Symmetry in some interpolation inequalities

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Universit´e Paris-Dauphine

December 13, 2018 Nonlinear PDEs at Valpara´ıso

a conference in honor of Patricio Felmer’s 60th anniversary (UFSM, December 10-14, 2018)

J. Dolbeault Symmetry and symmetry breaking in PDEs

Outline

Introduction

BSymmetry, symmetry breaking: the principle of Pierre Curie BThe issue of the symmetry for the minimizers

BA competition mechanism: potential or weight versus nonlinearity BThe flow point of view: connecting the issue of the symmetry with asymptotic states

Symmetry in some interpolation inequalities

BGagliardo-Nirenberg-Sobolev inequalities on the sphere BCaffarelli-Kohn-Nirenberg inequalities

Symmetry in interpolation inequalities involving Aharonov-Bohm magnetic fields

BAharonov-Bohm effect

BSubquadratic magnetic interpolation inequalities

BAharonov-Bohm magnetic interpolation inequalities inR²

The Cucker-Smale model: symmetry, phase transition, dynamics

Symmetry in some interpolation inequalities

Gagliardo-Nirenberg-Sobolev inequalities on the sphere Caffarelli-Kohn-Nirenberg inequalities

J. Dolbeault Symmetry and symmetry breaking in PDEs

A result of uniqueness on a classical example

On the sphereS^d, let us consider the positive solutions of

−∆u+λu=u^p−1 p∈[1,2)∪(2,2^∗]ifd≥3,2^∗=_d−2^2d

p∈[1,2)∪(2,+∞)ifd = 1,2 Theorem

Ifλ≤d , u≡λ^1/(p−2)is the unique solution

[Gidas & Spruck, 1981],[Bidaut-V´eron & V´eron, 1991]

Bifurcation point of view

2 4 6 8 10

2 4 6 8

Figure:(p−2)λ7→(p−2)µ(λ)with d= 3 k∇uk²L²(S^d)+λkuk²L²(S^d)≥µ(λ)kuk²L^p(S^d)

Taylor expansion ofu= 1 +ε ϕ1as ε→0with−∆ϕ1=dϕ1

µ(λ)< λ if and only if λ > d p−2

BThe inequality holds withµ(λ) =λ= _p−2^d [Bakry & Emery, 1985]

[Beckner, 1993],[Bidaut-V´eron & V´eron, 1991, Corollary 6.1]

J. Dolbeault Symmetry and symmetry breaking in PDEs

The Bakry-Emery method on the sphere

Entropy functional E^p[ρ] := _p−2¹ h´

S^dρ^2p dµ− ´

S^dρdµ²_pi

if p6= 2 E²[ρ] :=´

S^dρlog

ρ kρk_{L1 (Sd)}

dµ Fisher informationfunctional

I^p[ρ] :=´

S^d|∇ρ^1p|²dµ

[Bakry & Emery, 1985]carr´e du champmethod: use the heat flow

∂ρ

∂t = ∆ρ and observe that _dt^dE^p[ρ] =− I^p[ρ],

d dt

I^p[ρ]−dE^p[ρ]

≤0 =⇒ I^p[ρ]≥dE^p[ρ]

withρ=|u|^p, ifp≤2^#:= _(d−1)^2d2+12

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 = ∆ρ^m

(Demange),(JD, Esteban, Kowalczyk, Loss): for anyp∈[1,2^∗] K^p[ρ] := d

dt

I^p[ρ]−dE^p[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 and symmetry breaking in PDEs

Caffarelli-Kohn-Nirenberg, symmetry and symmetry breaking results, and weighted nonlinear flows

BThe critical Caffarelli-Kohn-Nirenberg inequality [JD, Esteban, Loss]

[BA family of sub-critical Caffarelli-Kohn-Nirenberg inequalities]

[JD. Esteban, Loss, Muratori]

