Jean Dolbeault
http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Universit´e Paris-Dauphine
16 mai 2017 Institut Camille Jordan Universit´e Claude Bernard Lyon 1
Outline
BSymmetry breaking and linearization
The critical Caffarelli-Kohn-Nirenberg inequality Linearization and spectrum
A family of sub-critical Caffarelli-Kohn-Nirenberg inequalities
BWithout weights: Gagliardo-Nirenberg inequalities and fast diffusion flows
The Bakry-Emery method on the sphere R´enyi entropy powers
Self-similar variables and relative entropies The role of the spectral gap
BWith weights: Caffarelli-Kohn-Nirenberg inequalities and weighted nonlinear flows
Large time asymptotics and spectral gaps A discussion of optimality cases
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Collaborations
Collaboration with...
M.J. Esteban and M. Loss(symmetry, critical case) M.J. Esteban, M. Loss and M. Muratori(symmetry, subcritical case) M. Bonforte, M. Muratori and B. Nazaret(linearization and large time asymptotics for the evolution problem) M. del Pino, G. Toscani (nonlinear flows and entropy methods) A. Blanchet, G. Grillo, J.L. V´azquez (large time asymptotics and linearization for the evolution equations) ...and also
S. Filippas, A. Tertikas, G. Tarantello, M. Kowalczyk ...
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Background references (partial)
Rigidity methods, uniqueness in nonlinear elliptic PDE’s: (B. Gidas, J. Spruck, 1981),(M.-F. Bidaut-V´eron, L. V´eron, 1991)
Probabilistic methods (Markov processes), semi-group theory and carr´e du champmethods (Γ2 theory): (D. Bakry, M. Emery, 1984), (Bentaleb),(Bakry, Ledoux, 1996),(Demange, 2008),(JD, Esteban, Kolwalczyk, Loss, 2014 & 2015)→D. Bakry, I. Gentil, and M.
Ledoux. Analysis and geometry of Markov diffusion operators (2014) Entropy methods in PDEs
BEntropy-entropy production inequalities: Arnold, Carrillo,
Desvillettes, JD, J¨ungel, Lederman, Markowich, Toscani, Unterreiter, Villani...,(del Pino, JD, 2001),(Blanchet, Bonforte, JD, Grillo, V´azquez)→A. J¨ungel, Entropy Methods for Diffusive Partial Differential Equations (2016)
BMass transportation: (Otto) →C. Villani, Optimal transport. Old and new (2009)
BR´enyi entropy powers (information theory)(Savar´e, Toscani, 2014), (Dolbeault, Toscani)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Symmetry and symmetry breaking results
BThe critical Caffarelli-Kohn-Nirenberg inequality BLinearization and spectrum
BA family of sub-critical Caffarelli-Kohn-Nirenberg inequalities
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Critical Caffarelli-Kohn-Nirenberg inequality
LetDa,b:=n
v ∈Lp Rd,|x|−bdx
: |x|−a|∇v| ∈L2 Rd,dx o Z
Rd
|v|p
|x|b p dx 2/p
≤ Ca,b
Z
Rd
|∇v|2
|x|2a dx ∀v ∈ Da,b holds under conditions ona andb
p= 2d
d−2 + 2 (b−a) (critical case) BAn optimal function among radial functions:
v?(x) =
1 +|x|(p−2) (ac−a)−p−22
and C?a,b= k |x|−bv?k2p
k |x|−a∇v?k22
Question: Ca,b= C?a,b (symmetry) orCa,b>C?a,b (symmetry breaking) ?
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Critical CKN: range of the parameters
Figure: d = 3 Z
Rd
|v|p
|x|b p dx 2/p
≤ Ca,b
Z
Rd
|∇v|2
|x|2a dx
a b
0 1
−1
b=a
b=a+ 1
a=d−22 p
a≤b≤a+ 1ifd≥3
a<b≤a+ 1ifd= 2,a+ 1/2<b≤a+ 1ifd= 1 anda<ac := (d−2)/2
p= 2d
d−2 + 2 (b−a)
(Glaser, Martin, Grosse, Thirring (1976)) (Caffarelli, Kohn, Nirenberg (1984)) [F. Catrina, Z.-Q. Wang (2001)]
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Linear instability of radial minimizers:
the Felli-Schneider curve
The Felli & Schneider curve bFS(a) := d(ac−a)
2p
(ac−a)2+d−1 +a−ac
a b
0
[Smets],[Smets, Willem],[Catrina, Wang],[Felli, Schneider]
The functional C?a,b
Z
Rd
|∇v|2
|x|2a dx− Z
Rd
|v|p
|x|b p dx 2/p
is linearly instable atv =v?
