Stability in Gagliardo-Nirenberg-Sobolev inequalities

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Jean Dolbeault

http://www.ceremade.dauphine.fr/_∼dolbeaul

Ceremade, Université Paris-Dauphine February 26, 2021

Applications of Optimal Transportation in the Natural Sciences Oberwolfach Workshop (21 - 27 February 2021)

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Matteo Bonforte

BUniversidad Autónoma de Madrid and ICMAT

Bruno Nazaret

BUniversité Paris 1 Panthéon-Sorbonne and Mokaplan team

Nikita Simonov

BCeremade, Université Paris-Dauphine (PSL)

J. Dolbeault Stability in Gagliardo-Nirenberg-Sobolev inequalities

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The stability result of G. Bianchi and H. Egnell

In Sobolev’s inequality (with optimal contantSd), S_d_k∇f_k²_L₂₍

R^d)− kf_k²

L^2∗(R^d)≥0

is there a natural way to bound the l.h.s. from below in terms of a “distance”

to the set of optimal [Aubin-Talenti] functionswhend_≥3? A question raised in[Brezis, Lieb (1985)]

B[Bianchi, Egnell (1991)]There is a positive constantαsuch that S_d_k∇f_k²

L²(R^d)− kf_k²

L^2∗(R^d)≥α inf

ϕ∈Mk∇f_{− ∇}ϕk²_L2(R^d)

BVarious improvements,e.g.,[Cianchi, Fusco, Maggi, Pratelli (2009)]

there are constantsαandκandf_7→λ(f)such that Sdk∇f_k²

L²(R^d)≥(1₊κ λ(f)^α)_kf_k²

L²^∗(R^d)

However, the question ofconstructiveestimates is still widely open

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From the carré du champ method to stability results

Carré du champ method(adapted from D. Bakry and M. Emery)

∂u

∂t ⁼^∆u^m, d_F dt ^{= −}^I^,

d_I

dt ^{≤ −}^Λ^I deduce thatI−ΛFis monotone non-increasing with limit0

I[u]_≥ΛF[u]

BImproved constantmeansstability

Under some restrictions on the functions, there is someΛ?≥Λsuch that I−ΛF≥(Λ?−Λ)_F

BImproved entropy – entropy production inequality(weaker form) I≥Λψ(_F)

for someψsuch thatψ(0)₌0,ψ⁰(0)₌1andψ⁰⁰>0 I−ΛF≥Λ(ψ(_F)_−F)_≥0

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An application in the Natural Sciences

[Carlen, Figalli, 2013]Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation Bthe log-HLS (Hardy-Littlewood-Sobolev) inequality

H[u] :₌ Z

R²ulog^³u M

´ dx₊2

M Ï

R²×R²u(x)u(y)log_|x₋y_|dx dy₊M(1₊logπ)_≥0 withM_{= k}u_k₁is linked with the8πcritical mass in the Keller-Segel model Bthe GNS (Gagliardo-Nirenberg-Sobolev) inequality (special case)

Z

R²|∇f_|²dx Z

R²|f_|⁴dx_≥π Z

R²|f_|⁶dx [Carlen, Carrillo, Loss, 2010]Ifusolves^∂_∂^u_t =∆p

uandf ₌u^1/4, then

−1 8

d

dt^H[u]₌ Z

R²|∇f_|²dx₋π R

R²|f_|⁶dx R

R²|f_|⁴dx Stability for log-HLS arises from the stability for GNS

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Optimal transportation and gradient flows

The fast diffusion flow seen as a gradient flow with respect to Wasserstein’s distance

[McCann, 1997],[Otto, 2001]

[Cordero-Erausquin, Nazaret, Villani, 2004]

[Agueh, Ghoussoub, Kang, 2004]

[Carrillo, Lisini, Savaré, Slepˇcev, 2009]

[Zinsl, Matthes, 2015],[Zinsl, 2019]

[Iacobelli, Patacchini, Santambrogio, 2019]

[Ambrosio, Mondino, Savaré, 2019]