Critical Caffarelli-Kohn-Nirenberg inequality

LetD^a,b:=n

v ∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d,dxo ˆ

R^d

|v|^p

|x|^{b p} dx 2/p

≤ Ca,b

ˆ

R^d

|∇v|²

|x|^2a dx ∀v ∈ D^a,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 and symmetry breaking in PDEs

Critical CKN: range of the parameters

Figure: d = 3 ˆ

R^d

|v|^p

|x|^{b p} dx 2/p

≤ Ca,b

ˆ

R^d

|∇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)]

Linear instability of radial minimizers:

the Felli-Schneider curve

The Felli & Schneider curve b_FS(a) := d(ac−a)

2p

(ac−a)²+d−1 +a−a_c

a b

0

[Smets],[Smets, Willem],[Catrina, Wang],[Felli, Schneider]

The functional C^?_a,b

ˆ

R^d

|∇v|²

|x|^2a dx− ˆ

R^d

|v|^p

|x|^{b p} dx 2/p

is linearly instable atv =v?

J. Dolbeault Symmetry and symmetry breaking in PDEs

Symmetry versus symmetry breaking:

the sharp result in the critical case

[JD, Esteban, Loss (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

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) =r^a−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)kϕk²L^p(C) ∀ϕ∈H¹(C)

whereΛ := (ac−a)²,C=R×S^d−1 and the optimal constantµ(Λ)is µ(Λ) = 1

Ca,b

with a=ac±√

Λ and b= d p ±√

Λ

J. Dolbeault Symmetry and symmetry breaking in PDEs

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 onkϕk²L^p(C)

ϕ? is notoptimal for (CKN) if the P¨oschl-Teller operator

−∂s²−∆ω+ Λ−ϕ^p−2_? =−∂s²−∆ω+ Λ− 1 (coshs)² hasa negative eigenvalue, i.e., forΛ>Λ1(explicit)

The variational problem on the cylinder

Λ7→µ(Λ) := min

ϕ∈H¹(C)

k∂sϕk²L²(C)+k∇^ωϕk²L²(C)+ Λkϕk²L²(C)

kϕk²L^p(C)

is a concave increasing function Restricted to symmetric functions, the variational problem becomes

µ?(Λ) := min

ϕ∈H¹(R)

k∂sϕk²L²(R^d)+ Λkϕk²L²(R^d)

kϕk²L^p(R^d)

=µ?(1) Λ^α

Symmetry meansµ(Λ) =µ?(Λ)

Symmetry breaking meansµ(Λ)< µ?(Λ)

J. Dolbeault Symmetry and symmetry breaking in PDEs

Numerical results

20 40 60 80 100

10 20 30 40 50

symmetric non-symmetric asymptotic

bifurcation µ

Λ^α µ(Λ)

(Λ) =µ

(1) Λ^α

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

Symmetry in one slide: 3 steps

Achange of variables: v(|x|^α−1x) =w(x),Dαv= α^∂v_∂s,¹_s∇^ωv kvk^L2p,d−n(R^d) ≤Kα,n,pkDαvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d) ∀v ∈H^p_d−n,d−n(R^d)

Concavity of the R´enyi entropy power: with L^α=−D^∗_αDα=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆ωuand ^∂u_∂t =L^αu^m

−dt^d G[u(t,·)] ´

R^du^mdµ1−σ

≥(1−m) (σ−1)´

R^du^m L^αP−

´

Rd´u|DαP|²dµ

Rdu^mdµ

2

dµ + 2´

R^d

α⁴ 1−¹n

P⁰⁰−^Ps⁰ −α²(n−1)^∆ωPs²

2

+^2α_s2²

∇^ωP⁰−^∇ωs^P

2 u^mdµ + 2´

R^d

(n−2) α²_FS−α²

|∇^ωP|²+c(n,m,d)^|∇_P^ω2^P|4

u^mdµ Elliptic regularity and the Emden-Fowler transformation: justifying theintegrations by parts

J. Dolbeault Symmetry and symmetry breaking in PDEs

Three references

Lecture notes onSymmetry and nonlinear diffusion flows...

a course on entropy methods (see webpage)

[JD, Maria J. Esteban, and Michael Loss]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs

... the elliptic point of view: Proc. Int. Cong. of Math., Rio de Janeiro, 3: 2279-2304, 2018.

