Sym´ etrisation bas´ ee sur des m´ ethodes d’entropie et des flots non-lin´ eaires
Jean Dolbeault
http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Universit´e Paris-Dauphine
March 17, 2017 LMJL, Nantes
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Outline
BSymmetry breaking and linearization
The critical Caffarelli-Kohn-Nirenberg inequality
A family of sub-critical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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
Towards a parabolic proof
Large time asymptotics and spectral gaps A discussion of optimality cases
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
Symmetry and symmetry breaking results
BThe critical Caffarelli-Kohn-Nirenberg inequality
BA family of sub-critical Caffarelli-Kohn-Nirenberg inequalities BLinearization and spectrum
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
Range of the parameters
Herepis given
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
A useful change of variables
With
α= 1 + β−γ
2 and n= 2 d−γ β+ 2−γ, (CKN) can be rewritten for a functionv(|x|α−1x) =w(x)as
kvkL2p,d−n(Rd)≤Kα,n,pkDαvkϑL2,d−n(Rd)kvk1−ϑLp+1,d−n(Rd)
with the notationss=|x|,Dαv= α∂v∂s,1s∇ωv
. Parameters are in the range
d≥2, α >0, n>d and p∈(1,p?], p?:= n n−2 By our change of variables,w? is changed into
v?(x) := 1 +|x|2−p−11
∀x∈Rd
The symmetry breaking condition (Felli-Schneider) now reads α < αFS with αFS:=
rd−1 n−1
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
The second variation
J[v] :=ϑlog kDαvkL2,d−n(Rd)
+ (1−ϑ) log kvkLp+1,d−n(Rd)
+ log Kα,n,p−log kvkL2p,d−n(Rd)
Let us definedµδ:=µδ(x)dx, where µδ(x) := (1 +|x|2)−δ. Sincev? is a critical point ofJ, a Taylor expansion at orderε2shows that
kDαv?k2L2,d−n(Rd)J
v?+ε µδ/2f
=12ε2ϑQ[f]+o(ε2) withδ=p−12p and
Q[f] = Z
Rd
|Dαf|2|x|n−ddµδ− 4pα2 p−1
Z
Rd
|f|2|x|n−ddµδ+1
We assume thatR
Rdf|x|n−ddµδ+1= 0(mass conservation)
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum
Symmetry breaking: the proof
Proposition (Hardy-Poincar´e inequality)
Let d≥2,α∈(0,+∞), n>d andδ≥n. If f has0 average, then Z
Rd
|Dαf|2|x|n−ddµδ ≥Λ Z
Rd
|f|2|x|n−ddµδ+1
with optimal constantΛ = min{2α2(2δ−n),2α2δ η} whereη is the unique positive solution toη(η+n−2) = (d−1)/α2. The corresponding eigenfunction is not radially symmetric ifα2> (d−1)δ2
n(2δ−n) (δ−1) Q ≥0iff 4pp−1α2 ≤Λand symmetry breaking occurs in (CKN) if
2α2δ η < 4pα2
p−1 ⇐⇒ η <1
⇐⇒ d−1
α2 =η(η+n−2)<n−1 ⇐⇒ α > αFS
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
Inequalities without weights and fast diffusion equations
BThe Bakry-Emery method for a Fokker-Planck equation on a domain of the Euclidean space, on the sphere and its extension by non-linear diffusion flows
B[R´enyi entropy powers]
BSelf-similar variables and relative entropies BThe role of the spectral gap
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
The Fokker-Planck equation: ϕ -entropies and Bakry-Emery method
The linear Fokker-Planck (FP) equation
∂u
∂t = ∆u+∇ ·(u∇φ)
on a domainΩ⊂Rd, with no-flux boundary conditions (∇u+u∇φ)·ν = 0 on ∂Ω is equivalent to the Ornstein-Uhlenbeck (OU) equation
∂v
∂t = ∆v− ∇φ· ∇v =:Lv
(Bakry, Emery, 1985), (Arnold, Markowich, Toscani, Unterreiter, 2001)
The unique stationary solution of (FP) (with mass normalized to1) is e−φ
R
Ωe−φdx
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
The Bakry-Emery method
Withv such thatR
Ωv dγ= 1,q∈(1,2], theq-entropyis defined by Eq[v] := 1
q−1 Z
Ω
(vq−1−q(v−1))dγ Under the action of (OU), withw =vq/2,Iq[v] := 4qR
Ω|∇w|2dγ, d
dtEq[v(t,·)] =− Iq[v(t,·)] and d
dt (Iq[v]−2λEq[v])≤0 with λ:= inf
w∈H1(Ω,dγ)\{0}
R
Ω
2q−1q kHesswk2+Hessφ: ∇w⊗ ∇w R dγ
Ω|w|2dγ Proposition
(Bakry, Emery, 1984) (JD, Nazaret, Savar´e, 2008)LetΩbe convex.
