Relative equilibria in continuous stellar dynamics

Relative equilibria in continuous stellar dynamics

Jean Dolbeault (Ceremade, Universit´e Paris-Dauphine) (with J. Campos and M. del Pino)

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

http://www.ceremade.dauphine.fr/∼dolbeaul/Oberwolfach1049

December 6-10, 2010 Oberwolfach

Introduction

Relative equilibria: [J. Campos, M. del Pino, and J. Dolbeault. Relative equilibria in continuous stellar dynamics. Communications in

Mathematical Physics, Doi:300:765788, 2010, Online First]

Flat systems: [J. Dolbeault and J. Fern´andez, Localized minimizers of flat rotating gravitational systems, Annales de l’Institut Henri Poincar´e (C) Non Linear Analysis, 25 (2008), pp. 1043-1071]

Symmetry breaking issues in functional inequalities:

[J. Dolbeault and M.J. Esteban. About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities, Preprint (2010)]

[J. Dolbeault and M.Esteban. Extremal functions in some interpolation inequalities: Symmetry, symmetry breaking and estimates of the best constants , Preprint (2010)]

[J. Dolbeault, M.J. Esteban, G. Tarantello, and A. Tertikas. Radial symmetry and symmetry breaking for some interpolation inequalities, Preprint (2010)]

[J. Dolbeault and M.J. Esteban. Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, Preprint (2010)]

[J. Dolbeault, M. J. Esteban, M. Loss, and G. Tarantello. On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities. Advanced Nonlinear Studies, 9:713-727, 2009]

[J. Dolbeault, M. J. Esteban, and G. Tarantello, The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann. Sc. Norm. Super. Pisa Cl.

Sci. (5), 7 (2008), pp. 313-341]

Multiple components configurations in continuous stellar dynamics

Introduction: N-body problems and mean-field kinetic equations Outline

◮ A first statement: there are non radially symmetric critical points

◮ Relative equilibria: examples and (partial) classification for systems of point particles

◮ Kinetic equations: extending the notion of relative equilibria to continuum mechanics

◮ Results for kinetic / diffusion equations

◮ The variational approach: heuristics

◮ The variational approach: a sketch of the proofs

◮ Flat systems: results and numerical computation

◮ Concluding remarks: symmetry breaking

A first statement

Gravitational (non-relativistic) Vlasov-Poisson system in R³







∂tF+w· ∇^zF− ∇^zΦ· ∇^wF= 0

∆Φ = Z

R³

F dw (1)

Theorem

For any N ≥2, any p∈(1,5), any positive numbersλ1,λ2, . . .λN and anyω >0 small enough, there is a solution F^ω of (1)which is a relative equilibriumwith angular velocityω whose support has N disjoint

connected components, each of them with mass m^ω_i such that

ω→0lim+

m^ω_i =λ^(3−p)/2_i m∗=:mi

for some positive constant m∗. The center of mass z_i^ω(t)of each component is such thatlimω→0+ω^2/3z_i^ω(t) =:zi(t)is a relative equilibrium of the N-body Newton’s equations with gravitational interaction

Systems of discrete particles:

the N-body problem in

gravitation

Solutions of the N-body problem in gravitation ... many solutions are known

◮ No stationary (time independent) solutions

◮ Periodic solutions in Hamiltonian dynamics: [Ekeland et al.]

◮ Choregraphies: [Chenciner et al.],[Terracini et al.]

Figure: Choregraphies, pictures taken from S. Terracini’s web page http://www.matapp.unimib.it/∼suster/files/index.html

Newton’s equations: basics

ConsiderN point particles with massesmi located at zi(t)∈R³subject to Newton’s equations

mi

d²zi

dt² =

N

X

i6=j=1

mimj

4π

zj−zi

|zj−zi|³ (2) Ansatz: the system is stationary in a reference frame rotating at constant angular velocity Ω =ωe3