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Symmetry versus symmetry breaking:
the sharp result in the critical case
[JD, Esteban, Loss (Inventiones 2016)]
a b
0
Theorem
Let d≥2and p<2∗. If either a∈[0,ac)and b>0, or a<0 and b≥bFS(a), then the optimal functions for the critical
Caffarelli-Kohn-Nirenberg inequalities are radially symmetric
J. Dolbeault Symmetry, entropy methods and nonlinear flows
The Emden-Fowler transformation and the cylinder
BWith an Emden-Fowler transformation, critical the
Caffarelli-Kohn-Nirenberg inequality on the Euclidean space are equivalent to Gagliardo-Nirenberg inequalities on a cylinder
v(r, ω) =ra−acϕ(s, ω) with r =|x|, s=−logr and ω= x r With this transformation, the Caffarelli-Kohn-Nirenberg inequalities can be rewritten as thesubcriticalinterpolation inequality
k∂sϕk2L2(C)+k∇ωϕk2L2(C)+ Λkϕk2L2(C)≥µ(Λ)kϕk2Lp(C) ∀ϕ∈H1(C) whereΛ := (ac−a)2,C=R×Sd−1 and the optimal constantµ(Λ)is
µ(Λ) = 1 Ca,b
with a=ac±√
Λ and b= d p ±√
Λ
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Linearization around symmetric critical points
Up to a normalization and a scaling
ϕ?(s, ω) = (coshs)−p−21
is a critical point of
H1(C)3ϕ7→ k∂sϕk2L2(C)+k∇ωϕk2L2(C)+ Λkϕk2L2(C)
under a constraint onkϕk2Lp(C)
ϕ? is notoptimal for (CKN) if the P¨oschl-Teller operator
−∂s2−∆ω+ Λ−ϕp−2? =−∂s2−∆ω+ Λ− 1 (coshs)2 hasa negative eigenvalue, i.e., forΛ>Λ1(explicit)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
The variational problem on the cylinder
Λ7→µ(Λ) := min
ϕ∈H1(C)
k∂sϕk2L2(C)+k∇ωϕk2L2(C)+ Λkϕk2L2(C)
kϕk2Lp(C)
is a concave increasing function Restricted to symmetric functions, the variational problem becomes
µ?(Λ) := min
ϕ∈H1(R)
k∂sϕk2L2(Rd)+ Λkϕk2L2(Rd)
kϕk2Lp(Rd)
=µ?(1) Λα
Symmetry meansµ(Λ) =µ?(Λ)
Symmetry breaking meansµ(Λ)< µ?(Λ)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Numerical results
20 40 60 80 100
10 20 30 40 50
symmetric non-symmetric asymptotic
bifurcation µ
Λα µ(Λ)
(Λ) =µ
(1) Λα
Parametric plot of the branch of optimal functions forp= 2.8, d= 5.
Non-symmetric solutions bifurcate from symmetric ones at a bifurcation pointΛ1computed by V. Felli and M. Schneider. The branch behaves for large values ofΛ as predicted by F. Catrina and Z.-Q. Wang
J. Dolbeault Symmetry, entropy methods and nonlinear flows
what we wan to discard...
non-symmetric symmetric
bifurcation Jθ(µ)
Λθ(µ) Λθ(µ)
Jθ(µ)
symmetric non-symmetric bifurcation
Jθ(µ) bifurcation
symmetric
non-symmetric
symmetric
Λθ(µ) KGN
KCKN(ϑ(p,d), ΛFS(p,θ),p) 1/
1/
∗
When the local criterion (linear stability) differs from global results in a larger family of inequalities (center, right)...
