamically control the plasma location by changing the waveform of the transmitted signal to an overmoded cavity. It is then the behavior of the waves inside the cavity that dynamically control the plasma location, hence the name of Space-Time Plasma Steering Source. Since its introduction in 2019, we have produced three main papers, aimed at different scientific communities: applied physics, physics of plasma, signal processing and mathematical and numerical methods. An overview of these works is first presented in this paper. Then, the promising outlooks for this unusual plasma source are identified and classified into four domains, among which the most important and essential one being the unhooking of the plasmas from the initiators in order to control their location anywhere in the cavity. Several solutions for this "unhooking issue" are then proposed, the most straightforward one being the use of non-intrusive probes.

Introduction

Plasma, or ionized gas, is a very interesting medium with various and exotic physical properties. It finds applications in different area of daily life and industry. The key issue to ignite a plasma is to obtain an electric field high enough to trigger an electron avalanche phenomenon in a medium. There exist a lot of methods to do it, which are based on quite different technologies [START_REF] Samukawa | The 2012 Roadmap[END_REF]. One of the challenges that face these plasma technologies is the development of "plasma source concepts that can be scaled to larger dimensions" [START_REF] Samukawa | The 2012 Roadmap[END_REF]. Indeed, there is an increasing need to be able to process large area which comes, among other things, from the need to process solar panel, flat panel displays, and larger wafers [START_REF] Samukawa | The 2012 Roadmap[END_REF]. However, in material processing, all of these technologies share a common feature: the more the area to be processed is large, the more it is difficult to process it. In fact, when increasing the area these technologies face physical and technological limitations [START_REF] Samukawa | The 2012 Roadmap[END_REF][START_REF] Sturm | Microwaves and microreactors: Design challenges and remedies[END_REF][START_REF] Liu | Electromagnetic effects in high-frequency large-area capacitive discharges[END_REF][START_REF] Lieberman | Standing wave and skin effects in large-area, high-frequency capacitive discharges[END_REF]. Microwave plasmas are usually ignited by exciting a resonance mode of a metallic chamber whose geometry is specifically designed to obtain a plasma discharge at the desired location [START_REF] Achard | High quality MPACVD diamond single crystal growth: High microwave power density regime[END_REF]. However, when the cavity dimensions are increased, it becomes overmoded and the control of the spatial distribution of the electric field (and thus of the plasma) is very difficult with these technologies [START_REF] Sturm | Microwaves and microreactors: Design challenges and remedies[END_REF]. To address this challenge of controlling plasmas in large cavities (overmoded), we proposed in 2019 an innovative concept of microwave plasma source [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF], that we name in this paper "Space-Time Plasma Steering Source" (STPSS). Contrary to the existing microwave plasma technologies, the idea is not to generate a plasma at a fixed location, but to ignite a plasma whose location can be dynamically controlled in a large cavity. To do that, one need to unlock a usually fixed parameter, namely the temporal shape of the exciting signal, which will be more complex than a continuous wave. It then becomes an additional degree of freedom of the system. This paper is organized as follow. First we summarize the experimental results that we obtained since our pioneer paper of 2019 [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF]. We will show how the STPSS was first developed using Time Reversal (TR) [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF][START_REF] Mazières | Spatio-temporal dynamics of a nanosecond pulsed microwave plasma ignited by time reversal[END_REF] and then using a more elaborated technique: the Linear Combination of Configuration Field (LCCF) [START_REF] Mazières | Transient Electric Field Shaping With the Linear Combination of Configuration Field Method for Enhanced Spatial Control of Microwave Plasmas[END_REF]. Then we will discuss about promising outlooks for this innovative plasma source, the main and essential one being the "unhooking" of the plasma from the initiators.