Notation: x^′ = (x¹,x²,0) =x−(x·e3) e3, a change of coordinates x³=z³, x¹+i x²=e^iωt(z¹+i z²)

provides Newton’s equations in a rotating frame d²xi

dt² =

N

X

i6=j=1

mj

4π

xj−xi

|xj−xi|³+ω²x_i^′+ 2 Ω∧dxi

dt

We look for stationary solutions in the rotating frame: relative equilibria The configuration is centraland planar: critical points of the function

Vω(x₁^′,x₂^′, . . .x_N^′) :=− 1 8π

N

X

i6=j=1

mimj

|x_j^′−x_i^′|−ω² 2

N

X

i=1

mi|x_i^′|²

Relative equilibria: example 1

◮ All massesmi are equal to somem>0 andx_i^′ are located at the summits of a regular polygon, wherer =|x_i^′|is adjusted so that

d dr

haN

4π m

r +1 2ω²r²i

= 0 with aN := 1

√2

N−1

X

j=1

1

p1−cos (2πj/N) gives a Lagrange solutionwithr =r(N, ω) := _{4π ω}^aNm2

1/3

[Perko-Walter]: all masses have to be equal ifN≥4 Scale invariance:

r(N, ε^3/2ω) =1

εr(N, ω) ∀ε >0 If∇Vω(x₁^′,x₂^′, . . .x_N^′) = 0,

then∇Vε^3/2ω(ε⁻¹x₁^′, ε⁻¹x₂^′, . . . ε⁻¹x_N^′) = 0:

the study of the critical points ofVωcan be

reduced to the caseω= 1

Relative equilibria: other examples

◮ N−1 points particles of same mass are located at the summits of a regular centered polygon and one more point particle stands at the center (with not necessarily the same mass as the other ones). A solution is then found again by adjusting the size of the polygon

◮ TheEuler–Moulton solutionsare made of aligned points

Relative equilibria: classification (1/3)

Relative equilibriaare critical points of the functionVω: (R²)^N→R V^ω(x₁^′,x₂^′, . . .x_N^′) :=− 1

8π

N

X

i6=j=1

mimj

|x_j^′−x_i^′|−¹2ω²

N

X

i=1

mi|x_i^′|²

Generic case: all masses are different

◮ N= 2:

|x1−x2|= M

4π ω² 1/3

and m1x1+m2x2= 0, withM=m1+m2

◮ N= 3:

-Lagrange solutions: masses are located at the vertices of an equilateral triangle, and the distance between each point is (M/(4π ω²))^1/3withM =m1+m2+m3: two classes of solutions corresponding to the two orientations of the triangle when labeled by the masses

-Euler solutionsare made of aligned points and provide three classes of critical points, one for each ordering of the masses on the line

Relative equilibria: classification (2/3)

N≥4: solutions made of aligned points are Moulton’s solutions N≥4: Lagrange solutions (all particles are located at the vertices of a regularN-polygon) exists if and only if all masses are equal

Standard variational setting[Smale]: form= (m1, . . .mN)∈R^N+, consider the manifold (q1, . . .qN)∈R^2N such that

PN

i=1miqi= 0, ¹₂PN

i=1mi|qi|²= 1, qi 6=qj ifi6=j quotiented by the equivalence classes associated to the invariances:

rotations and scalings

dim(Sm)= 2N−3, relative equilibria are critical points onSm of the potential

Um(q1, . . .qN) =− 1 8π

N

X

i6=j=1

mimj

|qj−qi|

Relative equilibria: classification (3/3)

For N≥4, various classification results have been achieved by[Palmore]

◮ ForN≥3, the index of a relative equilibrium is always greater or equal thanN−2. This bound is achieved by Moulton’s solutions

◮ ForN≥3, there are at leastµi(N) := ^N_i

(N−1−i) (N−2) ! distinct relative equilibria inSm of index 2N−4−i ifUmis a Morse function. As a consequence, there are at least

N−2

X

i=0

µi(N) =[2^N−1(N−2) + 1] (N−2) ! distinct relative equilibria inSm ifUm is a Morse function

◮ For every N≥3 and for almost all masses m∈R^N+,Umis a Morse function

◮ There are only finitely many classes of relative equilibria for every N≥3 and for almost all massesm∈R^N₊