... already a long story (with apologizes for not quoting all relevant papers) Bthe PDE point of view:

– decay of the entropies

– regularization properties of the parabolic equations – carré du champ method

– some spectral analysis

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Outline

Gagliardo-Nirenberg-Sobolev inequalities by variational methods BA special family of Gagliardo-Nirenberg-Sobolev inequalities BConcentration-compactness

BStability results by variational methods

The fast diffusion equation and the entropy methods BRényi entropy powers

BSpectral gaps and asymptotics BInitial time layer

Regularity and stability

BUniform convergence in relative error BThe threshold time

BImproved entropy-entropy production inequality BSome consequences

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Gagliardo-Nirenberg-Sobolev inequalities

k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GNS(p)_kf_k_2p (GNS) Up to translations, multiplications by a constant and scalings, there is a unique optimal function

g(x)₌^³1_{+ |}x_|²^´⁻

1 p−1

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Gagliardo-Nirenberg-Sobolev inequalities

We consider the inequalities

k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GNS(p)_kf_k₂_p (GNS)

θ=_(d₊₂^d₋^(p_p_(d⁻¹⁾₋_2))p, p_∈(1_,+∞)_ifd₌1or2, p_∈(1,p^∗]_ifd_≥3, p^∗₌_d^d₋₂

Theorem (del Pino, JD)

Equality case in(GNS)is achieved if and only if

f _∈M:₌ⁿg_λ,µ,_y : (λ,µ,y)_∈(0,+∞)_×_R_×R^d^o

Aubin-Talenti functions: g_λ,µ,_y(x) :₌µg((x₋y)/λ),g(x)₌^¡1_{+ |}x_|²^¢⁻

1 p−1

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Related inequalities

k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GNS(p)_kf_k₂_p (GNS) BSobolev’s inequality:d_≥3,p₌p^∗₌d/(d₋2)

S_d_k∇f_k₂_{≥ k}f_k_2p∗

BEuclidean Onofri inequality Z

R²e^h⁻^h ^dx

π(1+|x_|²)²≤e^16π¹ ^RR2|∇h_|²dx

d₌2,p_{→ +∞}withf_p(x) :₌g(x)^³1₊₂¹_p(h(x)₋h)^´,h₌^R_R2h(x) ^dx

π(1+|x_|²)²

BEuclidean logarithmic Sobolev inequality in scale invariant form d

2 log µ 2

πd e Z

R^d|∇f_|²dx

¶

≥ Z

R^d|f_|²log_|f_|²dx orR

R^d|∇f_|²dx_≥¹₂^R_Rd|f_|²log µ|f_|²

kf_k²₂

¶

dx₊^d₄log^¡2πe²^{¢ R}_Rd|f_|²dx

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Deficit, scale invariance

Deficit functional

δ[f] :₌(p₋1)²_k∇f_k²₂₊4d₋p(d₋2)

p₊1 ^kf_k^p_p⁺₊¹₁₋_K_GNS_kf_k^2p_2p^γ Lemma

(GNS)is equivalent toδ[f]_≥0if and only if KGNS=C(p,d)_C_GNS^2p^γ

whereγ=^d⁺_d₋²⁻_p(d^p^(d₋⁻₄₎²⁾ andC(p,d)is an explicit positive constant

Takef_λ(x)₌λ^2p^d f(λx)and optimize onλ>0 δ[f]_≥δ[f]₋inf

λ>0δ[f_λ]₌:δ?[f]_≥0

A simplification:δ[f]₌δ[_|f_|]so we shall assume thatf_≥0a.e.