[JD, Maria J. Esteban, and Michael Loss]Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization... the parabolic point of view

Journal of elliptic and parabolic equations, 2: 267-295, 2016.

Symmetry in Aharonov-Bohm magnetic fields

Aharonov-Bohm effect

Subquadratic magnetic interpolation inequalities Bon the circle: magnetic rings

Bon the torus

Aharonov-Bohm magnetic interpolation inequalities inR²

Joint work with D. Bonheure, M.J. Esteban, A. Laptev, & M. Loss

J. Dolbeault Symmetry and symmetry breaking in PDEs

Aharonov-Bohm effect

A major difference between classical mechanics and quantum mechanics is that particles are described by a non-local object, the wave function. Quantum particles can interact with an

electromagnetic field even if they are “localized” (from the

experimental point of view) in a region where the fields are zero, or if the fields are supported on zero-measure sets

In 1959 Y. Aharonov and D. Bohm proposed a series of experiments intended to put in evidence such phenomena which are nowadays calledAharonov-Bohm effects

One of the proposed experiments relies on a long, thin solenoid which produces a magnetic field such that the region in which the magnetic field is non-zero can be approximated by a line in dimensiond= 3 and by a point in dimensiond= 2

Notation

Themagnetic Laplacianis defined via amagnetic potentialAby

−∆Aψ=−∆ψ− 2iA· ∇ψ+|A|²ψ−i(divA)ψ Themagnetic fieldisB=curlA

H¹_A(R^d) :=

ψ∈L²(R^d) : ∇^Aψ∈L²(R^d) Themagnetic gradient takes the form

∇^A :=∇+iA Dimensiond = 2: polar coordinates(r, θ)

r =|x|=q

x₁²+x₂² and r e^iθ=x1+i x2

Dimensiond = 3: cylindrical coordinates(ρ, θ,z) ρ=

q

x₁²+x₂², ρe^iθ=x1+i x2 and z =x3

J. Dolbeault Symmetry and symmetry breaking in PDEs

Aharonov-Bohm magnetic fields

Dimensiond = 1:

∇^A = _∂θ^∂ − i a, −∆A=− ∂θ^∂ −i a² Dimensiond = 2: A= a

r²(−x2,x1) = a r²eθ

∇^A = _∂r^∂ , ¹_{r ∂θ}^∂ − i a

, −∆A=−∂r^∂22 −¹r

∂

∂r −r¹²

∂

∂θ−i a² {er= ^x_r,eθ}is the local orthogonal basis

Dimensiond = 3: A= a

ρ²(−x2,x1,0)

∇^A =

∂

∂ρ, _ρ¹ _∂θ^∂ −i a , _∂z^∂

, −∆A=−∂ρ^∂22−¹ρ

∂

∂ρ−ρ¹²

∂

∂θ−i a²

−∂z^∂22

Ais singular atx₁=x₂= 0and the magnetic fieldB=∇ ×Ais a measure supported in the setx1=x2= 0

Subquadratic magnetic interpolation inequalities

Inequalities involving L^pnorms with1<p<2 are generically designated assubquadratic inequalities

BMagnetic rings in the subquadratic range

BA result of symmetry and symmetry breaking on the torus

J. Dolbeault Symmetry and symmetry breaking in PDEs

Magnetic rings: interpolation inequalities on S

p∈[1,2), a non-magnetic interpolation inequality[Bakry, Emrey, 1984]