Concept of the Space-Time Plasma Steering Source

The concept of the Space-Time Plasma Steering Source is presented on Figure 1. It consists in dynamically controlling the position of the plasma in a cavity by playing on the waveform of the transmitted signal to the cavity. Indeed, this waveform determines the behavior of the waves inside the cavity and so the spatio-temporal structure of the field. In order to ignite a plasma at a desired location, the waveform must be controlled in order to obtain an intensification of the electric field at this location. This intensification will be obtained by making the waves constructively interfere at this location. Then, the waveform must be changed so that the waves will constructively interfere at the next desired location, to obtain the plasma at this novel location.

To reach this objective we can see that it is necessary to use an elaborate technique for controlling the waves, allowing the spatio-temporal control of the field inside the cavity. Furthermore, this technique must involve a phenomenon of intensification of the field at the desired time and position to ignite a plasma. We will see in the next section that Time Reversal (TR) fully meets these criteria and allows to ignite and control plasmas in cavity.

Experimental demonstrations of the Space-Time Plasma Steering Source

The experimental implementation of the STPSS requires in practice two phases. One first phase, the "probing phase", during which the behavior of the waves is evaluated between the exciting antenna and the desired location (at which we want to ignite a plasma). Then, during the second phase, also known as the "emitting phase", the transmitted signal (constructed based on the information collected during the first step) must allow the ignition of the plasma at the desired location. Our experimental setup consists of a main cavity of 0.6 × 0.6 × 0.3 m 3 -that is to say 5𝜆 𝑐 × 5𝜆 𝑐 × 2.5𝜆 𝑐 at our working microwave carrier frequency of about 𝑓 𝑐 = 2.4 GHz -in which the pressure and the gas can be controlled (typically with argon at 1 torr) and an appendix in which the air remains at atmospheric pressure to prevent gas breakdown in its vicinity during the high power emission of the complex waveforms (see Figure 2). This cavity includes several antennas: one of dimension 𝜆 𝑐 /4 located in the appendix and the other ones located in the main cavity. The antennas of the main cavity (which are electrically small metallic monopoles), act as emitting antennas during the probing phase of the STPSS process. During the emitting phase, they act as receiving antennas (allowing a measure of the field evolution at their locations) as well as initiators for plasma breakdown, as they produce a local enhancement of the electric field. Hence, they will be referred to as "initiator" in the following of this paper. The aim of our experiments is to control the plasma location on the initiators by playing on the waveform of the signal transmitted to the antenna in the appendix.

Space-Time Plasma Steering Source based on Time reversal

The first wave control technique that we used as a Space-Time Plasma Steering Source is TR [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF]. It has been shown to allow to focus electromagnetic waves [START_REF] Lerosey | Time reversal of electromagnetic waves[END_REF]. In a classical TR experiment, the probing phase consists in sending a short pulse that propagates through a complex medium, and the radiated field is recorded using an array of transducers, whose number can be reduced down to only one in a large reverberant chaotic cavity [START_REF] Draeger | One-Channel Time Reversal of Elastic Waves in a Chaotic 2D-Silicon Cavity[END_REF]. The recorded waveforms are then time-reversed and reemitted by this transducer such that the resulting wave converges back to its initial point of emission. An interesting property of TR for our application is that its focusing capability increases with the cavity volume, so large overmoded cavities are useable. The aim of our experiments is to focus the electromagnetic waves by TR on initiators 1, 2, or 3 represented on Figure 2 (with antenna 4 used as a transmitting antenna) in order to obtain an electric field high enough to reach breakdown conditions. In our experiments, the probing phase of TR is carried out in the frequency domain (with a Vector Network Analyzer) between antenna 4 and each of the three initiators 1, 2, and 3. Then using signal post-processing techniques, the impulse response 𝑠 1 (𝑡), 𝑠 2 (𝑡), and 𝑠 3 (𝑡) between antenna 4 and initiators 1, 2, and 3 respectively are constructed. They correspond to impulse response with an impulse of 8 ns (modulated at a carrier frequency 𝑓 𝑐 = 2.4 GHz), which is the shortest pulse that we are able to generate with our experimental setup [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF] and possess a reverberation time of about 𝑡 𝑟𝑒𝑣𝑒𝑟𝑏 = 300 ns. Then, emission of the time reversed signals 𝑠 1 (-𝑡), 𝑠 2 (-𝑡), or 𝑠 3 (-𝑡) at antenna 4 yields a focusing of the electromagnetic waves on initiator 1, 2, or 3 respectively. In the results presented in this subsection, the signal 𝑠 1 (-𝑡) , 𝑠 2 (-𝑡) or 𝑠 3 (-𝑡) is amplified (using a 2 kW amplifier) and transmitted periodically to the cavity with a repetition period of 𝑇 𝑟𝑒𝑝 = 166 μs. The pressure inside the main cavity is of 1 torr in argon, which are conditions that allows to ignite plasmas with our experimental setup [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF]. An example of our experimental setup used to ignite a plasma on initiator 1 is showed on Figure 2. The signals measured with an oscilloscope on initiators 1, 2 and 3 are plotted in red, green and blue respectively. We see that we obtain a focusing peak of 8 ns on initiator 1 and no peak on both initiators 2 and 3, as desired.