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Relative entropy, relative Fisher information

BFree energyorrelative entropy functional E[f_|g] :₌ 2p

1₋p Z

R^d

³

f^p⁺¹₋g^p⁺¹₋¹_2p⁺^pg¹⁻^p^³f^2p₋g²^p^´´dx If

Z

R^df²^p(1,x_,|x_|²)dx₌ Z

R^dg²^p(1,x,|x_|²)dx, g_∈M then E[f_|g]₌ 2p

1₋p Z

R^d

³

f^p⁺¹₋g^p⁺¹^´dx and δ?[f]_≈_E[f_|g]² BRelative Fisher information

J[f_|g] :₌p₊1 p₋1 Z

R^d

¯

¯(p₋1)_∇f₊f^p_∇g¹⁻^p^¯^¯_¯²dx BNonlinear extension of theHeisenberg uncertainty principle

µZ

R^df^p⁺¹dx

¶2

≤ Z

R^d|∇f_|²dx Z

R^d|x_|²f²^pdx

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Some inequalities

Lemma (Csiszár-Kullback inequality)

Letd_≥1andp_>1. There exists a constantC_p_>0such that

°

°f²^p₋g²^p^°^°_°²

L¹(_R^d)≤C_p_E[f_|g] if _kf_k₂_p_{= k}g_k_2p

Lemma (Entropy - entropy production inequality) p₊1

p₋1^δ[f]₌_J^£f_|g_f^¤₋4_E^£f_|g_f^¤_≥0

Lemma (A weak stability result)

δ[f]_≥δ?[f]_≈_E[f_|g]²

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Concentration-compactness

I_M₌infⁿ(p₋1)²_k∇f_k²₂₊4^d⁻^p_p^(d₊₁⁻²⁾_kf_k^p_p⁺₊¹₁:f_∈_Hp(_R^d), kf_k²₂^p_p₌M^o I₁₌_K_GNSandI_M₌I₁M^γfor anyM_>0

Lemma

Ifd_≥1 andp is an admissible exponent withp_<d/(d₋2), then I_M₁₊_M₂_<I_M₁₊I_M₂ _∀M₁,M₂_>0

Lemma

Letd_≥1andpbe an admissible exponent withp_<d/(d₋2)ifd_≥3. If (fn)n is minimizing and iflim sup

n_→+∞ sup

y_∈R^d Z

B(y)|fn|^p⁺¹dx₌0, then

nlim_→∞kf_n_k_2p₌0

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Existence of a minimizer, further properties

Proposition

Assume thatd_≥1is an integer and letpbe an admissible exponent with p_<d/(d₋2)ifd_≥3. Then there is an optimal function for (GNS) BPólya-Szeg˝o principle: there is a radial minimizer solving

−2(p₋1)²∆f₊4^¡d₋p(d₋2)^¢f^p₋C f^2p⁻¹₌0 Iff₌g, thenC₌8p

BA rigidity result: iff is a minimizer andP_{= −}^p_p⁺₋¹₁f¹⁻^p_{, then}

¡d₋p(d₋2)^¢ Z

R^df^p⁺¹

¯

¯∆P₊(p₊1)²

R

Rd|∇f_|²dx R

Rdf^p+1dx

¯

¯ 2

dx +2d p

Z

R^df^p⁺¹^°^°_°D²P₋_d¹∆PId°

°

° 2dx₌0

BWeak stability result: the minimizer isuniqueup to the invariances f(x)₌g(x)₌(1_{+ |}x_|²)⁻^1/(p⁻¹⁾

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An abstract stability result

Relative entropy

F[f] :₌₁²₋^p_p^R_Rd

³

f^p⁺¹₋g^p⁺¹₋¹_2p⁺^pg¹⁻^p^¡f^2p₋g^2p^¢^´dx Deficit functional

δ[f] :₌a_k∇f_k²₂₊b_kf_k^p_p⁺₊¹₁₋_K_GN_kf_k²₂^p_p^γ_≥0 Theorem

Letd_≥1andp_∈(1,p^∗). There is a C>0such that δ[f]_≥_{C F}[f]