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Subcritical Caffarelli-Kohn-Nirenberg inequalities
Norms: kwkLq,γ(Rd) := R
Rd|w|q|x|−γ dx1/q
,kwkLq(Rd) :=kwkLq,0(Rd)
(some)Caffarelli-Kohn-Nirenberg interpolation inequalities(1984) kwkL2p,γ(Rd)≤Cβ,γ,pk∇wkϑL2,β(Rd)kwk1−ϑLp+1,γ(Rd) (CKN) HereCβ,γ,p denotes the optimal constant, the parameters satisfy
d≥2, γ−2< β < d−2d γ , γ∈(−∞,d), p∈(1,p?] withp?:= d−β−2d−γ and the exponentϑis determined by the scaling invariance, i.e.,
ϑ= (d−γ) (p−1)
p d+β+2−2γ−p(d−β−2)
Is the equality case achieved by the Barenblatt / Aubin-Talenti type function
w?(x) = 1 +|x|2+β−γ−1/(p−1)
∀x ∈Rd ?
Do we know (symmetry) that the equality case is achieved among radial functions ?
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Range of the parameters
Herepis given
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Symmetry and symmetry breaking
(M. Bonforte, J.D., M. Muratori and B. Nazaret, 2016)Let us define βFS(γ) :=d−2−p
(d−γ)2−4 (d−1) Theorem
Symmetry breaking holds in (CKN) if
γ <0 and βFS(γ)< β < d−2 d γ
In the rangeβFS(γ)< β <d−2d γ,w?(x) = 1 +|x|2+β−γ−1/(p−1)
is notoptimal
(JD, Esteban, Loss, Muratori, 2016) Theorem
Symmetry holds in (CKN) if
γ≥0, or γ≤0 and γ−2≤β≤βFS(γ)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
The green area is the region of symmetry, while the red area is the region of symmetry breaking. The threshold is determined by the hyperbola
(d−γ)2−(β−d+ 2)2−4 (d−1) = 0
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Inequalities without weights and fast diffusion equations
BThe Bakry-Emery method on the sphere
BEuclidean space: self-similar variables and relative entropies BThe role of the spectral gap
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
The Bakry-Emery method on the sphere
Entropy functional Ep[ρ] := p−21 hR
Sdρ2p dµ− R
Sdρdµ2pi
if p6= 2 E2[ρ] :=R
Sdρlog
ρ kρkL1(Sd)
dµ
Fisher informationfunctional Ip[ρ] :=R
Sd|∇ρ1p|2dµ
Bakry-Emery (carr´e du champ) method: use the heat flow
∂ρ
∂t = ∆ρ
and compute dtdEp[ρ] =− Ip[ρ]and dtdIp[ρ]≤ −dIp[ρ]to get d
dt(Ip[ρ]−dEp[ρ])≤0 =⇒ Ip[ρ]≥dEp[ρ]
withρ=|u|p, ifp≤2#:= (2d−1)d2+12 J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
The evolution under the fast diffusion flow
To overcome the limitationp≤2#, one can consider a nonlinear diffusion of fast diffusion / porous medium type
∂ρ
∂t = ∆ρm. (1)
(Demange),(JD, Esteban, Kowalczyk, Loss): for anyp∈[1,2∗] Kp[ρ] := d
dt
Ip[ρ]−dEp[ρ]
≤0
1.0 1.5 2.5 3.0
0.0 0.5 1.5 2.0
(p,m)admissible region,d = 5
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
R´ enyi entropy powers and fast diffusion
The Euclidean space without weights
BR´enyi entropy powers, the entropy approach without rescaling:
(Savar´e, Toscani): scalings, nonlinearity and a concavity property inspired by information theory
BFaster rates of convergence: (Carrillo, Toscani),(JD, Toscani)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
The fast diffusion equation in original variables
Consider the nonlinear diffusion equation inRd,d ≥1
∂v
∂t = ∆vm
with initial datumv(x,t = 0) =v0(x)≥0such thatR
Rdv0dx= 1 and R
Rd|x|2v0dx<+∞. The large time behavior of the solutions is governed by the source-type Barenblatt solutions
U?(t,x) := 1
κt1/µdB? x κt1/µ
where
µ:= 2 +d(m−1), κ:=2µm m−1 1/µ andB? is the Barenblatt profile
B?(x) :=
C?− |x|21/(m−1)
+ ifm>1 C?+|x|21/(m−1)
ifm<1
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
The R´ enyi entropy power F
Theentropy is defined by E :=
Z
Rd
vmdx and theFisher informationby
I :=
Z
Rd
v|∇p|2dx with p = m m−1vm−1 Ifv solves the fast diffusion equation, then
E0 = (1−m) I To computeI0, we will use the fact that
∂p
∂t = (m−1) p ∆p +|∇p|2 F := Eσ with σ= µ
d(1−m) = 1+ 2 1−m
1
d +m−1
= 2 d
1 1−m−1 has a linear growth asymptotically ast →+∞
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
The concavity property
Theorem
[Toscani-Savar´e]Assume that m≥1−1d if d>1 and m>0 if d= 1.