Pictures of the initiators 1, 2, and 3 are shown on Figure 3 (top picture). The three bottom pictures show the plasmas ignited when sending 𝑠 1 (-𝑡), 𝑠 2 (-𝑡), and 𝑠 3 (-𝑡). This demonstrates that TR can be used as a STPSS, as it allows to control the plasma location in the overmoded cavity by changing the waveform of the transmitted signal [START_REF] Mazières | Plasma generation using time reversal of microwaves[END_REF]. Note that for now these plasmas have only been ignited and controlled on initiators, which are necessary for the probing phase. -Very low duty cycle ~ 0.05, maintaining the discharges. In order to investigate this unusual source, in addition to the electrical measurements presented in the previous subsection, optical diagnostics of the plasma ignited by TR on one of the three monopoles have also been performed. It consists in imaging using a camera that enables to measure the spatio-temporal evolution of the light intensity emitted by the plasma during the signal period. These investigations have been made as a function of the pressure 𝑝 and the repetition period of the TR pulses 𝑇 𝑟𝑒𝑝 in our paper [START_REF] Mazières | Spatio-temporal dynamics of a nanosecond pulsed microwave plasma ignited by time reversal[END_REF] and three temporal intervals, with distinct mechanisms, have been identified: the discharge, the "early post-discharge" and the "late postdischarge", as represented on Figure 4. The interested reader must refer to [START_REF] Mazières | Spatio-temporal dynamics of a nanosecond pulsed microwave plasma ignited by time reversal[END_REF] to fully grasp these works.

Spatio-temporal description of the STPSS based on Time Reversal

Space-Time Plasma Steering Source based on LCCF

The second wave control technique that we used as a STPSS is the LCCF method [START_REF] Mazières | Transient Electric Field Shaping With the Linear Combination of Configuration Field Method for Enhanced Spatial Control of Microwave Plasmas[END_REF]. This technique allows to determine a temporal waveform that leads to predefined electromagnetic fields at one or more positions over a given interval of time (contrary to TR that allows only the control of the field at the time and location of the focusing). The probing phase is the same as the one of a TR experiment, meaning the impulse responses are recorded between the emitting point and the receiver points. Then, the LCCF finds a linear combination of these responses to compute the desired waveform. Emission of this waveform yields the desired target electromagnetic fields, and this for an arbitrary duration [START_REF] Benoit | Time-dependent current source identification for numerical simulations of Maxwell's equations[END_REF]. We have used this technique to improve the control of the plasmas in cavity [START_REF] Mazières | Transient Electric Field Shaping With the Linear Combination of Configuration Field Method for Enhanced Spatial Control of Microwave Plasmas[END_REF].