(2−p)ku⁰k²L²(S¹)+kuk²L^p(S¹)≥ kuk²L²(S¹) ∀u∈H¹(S¹) Lemma

Let a∈Rand p∈[1,2). Then there exists a concave monotone increasing functionµ7→λa,p(µ)onR⁺such that

kψ⁰−i aψk²L²(S¹)+µkψk²L^p(S¹)≥λa,p(µ)kψk²L²(S¹) ∀ψ∈H¹(S¹,C) Diamagnetic inequality and non-magnetic interpolation inequality Existence of an optimal function: Sobolev’s inequalities,

compactness and semi-continuity

Properties

A non-vanishing property: ifψ∈H¹(S¹)is a non-trivial optimal function, thenψ(s)6= 0for any s∈S¹. Takev1(s) +i v2(s) =ψ(s)e^ias and consider the Wronskianw = (v1v₂⁰−v₁⁰v2)

Use the Euler-Lagrange equation for the phase

Q^a,p,µ[u] := ku⁰k²L²(S¹)+a²ku⁻¹k⁻²L²(S¹)+µkuk²L^p(S¹)

kuk²L²(S¹)

The minimization problem is reduced to the study of the inequality ku⁰k²L²(S¹)+a²ku⁻¹k⁻²L²(S¹)+µkuk²L^p(S¹)≥λa,p(µ)kuk²L²(S¹) ∀u∈H¹(S¹) whereuis now a real valued function

J. Dolbeault Symmetry and symmetry breaking in PDEs

A rigidity result

ku⁰k²L²(S¹)+a²ku⁻¹k⁻²L²(S¹)+µkuk²L^p(S¹)≥λa,p(µ)kuk²L²(S¹) ∀u∈H¹(S¹) Proposition

Let p∈(1,2), a∈(0,1/2), andµ >0.

(i) Ifµ(2−p) + 4a²≤1, thenλa,p(µ) =a²+µand equality is achieved only by the constants

(ii) Ifµ(2−p) + 4a²>1, thenλa,p(µ)<a²+µand equality is not achieved by the constants

ku⁰k²L²(S¹)+a²ku⁻¹k⁻²L²(S¹)+µkuk²L^p(S¹)

= (1−4a²)

ku⁰k²L²(S¹)+ µ

1−4a²kuk²L^p(S¹)

+ 4a²

ku⁰k²L²(S¹)+1

4ku⁻¹k²L²(S¹)

Aharonov-Bohm magnetic interpolation inequalities on T

T²=S¹×S¹≈[−π, π)×[−π, π)3(x,y) dσthe uniform probability measure

∇^Aψ= ψx, ψy−i aψ ,

¨

T²

|∇^Aψ|²dσ=

¨

T²

|ψx|²+|ψy −i aψ|² dσ Lemma (A magnetic ground state estimate)

Assume that a∈(0,1/2). Then

¨

T²

|∇^Aψ|²dσ≥a²

¨

T²

|ψ|²dσ ∀ψ∈H¹A(T²) We make a Fourier decomposition on the basis(e<sup>i`x</sup>e<sup>i k y</sup>)k,`∈Z

Ifa∈(0,1/2), thenλ00 is the lowest mode k= 0, `= 0 : λ00=a²

k= 1, `= 0 : λ10= (1−a)<sup>2</sup>>a<sup>2</sup> k= 0, `= 1 : λ01= 1 +a²

J. Dolbeault Symmetry and symmetry breaking in PDEs

A magnetic interpolation inequality in the flat torus

By tensorization, for anyp∈[1,2)

(2−p)k∇uk²L²(T²)+kuk²L^p(T²)≥ kuk²L²(T²) ∀u∈H¹(T²) Lemma

Let p∈[1,2), a∈(0,1/2). Then

k∇^Auk²L²(T²)+µkuk²L^p(T²)≥Λa,p(µ)kuk²L²(T²) ∀u∈H¹_A(T²) µ7→Λa,p(µ)is concave increasing on(0,+∞),limµ→0+Λa,p(µ) =a²