for anyf_∈_W :₌^©f_∈L¹(_R^d,(1_{+ |}x_|)²dx) :_∇f_∈L²(_R^d, dx)^ªsuch that Z

R^df²^p(1,x)dx₌ Z

R^d|g_|^2p(1,x)dx

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A constructive result

Therelative entropy F[f] :₌₁²₋^p_p^R_Rd

³

f^p⁺¹₋g^p⁺¹₋¹_2p⁺^pg¹⁻^p^¡f^2p₋g^2p^¢^´dx Thedeficit functional

δ[f] :₌a_k∇f_k²₂₊b_kf_k^p⁺¹

p₊1−KGNkf_k²₂^p^γ

p ≥0 Theorem

Letd_≥1,p_∈(1,p^∗),A_>0andG_>0. There is a _C_>0such that δ[f]_≥_{C F}[f]

for anyf_∈_W :₌^©f_∈L¹(_R^d,(1_{+ |}x_|)²dx) :_∇f_∈L²(_R^d, dx)^ªsuch that Z

R^df^2pdx₌ Z

R^d|g_|²^pdx, Z

R^dx f^2pdx₌0 sup

r_>0

r^d−p^p−1^(d−4) Z

|x_|>r

f²^pdx_≤A and F[f]_≤G

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The fast diffusion equation and the entropy methods

∂u

∂t ⁼^∆u^m

The Rényi entropy powers and the Gagliardo-Nirenberg inequalities Self-similar solutions and the entropy-entropy production method Large time asymptotics, spectral analysis (Hardy-Poincaré inequality)

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The fast diffusion equation in original variables

Consider thefast diffusionequation inR^d,d_≥1,m_∈(0,1)

∂u

∂t ⁼^∆u^m (1)

with initial datumu(t₌0,x)₌u₀(x)_≥0such that Z

R^du₀dx₌_M_>0 and Z

R^d|x_|²u₀dx_{< +∞}

The large time behavior is governed bythe self-similar Barenblatt solutions

B(t,x) :₌ 1

¡κt^1/^µ^¢^d^B

µ x

κt^1/^µ

¶

whereµ:₌2₊d(m₋1),κ:₌

¯

¯ 2µm m₋1

¯

¯ 1/µ

andBis the Barenblatt profile B(x) :₌^¡C_{+ |}x_|²^¢⁻

1 1−m

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Entropy growth rate and Rényi entropy powers

Withp₌_2m¹₋₁_⇐⇒m₌^p_2p⁺¹, let us considerf such thatu₌f^2p u^m₌f^p⁺¹andu_|∇^m⁻¹u_|²₌(p₋1)²_|∇f_|²

M_{= k}f_k^2p_2p, E[u] :₌ Z

R^du^mdx_{= k}f_k^p_p⁺₊¹₁ and I[u] :₌(p₊1)²_k∇f_k²₂ By(GNS), ifusolves(1), then

E⁰₌p₋1

2p I₌p₋1

2p (p₊1)² Z

R^d|∇f_|²_dx

≥p₋1

2p (p₊1)²^³C_GNS_(p)´²_θ kf_k

2 θ

2pkf_k⁻

2(1−θ) θ

p₊1 ≥C₀E¹⁻^m−mc^1−m withC₀:₌^p₂⁻_p¹(p₊1)²(_C_GNS(p))²^θ M

(d+2)m−d d(1−m)

Z

R^du^m(t,x)dx_≥ µZ

R^du^m₀ dx₊⁽¹_m⁻₋^m)C_m ⁰

c t

¶_m−mc¹⁻^m

∀t_≥0

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Theentropyis defined by

E:₌ Z

R^du^mdx and theFisher informationby

I:₌ Z

R^du_|∇P_|²dx with P₌ m

m₋1u^m⁻¹is thepressure variable Ifusolves the fast diffusion equation, then

E⁰₌(1₋m)I

TheRényi entropy powerF:₌E^σ₌(^R_Rdu^mdx)^σwithσ=²_d ₁₋¹_m−1 applied to self-similar Barenblatt solutions has a linear growth int [Toscani, Savaré, 2014],[JD, Toscani, 2016]

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Nonlinear carré du champ method

I⁰₌ Z

R^d∆(u^m)_|∇P_|²dx₊2 Z

R^du_∇P_{· ∇}^³(m₋1)P∆P_{+ |∇}P_|²^´dx Ifuis a smooth and rapidly decaying function onR^d, then