Then F(t)is increasing,(1−m) F00(t)≤0 and
t→+∞lim 1
t F(t) = (1−m)σ lim
t→+∞Eσ−1I = (1−m)σEσ−1? I?
[Dolbeault-Toscani]The inequality
Eσ−1I≥Eσ−1? I?
is equivalent to the Gagliardo-Nirenberg inequality k∇wkθL2(Rd)kwk1−θLq+1(Rd)≥CGNkwkL2q(Rd)
if1−1d ≤m<1. Hint: vm−1/2= kwkw
L2q(Rd),q=2m−11
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
The proof
Lemma
If v solves ∂v∂t = ∆vm with 1d ≤m<1, then I0= d
dt Z
Rd
v|∇p|2dx =−2 Z
Rd
vm
kD2pk2+ (m−1) (∆p)2 dx Explicit arithmetic geometric inequality
kD2pk2− 1
d (∆p)2=
D2p− 1 d∆pId
2
There are no boundary terms in the integrations by parts
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Remainder terms
F00=−σ(1−m) R[v]. Thepressure variableisP = 1−mm vm−1
R[v] := (σ−1) (1−m) Eσ−1 Z
Rd
vm ∆P−
R
Rdv|∇P|2dx R
Rdvmdx
2
dx + 2 Eσ−1
Z
Rd
vmD2P−d1∆PId2dx Let
G[v] := F[v] σ(1−m) =
Z
Rd
vmdx σ−1Z
Rd
v|∇P|2dx The Gagliardo-Nirenberg inequality is equivalent toG[v0]≥G[v?] Proposition
G[v0]≥G[v?] + Z ∞
0
R[v(t,·)]dt
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Euclidean space: self-similar variables and relative entropies
The large time behavior of the solution of ∂v∂t = ∆vm is governed by the source-typeBarenblatt solutions
v?(t,x) := 1 κd(µt)d/µB?
x κ(µt)1/µ
where µ:= 2 +d(m−1) whereB?is the Barenblatt profile (with appropriate mass)
B?(x) := 1 +|x|21/(m−1)
A time-dependent rescaling: self-similar variables
v(t,x) = 1 κdRd u
τ, x κR
where dR
dt =R1−µ, τ(t) := 12 log R(t)
R0
Then the functionusolvesa Fokker-Planck type equation
∂u
∂τ +∇ ·h
u ∇um−1− 2x i
= 0
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Free energy and Fisher information
The functionusolves a Fokker-Planck type equation
∂u
∂τ +∇ ·h
u ∇um−1− 2x i
= 0 (Ralston, Newman, 1984)Lyapunov functional:
Generalized entropyor Free energy E[u] :=
Z
Rd
−um
m +|x|2u
dx− E0
Entropy production is measured by theGeneralized Fisher information
d
dtE[u] =−I[u], I[u] :=
Z
Rd
u∇um−1+ 2x2 dx
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Without weights: relative entropy, entropy production
Stationary solution: chooseC such thatku∞kL1=kukL1=M>0 u∞(x) := C+ |x|2−1/(1−m)
+
Relative entropy: FixE0 so thatE[u∞] = 0
Entropy – entropy production inequality(del Pino, J.D.)