Λa,p(µ)≥µ+ 1−µ(2−p)

a² for any µ≤ 1 2−p

A symmetry result in the subquadratic regime

λa,p(µ): the optimal constant onS¹ Λa,p(µ): the optimal constant on T²

We recall that the magnetic energy onT²is

¨

T²

|∇^Aψ|²dσ=

¨

T²

|ψx|²+|ψy−i aψ|² dσ

Proposition

Let p∈[1,2), a∈(0,1/2). Then

Λa,p(µ) =λa,p(µ) if µ≤ 1 p−2 and any optimal function is then constant w.r.t. x Λa,p(µ) =a²+µif and only ifµ(2−p) + 4a²≤1

J. Dolbeault Symmetry and symmetry breaking in PDEs

Proof

Notationffl

f dx:= _2π¹ ´π

−πf dx k∇^Auk²L²(T²)+µkψk²L^p(T²)

≥ k∂xψk²L²(T²)+λa,p(µ)kψk²L²(T²)+µkψk²L^p(T²)−µ |ψ|^pdy _p²

dx Let us defineu:=|ψ|,v(x) := ffl

|u(x,y)|^pdy1/p

. By H¨older (p≤2)

|v_x|=v^1−p u^p−1u_xdy ≤v^1−p u^pdy ^p−1_p

|u_x|²dy ¹₂

that is,|vx|²≤ffl

|ux|²dy ≤ffl

|∂xψ|²dy. With µ≤1/(2−p), ˆ

S¹

|v_x|²dσ+µ ˆ

S¹

|v|^pdσ 2/p

−µ ˆ

S¹

|v|²dσ+λa,p(µ)kψk²L²(T²)

≥λa,p(µ)kψk²L²(T²)

Aharonov-Bohm magnetic interpolation inequalities in R ²

BOnR²without weights, there is a loss of compactness BOnR²with weights, optimal functions exist

Bthere is a range of symmetry breaking and a range of symmetry

J. Dolbeault Symmetry and symmetry breaking in PDEs

Magnetic interpolation inequalities without weights

k∇^Aψk²L²(R²)+λkψk²L²(R²)≥µa,p(λ)kψk²L^p(R²) ∀ψ∈H¹a(R²) Aharonov-Bohm magnetic potentialA(x) =a|x|⁻²e_θ

Proposition

Let a∈R\Zand p∈(2,∞). The optimal constant is µa,p(λ) = Cpλ^p2 ∀λ >0 and equality is not achieved onH¹(R²)∩L^p(R²)

Magnetic Hardy-Sobolev interpolation inequalities (d = 2)

TheCaffarelli-Kohn-Nirenberg inequalitywithb = c + 2/p ˆ

R²

|∇v|²

|x|^2c dx ≥Cc

ˆ

R²

|v|^p

|x|^bpdx 2/p

By consideringv(x) =|x|^cu(x): theHardy-Sobolev inequality ˆ

R²

|∇u|²dx+ c² ˆ

R²

|u|²

|x|²dx≥Cc

ˆ

R²

|u|^p

|x|²dx 2/p

The optimal functions are radially symmetric if and only if b≥bFS(c) := c− c

√1 + c²

according to[Felli-Schneider (2003)],[D.-Esteban-Loss (2015)]

J. Dolbeault Symmetry and symmetry breaking in PDEs

Theorem (magnetic Hardy-Sobolev type inequality) Let a∈[0,1/2)and p>2. For anyλ >−a²

ˆ

R²

|∇^Aψ|²dx+λ ˆ

R²

|ψ|²

|x|² dx≥µ(λ) ˆ

R²

|ψ|^p

|x|² dx 2/p

∀ψ∈H¹A(R²)

with optimal functionψ(x) = (|x|^α+|x|^−α)^−p−22 ,α= ^p−2₂ √

λ+a²if λ≤λ?:= 41−4a²

p²−4 −a² Conversely, there is symmetry breaking if

λ > λFS(a) := 4

p²−4 −a² Ifλ≤λ?,µ(λ) = ^p₂(2π)^1−2p λ+a²¹⁺_p²

2√ πΓ _p−2^p

(p−2) Γ _p−2^p +¹₂ ¹⁻²p

Corollary 1

Corollary (A magnetic Caffarelli-Kohn-Nirenberg inequality)