Z

R^d∆(u^m)_|∇P_|²dx₊2 Z

R^du_∇P_{· ∇}^³(m₋1)P∆P_{+ |∇}P_|²^´dx

= −2 Z

R^du^m

°

°D²P₋_d¹∆PId°

°

2dx₋2(m₋m₁) Z

R^du^m(∆P)²dx Lemma

Letd_≥1and assume thatm_∈(m₁,1). Ifu solves (1)with initial datum u₀_∈L¹₊(_R^d)such that ^R_Rd|x_|²u₀dx_{< +∞}and if, for anyt_≥0, u(t,·)is a smooth and rapidly decaying function onR^d, then for anyt_≥0we have

−_dt^d log µ

I¹²E^θ(p+1)^1−θ

¶

=R

R^du^m^°°D²P₋¹_d∆PId°

°

2dx₊(m₋m₁)^R_Rdu^m^¯¯∆P₊_E^I^¯¯ 2dx

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Self-similar variables: entropy-entropy production inequality

With a time-dependent rescaling based onself-similar variables u(t,x)₌ 1

κ^dR^dv^³τ, x κR

´ where dR

dt ⁼R¹⁻^µ, τ(t) :₌¹₂logR(t)

∂u

∂t =∆u^mis changed intoa Fokker-Planck type equation

∂v

∂τ+ ∇ ·h

v^³_∇u^m⁻¹₋2x^{´ i}₌0 (2) Generalized entropy (free energy)andFisher information

F[v] :_{= −}1 m

Z

R^d

³

v^m_−B^m₋m_B^m⁻¹(v_−B)^´dx I[v] :₌

Z

R^dv

¯

¯¯∇v^m⁻¹₊2x^¯^¯_¯²dx

are such thatI[v]_≥4_F[v]by(GNS)[del Pino, JD, 2002]so that F[v(t,·)]_≤_F[v₀]e⁻⁴^t

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Spectral gap: sharp asymptotic rates of convergence

[Blanchet, Bonforte, JD, Grillo, Vázquez, 2009]

¡C₀_{+ |}x_|²^¢⁻

1

1−m≤v₀_≤^¡C₁_{+ |}x_|²^¢⁻

1

1−m (H)

F[v(t_,·)]_≤C e⁻²^γ^(m)t _∀t_≥0, γ(m) :₌(1₋m)Λα,d whereΛα,d>0is the best constant in the Hardy–Poincaré inequality

Λ_α,d Z

R^df²dµα−1≤ Z

R^d|∇f_|²dµα ∀f_∈H¹(dµα), Z

R^dfdµα−1=0 withα:₌_m¹₋₁_<0, dµα:₌h_αdx,h_α(x) :₌(1_{+ |}x_|²)^α

Lemma (already a stability result)

Under assumption(H), _I[v]_≥(4₊η)_F[v]for someη∈(0,2(γ(m)₋2)) Much more is know,e.g.,[Denzler, Koch, McCann, 2015]

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Spectral gap and the asymptotic time layer

mc=^d−2_d 0

m1=^d−1_d m2=^d+1_d+2

! m2:=^d+4_d+6

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

! m1:=_d+2^d

F[v(t,·)]_≤C e⁻²^γ^(m)t _∀t_≥0

[BBDGV, 2009] [BDGV, 2010] [JD, Toscani, 2015]

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The asymptotic time layer improvement

Linearized free energyandlinearized Fisher information F[g] :₌m

2 Z

R^dg²_B²⁻^m_dx _and I[g] :₌m(1₋m) Z

R^d|∇g_|²_B_dx Hardy-Poincaré inequality. Letd_≥1,m_∈(m₁,1)andg_∈L²(_R^d_,B²⁻^mdx) such that∇g_∈L²(_R^d_,Bdx),R

R^dg_B²⁻^mdx₌0andR

R^dx g_B²⁻^mdx₌0 I[g]_≥4αF[g] where α=2₋d(1₋m)