Theorem
d≥3, m∈[d−1d ,+∞), m>12, m6= 1 I[u]≥4E[u]
Corollary
(del Pino, J.D.)A solution u with initial data u0∈L1+(Rd)such that
|x|2u0∈L1(Rd), u0m∈L1(Rd)satisfies
E[u(t,·)]≤ E[u0]e−4t
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
A computation on a large ball, with boundary terms
∂u
∂τ +∇ ·h
u ∇um−1− 2x i
= 0 τ >0, x ∈BR
whereBR is a centered ball in Rd with radiusR>0, and assume that usatisfies zero-flux boundary conditions
∇um−1−2x
· x
|x| = 0 τ >0, x ∈∂BR. Withz(τ,x) :=∇Q(τ,x) :=∇um−1− 2x, therelative Fisher informationis such that
d dτ
Z
BR
u|z|2dx+ 4 Z
BR
u|z|2dx + 21−mm
Z
BR
um
D2Q2− (1−m) (∆Q)2 dx
= Z
∂BR
um ω· ∇|z|2
dσ≤0 (by Grisvard’s lemma)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Entropy – entropy production, Gagliardo-Nirenberg ineq.
4E[u]≤ I[u]
Rewrite it withp=2m−11 ,u=w2p,um=wp+1 as 1
2 2m
2m−1 2Z
Rd
|∇w|2dx+ 1
1−m −d Z
Rd
|w|1+pdx−K ≥0 for some γ,K =K0 R
Rdu dx =R
Rdw2pdxγ w =w∞=v∞1/2p is optimal
Theorem
[Del Pino, J.D.]With1<p≤ d−2d (fast diffusion case) and d ≥3 kwkL2p(Rd) ≤ Cp,dGNk∇wkθL2(Rd)kwk1−θLp+1(Rd)
Cp,dGN=y(p−1)2
2πd
θ22y−d
2y
2p1 Γ(y)
Γ(y−d2)
θd
,θ= p(d+2−(d−2)p)d(p−1) , y =p−1p+1
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Spectral gap: sharp asymptotic rates of convergence
Assumptions on the initial datumv0 (H1)VD0 ≤v0≤VD1 for someD0>D1>0
(H2)if d≥3 andm≤m∗,(v0−VD)is integrable for a suitable D∈[D1,D0]
Theorem
(Blanchet, Bonforte, J.D., Grillo, V´azquez) Under Assumptions (H1)-(H2), if m<1 and m6=m∗:= d−4d−2, the entropy decays according to
E[v(t,·)]≤C e−2 (1−m) Λα,dt ∀t ≥0
whereΛα,d >0is the best constant in the Hardy–Poincar´e inequality Λα,d
Z
Rd
|f|2dµα−1≤ Z
Rd
|∇f|2dµα ∀f ∈H1(dµα) withα:= 1/(m−1)<0, dµα:=hαdx , hα(x) := (1 +|x|2)α
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows and CKN inequalities The role of the spectral gap
Spectral gap and best constants
mc=d−2d 0
m1=d−1d
m2=d+1d+2
! m2:=d+4d+6
m 1
2 4
Case 1 Case 2 Case 3
γ(m)
(d= 5)
! m1:=d+2d
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Weighted nonlinear flows:
Caffarelli-Kohn-Nirenberg inequalities
BEntropy and Caffarelli-Kohn-Nirenberg inequalities BLarge time asymptotics and spectral gaps
BOptimality cases
J. Dolbeault Symmetry, entropy methods and nonlinear flows
CKN and entropy – entropy production inequalities
When symmetry holds, (CKN) can be written as anentropy – entropy productioninequality
1−m
m (2 +β−γ)2E[v]≤ I[v]