Assume that p∈(2,+∞),A(x) =a|x|⁻²e_θ for some a∈[0,1/2)and c≤0. Withµas in Theorem 2, for any γ <c²+a², we have that

ˆ

R²

|∇^Aφ|²

|x|^2c dx ≥γ ˆ

R²

|φ|²

|x|^2c+2dx+µ(c²−γ) ˆ

R²

|φ|^p

|x|^cp+2dx 2/p

andµ(c²−γ)is the optimal constant Takeφ(x) =|x|^cψ(x)

ˆ

R²

|∇^Aφ|²

|x|^2c dx= ˆ

R²

|∇^Aψ|²dx+ ˆ

R²

|ψ|²

|x|²dx

J. Dolbeault Symmetry and symmetry breaking in PDEs

Corollary 2

Proposition (A magnetic Hardy inequality inR²)

Assume that q∈(1,2),A(x) =a|x|⁻²e_θfor some a∈[0,1/2). Then for any functionφ∈L^q R²|x|⁻²dx

, we have ˆ

R²

|∇^Aψ|²dx≥µ(0) ˆ

R²

|φ|^q

|x|²dx −¹_q ˆ

R²

φ

|x|²|ψ|²dx ∀ψ∈H¹_A(R²) Moreover,µ(0)is the optimal constant and

µ(0) = p

2(2π)^1−2pa^2+p4 2√πΓ _p−2^p (p−2) Γ _p−2^p +¹₂

!1−²_p

if a²< 4 12 +p²

The Cucker-Smale model

The homogeneous model Phase transition

Dynamics

Xingyu Li, in preparation

J. Dolbeault Symmetry and symmetry breaking in PDEs

A simple version of the Cucker-Smale model

A model for bird flocking (simplified version)

∂f

∂t =D∆vf +∇^v·(∇^vΦ(v)f − U^f f) whereU^f =´

v f dv is the average velocity (f is a probability measure) Φ(v) = 1

4v⁴−1 2v²

-2 -1 1 2

-1.0 -0.5 0.5 1.0 1.5 2.0

-2 -1 1 2

-2 2 4 6

[J. Tugaut, 2014]

[A. Barbaro, J. Ca˜nizo, J.A. Carrillo, and P. Degond, 2016]

Stationary solutions: phase transition

0.1 0.2 0.3 0.4 0.5

-1.0 -0.5 0.5 1.0

d = 1: there exists a bifurcation pointD=D_∗ such that the only stationary solution corresponds toU^f = 0 ifD>D_∗ and there are three solutions corresponding toU^f = 0,±u(D)ifD<D_∗

U^f = 0is linearly unstable if D<D_∗

Notation: f_?⁽⁰⁾,f_?⁽⁺⁾,f_?⁽⁻⁾

J. Dolbeault Symmetry and symmetry breaking in PDEs

Dynamics

The free energy F[f] :=D

ˆ

R^d

f logf dv+ ˆ

R^d

fΦdv−1 2|U^f|² decays according to

d

dtF[f(t,·)]− ˆ

R^d

D∇^vf

f +∇^vΦ− U^f

2

f dv d = 1: IfF[f(t= 0,·)]<F[f?⁽⁰⁾]andD<D∗, then

F[f(t,·)] =−Fh f_?^(±)i

≤C e^−λt

λis the eigenvalue of the linearized problem atf_?^(±) in the weighted space L²

(f?^(±))⁻¹

with scalar product

hf,gi^±:=D ˆ

R^d

f g

f_?^(±)−1

dv− U^f U^g

These slides can be found at

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

BLectures The papers can be found at

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J. Dolbeault Symmetry and symmetry breaking in PDEs