Proposition

Letm_∈(m₁,1)ifd_≥2,m_∈(1/3,1)ifd₌1,η=2d(m₋m₁)and χ=m/(266₊56m). If^R_Rdvdx₌_M,^R_Rdx vdx₌0and

(1₋ε)_B_≤v_≤(1₊ε)_B

for someε∈(0,χ η), then

Q[v] :₌^I[v] F[v]^≥4₊η

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The initial time layer improvement: backward estimate

Rephrasing thecarré du champmethod,Q[v] :₌^I^[v]

F[v]is such that

d_Q

dt ^≤^Q(_Q₋4)

Lemma

Assume thatm_>m₁ andv is a solution to (2)with nonnegative initial datumv₀. If for someη>0 andT_>0, we have_Q[v(T,·)]_≥4₊η, then

Q[v(t_,·)]_≥4₊ 4ηe⁻^4T

4₊_η−ηe⁻^4T ^∀t_∈[0,T]

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Regularity and stability The threshold time

Improved entropy-entropy production inequality

Regularity and stability

Our strategy

Regularity and stability

Our strategy

Choose">0, small enough

Get a threshold timet?(")

0 t?(") t

Backward estimate by entropy methods

Forward estimate based on a spectral gap

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

E s

⇐

# ^↳

Initial time layer Asymptotic time layer

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Uniform convergence in relative error

Theorem

Assume thatm_∈(m₁,1)ifd_≥2,m_∈(1/3,1)ifd₌1 and letε∈(0,1/2), small enough,A_>0, andG_>0 be given. There exists an explicit time t_?_≥0 such that, ifu is a solution of

∂u

∂t ⁼^∆u^m (2)

with nonnegative initial datumu₀_∈L¹(_R^d)satisfying

sup

r_>0

r

d(m−mc) (1−m)

Z

|x_|>r

u₀dx_≤A_{< ∞} (H_A)

R

R^du₀dx₌^R_RdBdx₌_M andF[u₀]_≤G, then

sup

x_∈R^d

¯

¯ u(t,x) B(t,x)⁻1

¯

¯≤ε ∀t_≥t?

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The threshold time

Proposition

Letm_∈(m₁,1)ifd_≥2,m_∈(1/3,1)ifd₌1,ε∈(0,εm,d),A_>0andG_>0

t?=c?

1₊A¹⁻^m₊G^α² ε^a wherea₌^α_ϑ²₁⁻₋^m_m andϑ=ν/(d₊ν)

c?=c?(m,d)₌ sup

ε∈(0,εm,d)

κ₁(ε,m) :₌maxⁿ 8c (1₊ε)¹⁻^m₋1^,

2³⁻^mκ?

1₋(1₋ε)¹⁻^m o

κ2(ε,m) :₌(4α)^α⁻¹K^α^ϑ ε^2−m¹⁻^m^α^ϑ

and κ3(ε,m) :₌ 8α⁻¹ 1₋(1₋ε)¹⁻^m

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Improved entropy-entropy production inequality

Theorem

Letm_∈(m₁,1)ifd_≥2,m_∈(1/2,1)ifd₌1,A_>0andG_>0. Then there is a positive numberζsuch that

I[v]_≥(4₊ζ)_F[v]

for any nonnegative functionv_∈L¹(_R^d)such thatF[v]₌G, R

R^dvdx₌_M, ^R_Rdx vdx₌0andv satisfies (HA) We have theasymptotic time layer estimate

ε∈(0,2ε?), ε?:₌1

with T₌1

2logR(t?) (1_−ε)_B_≤v(t_,·)_≤(1₊ε)_B _∀t_≥T

and, as a consequence, theinitial time layer estimate

I[v(t,.)]_≥(4₊ζ)_F[v(t,.)] _∀t_∈[0,T], where ζ= 4ηe⁻^4T 4_+η₋ηe⁻⁴^T

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Two consequences

ζ=Z^¡A_,F[u₀]^¢, Z(A,G) :₌ ^ζ^?

1₊A⁽¹⁻^m)^α²₊G^, ^ζ^? :₌ 4η

4₊η µ ε^a_?