and equality is achieved byBβ,γ(x) := 1 +|x|2+β−γm−11
Here thefree energyand therelative Fisher informationare defined by E[v] := 1
m−1 Z
Rd
vm−Bmβ,γ−mBm−1β,γ (v−Bβ,γ) dx
|x|γ I[v] :=
Z
Rd
v∇vm−1− ∇Bm−1β,γ 2 dx
|x|β Ifv solves theFokker-Planck type equation
vt+|x|γ∇ ·h
|x|−βv∇ vm−1− |x|2+β−γi
= 0 (WFDE-FP) then d
dtE[v(t,·)] =− m
1−mI[v(t,·)]
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Proposition
Let m=p+12p and consider a solution to (WFDE-FP) with nonnegative initial datum u0∈L1,γ(Rd)such that ku0mkL1,γ(Rd) and
R
Rdu0|x|2+β−2γdx are finite. Then
E[v(t,·)]≤ E[u0]e−(2+β−γ)2t ∀t≥0 if one of the following two conditions is satisfied:
(i) either u0 is a.e. radially symmetric (ii) or symmetry holds in (CKN)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Proof of symmetry (1/3: changing the dimension)
We rephrase our problem in a space of higher,artificial dimension n>d (herenis a dimension at least from the point of view of the scaling properties), or to be precise we consider a weight|x|n−d which is the same in all norms. With
v(|x|α−1x) =w(x), α= 1 +β−γ
2 and n= 2 d−γ β+ 2−γ, we claim that Inequality (CKN) can be rewritten for a function v(|x|α−1x) =w(x)as
kvkL2p,d−n(Rd) ≤Kα,n,pkDαvkϑL2,d−n(Rd)kvk1−ϑLp+1,d−n(Rd) ∀v ∈Hpd−n,d−n(Rd) with the notationss=|x|,Dαv= α∂v∂s,1s∇ωv
and d≥2, α >0, n>d and p∈(1,p?] . By our change of variables,w? is changed into
v?(x) := 1 +|x|2−1/(p−1)
∀x∈Rd
J. Dolbeault Symmetry, entropy methods and nonlinear flows
The strategy of the proof (2/3: R´ enyi entropy)
The derivative of the generalizedR´enyi entropy powerfunctional is
G[u] :=
Z
Rd
umdµ σ−1Z
Rd
u|DαP|2dµ
whereσ= 2d 1−m1 −1. Heredµ=|x|n−ddx and the pressure is P := m
1−mum−1
Looking for an optimal function in (CKN) is equivalent to minimizeG under a mass constraint
J. Dolbeault Symmetry, entropy methods and nonlinear flows
WithLα=−D∗αDα=α2 u00+n−1s u0
+s12∆ωu, we consider the fast diffusion equation
∂u
∂t = Lαum
in the subcritical range1−1/n<m<1. The key computation is the proof that
−dtd G[u(t,·)] R
Rdumdµ1−σ
≥(1−m) (σ−1)R
Rdum LαP−
R
Rdu|DαP|2dµ R
Rdumdµ
2dµ + 2R
Rd
α4 1−1n P00−Ps0 −α2(∆n−1)ωPs2
2+2αs22
∇ωP0−∇ωsP
2 umdµ + 2R
Rd
(n−2) α2FS−α2
|∇ωP|2+c(n,m,d)|∇Pω2P|4
umdµ=:H[u]
for some numerical constantc(n,m,d)>0. Hence ifα≤αFS, the r.h.s.H[u] vanishes if and only ifPis an affine function of|x|2, which proves the symmetry result. A quantifier elimination problem (Tarski, 1951)?