Ifλ >0and v is a solution of (OU), thenIq[v(t,·)]≤ Iq[v(0,·)]e−2λt andEq[v(t,·)]≤ Eq[v(0,·)]e−2λt for any t ≥0 and, as a consequence,
Iq[v]≥ 2λEq[v] ∀v ∈H1(Ω,dγ)
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
The interpolation inequalities
On thed-dimensional sphere, let us consider the interpolation inequality
k∇uk2L2(Sd)+ d
p−2kuk2L2(Sd) ≥ d
p−2kuk2Lp(Sd) ∀u∈H1(Sd,dµ) where the measuredµis the uniform probability measure on Sd⊂Rd+1 corresponding to the measure induced by the Lebesgue measure onRd+1, and the exposantp≥1, p6= 2, is such that
p≤2∗:= 2d d−2
ifd ≥3. We adopt the convention that2∗=∞ifd = 1ord= 2.
The casep= 2 corresponds to the logarithmic Sobolev inequality
k∇uk2L2(Sd) ≥d 2
Z
Sd
|u|2log |u|2 kuk2L2(Sd)
!
dµ ∀u∈H1(Sd,dµ)\ {0}
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
The Bakry-Emery method
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#:= (d−1)2d2+12 J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
The rigidity point of view (elliptic PDEs)
In cylindrical coordinates withz ∈[−1,1], let Lf := (1−z2)f00−d z f0=νf00+d
2 ν0f0 be theultraspherical operatorand consider
− Lu−(β−1)|u0|2
u ν+ λ
p−2u= λ p−2uκ Multiply byLuand integrate
...
Z 1
−1Lu uκdνd=−κ Z 1
−1
uκ|u0|2 u dνd
Multiply byκ|uu0|2 and integrate
...= +κ Z 1
−1
uκ|u0|2 u dνd
Z 1
−1
u00−p+ 2 6−p
|u0|2 u
2
ν2dνd= 0 ifp= 2∗andβ = 4 6−p
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
Consequences, improvements, related problems
BImproved inequalities in the subcritical range
BImproved constants under orthogonality constraints forp≤2# BImproved constants under antipodal symmetry forp≤2∗ BThe extension to Riemannian manifolds
BOnofri inequality(JD, Esteban, Jankowiak, 2015) BLin-Ni problems(JD, Kowalczyk)
BKeller-Lieb-Thirring inequalities on manifolds: estimates for Schr¨odinger operator that (really) differ from the semi-classical estimates(JD, Esteban, Laptev, Loss)
[...]
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
R´ enyi entropy powers and fast diffusion
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
Another improvement of the GN inequalities
Let us define therelative entropy E[u] :=−1
m Z
Rd
um− Bm? − mBm−1? (u− B?) dx therelative Fisher information
I[u] :=
Z
Rd
u|z|2dx= Z
Rd
u ∇um−1−2x2dx
and R[u] := 21−m m
Z
Rd
umD2Q2− (1−m) (∆Q)2 dx
Proposition
If1−1/d≤m<1and d ≥2, then I[u0]− 4E[u0]≥
Z ∞
0 R[u(τ,·)]dτ
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies 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
θ2
2y−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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
Spectral gaps 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The Bakry-Emery method, the sphere Self-similar variables and relative entropies The role of the spectral gap
Comments
The spectral gap corresponding to the red curves relies on a refined notion of relative entropy with respect tobest matching Barenblatt profiles(J.D., Toscani)
A result by(Denzler, Koch, McCann)Higher order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach
The constantC in
E[v(t,·)]≤Ce−2γ(m)t ∀t≥0
can be made explicit, under additional restrictions on the initial data(Bonforte, J.D., Grillo, V´azquez)
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
Weighted nonlinear flows:
Caffarelli-Kohn-Nirenberg inequalities
BA parabolic proof ?
BEntropy and Caffarelli-Kohn-Nirenberg inequalities BLarge time asymptotics and spectral gaps
BOptimality cases
J. Dolbeault Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
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
2
dµ + 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 Sym´etrisation, entropie, flots non-lin´eaires
Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities
The strategy of the proof
Large time asymptotics and spectral gaps Linearization and optimality
(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 Sym´etrisation, entropie, flots non-lin´eaires