2αc_?

¶_α² cα

BImproved decay rate for the fast diffusion equation in rescaled variables Corollary

Letm_∈(m₁,1)ifd_≥2,m_∈(1/2,1)ifd₌1,A_>0andG_>0. Ifv is a solution of (2) with nonnegative initial datumv₀_∈_L¹(_R^d)such that F[v₀]₌G,^R_Rdv₀_dx₌_M, ^R_Rdx v₀_dx₌0 andv₀ satisfies (HA), then

F[v(t,.)]_≤_F[v₀]e⁻⁽⁴⁺^ζ^)t _∀t_≥0 BThe stability in the entropy - entropy production estimate I[v]₋4_F[v]_≥ζF[v]also holds in a stronger sense

I[v]₋4F[v]_≥ ^ζ 4₊ζI[v]

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A general stability result

λ[f] :₌

µ2dκ[f]^p−1 p²₋1

kf_k^p+_p+¹₁ k∇f_k²₂

¶

2p d−p(d−4)

, κ[f] :₌^M

1 2p kf_k2p

A[f] :₌ ^M

λ[f]

d−p(d−4) p−1 kf_k²₂^p_p

sup_r_>₀r^d−p(d−

4) p−1 R

|x_|>r|f(x₊x_f)_|²^pdx

E[f] :₌₁^2p₋_p^R_Rd

Ã

κ[f]^p+¹ λ[f]^d

p−1 2p

f^p⁺¹₋g^p⁺¹₋¹₂⁺_p^pg¹⁻^p µκ[f]^2p

λ[f]² f²^p₋g²^p

¶! dx S[f]:₌^M

p−1 2p

p²₋1 1

C(p,d)Z(A[f],E[f]) Theorem

Letd_≥1andp_∈(1,p^∗). For anyf_∈_W, we have

³k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁^´²^p^γ₋(_C_GN_kf_k_2p)^2p^γ_≥S[f]_kf_k²₂^p_p^γE[f]

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References

M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov.Stability in Gagliardo-Nirenberg inequalities.

Preprinthttps://hal.archives-ouvertes.fr/hal-02887010

M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov.Explicit constants in Harnack inequalities and regularity estimates, with an application to the fast diffusion equation(supplementary material).

Preprinthttps://hal.archives-ouvertes.fr/hal-02887013

M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov. A memoir...

work in progress ! + some results also in the PhD thesis of N. Simonov (UAM, february 2020)

http://www.ceremade.dauphine.fr/∼dolbeaul

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These slides can be found at

http://www.ceremade.dauphine.fr/∼dolbeaul/Conferences/

B^Lectures The papers can be found at

http://www.ceremade.dauphine.fr/∼dolbeaul/Preprints/list/

BPreprints and papers

For final versions, useDolbeaultas login andJeanas password

E-mail:dolbeault@ceremade.dauphine.fr

Thank you for your attention !

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Stability in Gagliardo-Nirenberg-Sobolev inequalities

The stability result of G. Bianchi and H. Egnell

From the carré du champ method to stability results

An application in the Natural Sciences

Optimal transportation and gradient flows

Outline

Gagliardo-Nirenberg-Sobolev inequalities

Gagliardo-Nirenberg-Sobolev inequalities

Related inequalities

Deficit, scale invariance

Relative entropy, relative Fisher information

Some inequalities

Concentration-compactness

Existence of a minimizer, further properties

An abstract stability result

A constructive result

The fast diffusion equation and the entropy methods

The fast diffusion equation in original variables

Entropy growth rate and Rényi entropy powers

Nonlinear carré du champ method

Self-similar variables: entropy-entropy production inequality

Spectral gap: sharp asymptotic rates of convergence

Spectral gap and the asymptotic time layer

The asymptotic time layer improvement

The initial time layer improvement: backward estimate

Regularity and stability

Regularity and stability

E s

# ↳

Uniform convergence in relative error

The threshold time

Improved entropy-entropy production inequality

Two consequences

A general stability result

References

Thank you for your attention !

# ^↳