J. Dolbeault Symmetry, entropy methods and nonlinear flows
(3/3: elliptic regularity, boundary terms)
This method has a hidden difficulty: integrations by parts ! Hints:
use elliptic regularity: Moser iteration scheme, Sobolev regularity, local H¨older regularity, Harnack inequality, and get global regularity using scalings
use the Emden-Fowler transformation, work on a cylinder, truncate, evaluate boundary terms of high order derivatives using Poincar´e inequalities on the sphere
Summary: ifusolves the Euler-Lagrange equation, we test byLαum 0 =
Z
Rd
dG[u]·Lαumdµ≥ H[u]≥0
H[u]is the integral of a sum of squares (with nonnegative constants in front of each term)... or test by|x|γdiv |x|−β∇w1+p
the equation (p−1)2
p(p+ 1)w1−3pdiv |x|−βw2p∇w1−p
+|∇w1−p|2+|x|−γ c1w1−p−c2
= 0
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Fast diffusion equations with weights: large time asymptotics
Relative uniform convergence Asymptotic rates of convergence From asymptotic to global estimates
Herev solves the Fokker-Planck type equation vt+|x|γ∇ ·h
|x|−βv∇ vm−1− |x|2+β−γi
= 0 (WFDE-FP)
Joint work with M. Bonforte, M. Muratori and B. Nazaret
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Relative uniform convergence
ζ:= 1− 1−(1−m)2−mq
1−2−m1−mθ θ:= (1−m) (2+β−γ)
(1−m) (2+β)+2+β−γ is in the range0< θ < 1−m2−m <1 Theorem
For “good” initial data, there exist positive constantsKand t0such that, for all q∈2−m
1−m,∞
, the function w=v/Bsatisfies
kw(t)−1kLq,γ(Rd) ≤ Ke−2(1−m)22−m Λζ(t−t0) ∀t ≥t0 in the caseγ∈(0,d), and
kw(t)−1kLq,γ(Rd)≤ Ke−2(1−m)22−m Λ (t−t0) ∀t ≥t0 in the caseγ≤0
J. Dolbeault Symmetry, entropy methods and nonlinear flows
0
Λ0,1
Λ1,0
Λess
Essential spectrum
δ4 δ
δ1 δ2 δ5
Λ0,1
Λ1,0 Λess Essential spectrum
δ4 δ5:=n 2−η
The spectrum ofLas a function ofδ=1−m1 , withn= 5. The essential spectrum corresponds to the grey area, and its bottom is determined by the parabolaδ7→Λess(δ). The two eigenvaluesΛ0,1and Λ1,0are given by the plain, half-lines, away from the essential
spectrum. The spectral gap determines the asymptotic rate of convergence to the Barenblatt functions
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Global vs. asymptotic estimates
Estimates on the global rates. When symmetry holds (CKN) can be written as anentropy – entropy production inequality
(2 +β−γ)2E[v]≤ m 1−mI[v]
so that
E[v(t)]≤ E[v(0)]e−2 (1−m) Λ?t ∀t ≥0 with Λ?:= (2 +β−γ)2 2 (1−m)
Optimal global rates. Let us consider again theentropy – entropy productioninequality
K(M)E[v]≤ I[v] ∀v ∈L1,γ(Rd) such that kvkL1,γ(Rd)=M, whereK(M)is the best constant: withΛ(M) := m2(1−m)−2K(M)
E[v(t)]≤ E[v(0)]e−2 (1−m) Λ(M)t
∀t≥0
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Linearization and optimality
Joint work with M.J. Esteban and M. Loss
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Linearization and scalar products
Withuεsuch that
uε=B? 1 +εfB?1−m
and Z
Rd
uεdx=M? at first order inε→0 we obtain thatf solves
∂f
∂t =Lf where Lf := (1−m)Bm−2? |x|γD∗α |x|−βB?Dαf Using the scalar products
hf1,f2i= Z
Rd
f1f2B2−m? |x|−γ dx and hhf1,f2ii= Z
Rd
Dαf1·Dαf2B?|x|−βdx we compute
1 2
d
dthf,fi=hf,Lfi= Z
Rd
f (Lf)B?2−m|x|−γ dx=− Z
Rd
|Dαf|2B?|x|−β dx for anyf smooth enough: with hf,Lfi=− hhf,fii
1 2
d
dthhf,fii= Z
Rd
Dαf ·Dα(Lf)B?|x|−βdx=− hhf,Lfii
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Linearization of the flow, eigenvalues and spectral gap
Now let us consider an eigenfunction associated with the smallest positive eigenvalueλ1ofL
− Lf1=λ1f1
so thatf1realizes the equality case in theHardy-Poincar´e inequality hhg,gii=− hg,Lgi ≥λ1kg−g¯k2, g¯ :=hg,1i/h1,1i
− hhg,Lgii ≥λ1hhg,gii Proof: expansion of the square :
− hh(g−g¯),L(g −g¯)ii=hL(g−g¯),L(g −g¯)i=kL(g −g¯)k2 Key observation:
λ1≥4 ⇐⇒ α≤αFS :=
rd−1 n−1
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Why is this method optimal ?