Outlooks

Unhooking issue

The most important and essential step that needs to be taken is the "unhooking" of the plasma from the initiators, in order to obtain a "ball" of plasma that could be moved anywhere in the overmoded cavity. The difficulty lies in evaluating, during the probing phase, the behavior of the waves between the exciting antenna and the location at which we want to ignite a plasma. Indeed, removing an antenna (initiator) from the cavity changes significantly the behavior of the waves. Thus the information collected during the probing phase does not hold during the emitting phase if an antenna (initiator) is removed, so no TR focusing would occur. To address this issue, several ways can be envisioned: -The probing phase could be obtained by simulating the behavior of the waves in the cavity. This would be very complicated as it would require an accurate representation of the experimental cavity and long simulation times [START_REF] Xiao | Focusing waves at arbitrary locations in a ray-chaotic enclosure using time-reversed synthetic sonas[END_REF]. -We could also think about using elaborated techniques for controlling the waves, in order for example to interpolate the information of the behavior of the waves between two initiators. -The more promising and straightforward way would be the use of a non-intrusive probe. A lot of work have been done on TR with electro-optic probes at microwave frequencies in reverberant room [START_REF] Monsef | Analysis of time-reversalbased propagation for spatial focusing and multiplexing[END_REF]. Thus, the use of an electro-optic probe to "unhook" the plasma from the initiators seems to be very promising.

Plasma diagnostics

In order to understand the rich physics of these unusual plasmas, further characterizations are planned in the future, for example using a time resolved spectrometer.

Modeling of the multi-physics problem

It would be useful to be able to describe theses plasmas ignited by TR with a numerical model, as it would help in grasping the physical mechanisms and it would allow to follow quantities that are not easily accessible experimentally, such as the electron density. The difficulties lie in the multi-physics modelling, along with the long simulation times that are required. Another interesting aspect would be a more fundamental study of the physics of these unusual plasmas. A tool seems to lend itself to this purpose: the space-time diagrams. A good example of how these diagrams can be effectively used to give "visual perspective of the overall physics" is found in [START_REF] Caloz | Spacetime metamaterials[END_REF] for describing space-time metamaterials. As a plasma ignited with a Space-Time Plasma Steering Source consists a priori both of a timediscontinuity (the plasma is ignited at the focusing time) and a space-discontinuity (the plasma is spatially localized), space-time diagram seems to be the right tool for describing the physics of these plasmas.

Potential applications

We already mentioned that this innovative source could really be useful in surface processing as it would allow large area processing in overmoded cavities.

Another potential application could be found at higher pressure, for example for microwave plasma assisted combustion and ignition as proposed in [START_REF] Fragge | Etude de l'allumage d'un carburant liquide par des micro-onde focalisées[END_REF]. The ability to control in space and time the plasmas suggests also interesting outlooks in the field of spacetime modulated media [START_REF] Caloz | Spacetime metamaterials[END_REF].

Conclusion

This paper summarizes the work we have done these last few years on the new plasma source concept, the Space-Time Plasma Steering Source, and propose the interesting and challenging outlooks for this exotic plasma source. Further work on each of these outlooks, especially on the "unhooking issue", is planned in the future.

Figure 1 .

 1 Figure 1. Principle of the STPSS. The location of the plasma is controlled by the waveform of the transmitted signal 𝑠 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 (𝑡) to an overmoded cavity, which dynamically shapes the field in the cavity.

Figure 2 .

 2 Figure 2. Sketch of an experiment of plasma ignition by TR on initiator 1 (the presence of the plasma is denoted by the red star) [6].

Figure 3 .

 3 Figure 3. Picture of the three initiators 1, 2, and 3 of Figure 2 (top) and of the ignited plasmas by TR [6].

Figure 4 .

 4 Figure 4. Representation of the three time intervals: discharge, early post-discharge and late post-discharge. In blue is represented the absolute value of the TR signal and in red the integrated light intensity as a function of time.

Acknowledgements

The authors would like to acknowledge the DGA/AID for their funding support and the CEA-Gramat for the material support. The first author would also like to acknowledge the CNES for their postdoctoral funding.