The condition λ1<4 is sufficient for symmetry breaking Withλ1≥4, we prove that
H[v] := m
1−mI[v]−(2 +β−γ)2E[v]
is monotone decaying along the flow and that equality is achieved only on the stationary solution, which attracts all solutions. this has to do with the (formal) gradient flow structure of the problem
The condition λ1≥4 is enough to prove that
Bthe Fisher informationI[v] exponentially decays with ratee−4t Bthe functionalH[v]is decreasing
The decay of the Fisher informationI[v]“has to be given” by the conditionλ1≥4 because the problem degenerates into a sharp spectral gap problem in the asymptotic regime
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Symmetry breaking in CKN inequalities
Symmetry holds in (CKN) ifJ[w]≥ J[w?]with J[w] :=ϑlog kDαwkL2,δ(Rd)
+(1−ϑ) log kwkLp+1,δ(Rd)
−log kwkL2p,δ(Rd)
withδ:=d−nand
J[w?+εg] =ε2Q[g]+o(ε2) where
2
ϑkDαw?k2L2,d−n(Rd)Q[g]
=kDαgk2L2,d−n(Rd)+p(2+β−γ)(p−1)2
d−γ−p(d−2−β) Z
Rd
|g|21+|x||x|n−d2 dx
−p(2p−1)(2+β−γ)(p−1)22
Z
Rd
|g|2(1+|x|x|n−d|2)2 dx
is a nonnegative quadratic form if and only ifα≤αFS
Symmetry breaking holds ifα > αFS
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Information – production of information inequality
LetK[u]be such that d
dτI[u(τ,·)] =− K[u(τ,·)] =−(sum of squares) Ifα≤αFS, thenλ1≥4and
u7→ K[u]
I[u] −4 is a nonnegative functional
Withuε=B? 1 +εfB1−m?
, we observe that
4≤ C2:= inf
u
K[u]
I[u] ≤ lim
ε→0inf
f
K[uε] I[uε] = inf
f
hhf,Lfii
hhf,fii = hhf1,Lf1ii hhf1,f1ii =λ1
ifλ1= 4, that is, if α=αFS, theninfK/I= 4is achieved in the asymptotic regime asu→ B?and determined by the spectral gap ofL
ifλ1>4, that is, if α < αFS, thenK/I >4
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Symmetry in Caffarelli-Kohn-Nirenberg inequalities
Ifα≤αFS, the fact thatK/I ≥4has an important consequence.
Indeed we know that d
dτ (I[u(τ,·)]− 4E[u(τ,·)])≤0 so that
I[u]−4E[u]≥ I[B?]−4E[B?] = 0
This inequality is equivalent toJ[w]≥ J[B?], which establishes that optimality in (CKN) is achieved among symmetric functions. In other words, the linearized problem shows that forα≤αFS, the function
τ7→ I[u(τ,·)]−4E[u(τ,·)]
is monotone decreasing
This explains why the method based on nonlinear flows provides theoptimal range for symmetry
J. Dolbeault Symmetry, entropy methods and nonlinear flows
Entropy – production of entropy inequality
Using dτd (I[u(τ,·)]− C2E[u(τ,·)])≤0, we know that I[u]− C2E[u]≥ I[B?]− C2E[B?] = 0 As a consequence, we have that
C1:= inf
u
I[u]
E[u] ≥C2= inf
u
K[u] I[u]
Withuε=B? 1 +εfB1−m?
, we observe that
C1≤ lim
ε→0inf
f
I[uε] E[uε] = inf
f
hf,Lfi
hf,fi = hf1,Lf1i hf1,fi1
=λ1= lim
ε→0inf
f
K[uε] I[uε] Iflimε→0inff K[uε]
I[uε] =C2, thenC1=C2=λ1
This happens ifα=αFS and in particular in the case without weights (Gagliardo-Nirenberg inequalities)
J. Dolbeault Symmetry, entropy methods and nonlinear flows
These slides can be found at
http://www.ceremade.dauphine.fr/∼dolbeaul/Conferences/
BLectures The papers can be found at
http://www.ceremade.dauphine.fr/∼dolbeaul/Preprints/list/
BPreprints / papers
For final versions, useDolbeaultas login andJeanas password
Thank you for your attention !
J. Dolbeault Symmetry, entropy methods and nonlinear flows