◮ IfN≥4, the set of masses for which there exist degenerate classes of relative equilibria has positivek-dimensional Hausdorff measure if 0≤k≤N−1

The kinetic problem

Vlasov-Poisson system in the rotating frame

Gravitational Vlasov-Poisson system (with centrifugal and Coriolis forces):

∂f

∂t +v· ∇xf − ∇xφ· ∇vf −ω²x^′· ∇vf + 2 Ω∧v· ∇vf = 0

∆φ=ρ:=

Z

R³

f dv Boundary conditions: φ=−_4π¹_|·|∗ρ

Change of coordinates: f(t,x,v) =F(t,z,w),φ(t,x) = Φ(t,z) x = exp(ωt A)z, v = Ω∧x+exp(ωt A)w with A:=





0 −1 0

1 0 0

0 0 0





For some arbitrary convex functionβ, critical points of thefree energy F[f] :=

Z Z

R³×R³

β(f)dx dv+1 2

Z Z

R³×R³

|v|²−ω²|x^′|²

f dx dv−1 2

Z

R³

|∇φ|²dx give stationary solutions under the constraintRR

R³×R³f dx dv =M For ω6= 0: no minimizers

Polytropic gases

[Binney-Tremaine]A typical example of such a function is β(f) =κf^q

A critical point ofF such thatRR

R³×R³f dx dv =M is given by f(x,v) =γ

λ+1

2|v|²+φ(x)−1 2ω²|x^′|²

where γ(s) = (β^′)⁻¹(−s): γ(s) = (−s)^1/(q−1)₊

The problem is reduced to solve a non-linear Poisson equation

∆φ=g λ+φ(x)−1

2ω²|x^′|²

χsupp^(ρ) g(µ) :=

Z

R³

γ µ+1 2|v|²

dv Variational approach:

J[φ] := 1 2

Z

R³

|∇φ|²dx+ Z

R³

G

λ+φ(x)−1 2ω²|x^′|²

dx−

Z

R³

λ ρdx whereλ=λ[x, φ] is now a functional which is constant on each

connected componentKi of the support ofρ(x)

Polytropic gases

The total mass is M=PN i=1mi

Z

Ki

g

λi+φ(x)−1 2ω²|x^′|²

dx =mi

J[φ] = 1 2 Z

R³|∇φ|²dx+

N

X

i=1

Z

Ki

G

λi+φ(x)−1 2ω²|x^′|²

dx−miλi

Heuristics. The various componentsKi are far away from each other so that the dynamics of their center of mass is described by theN-body point particles system, at first order. On each component Ki, the solution is a perturbation of an isolated minimizer ofF (without angular rotation) under the constraint that the mass is equal tomi. Alternatively, we consider a critical point of J obtained as the perturbation of a superposition of single components critical points ofJ of mass mi, provided the centers of massxi of each of the components are close enough of a critical point ofVω, withω >0, small

ω = 0: [Guo et al.],[Lemou-M´ehats-Rapha¨el],[Rein et al.], [S´anchez et al. ], [Soler et al. ],[Schaeffer], [Wolansky],[JD-Fern´andez]+[Kurth]

The first result

The spatial density ρ^ω:=R

R³f^ωdv has exactlyNdisjoint connected componentsK_i^ω and

m_i^ω= Z

K_i^ω

ρ^ωdx, z_i^ω(t) = exp (−ωt A)x_i^ω where x_i^ω:= 1 m_i^ω

Z

K_i^ω

xρ^ωdx ρi(x):= 1

λ^p_i ρ^ω

λ^(1−p)/2_i (x+x_i^ω) χK_i^ω

λ^(1−p)/2_i (x+x_i^ω) converges to a density functionρ∗= (w−1)^p₊given by

−∆w = (w −1)^p₊ inR³, lim

|x|→∞w(x) = 0

Theorem

For any N ≥2, any p∈(1,5), any positive numbersλ1,λ2, . . .λN and anyω >0 small enough, there is arelative equilibriumsolution F^ωs.t.

ω→0lim+

m^ω_i =λ^(3−p)/2_i m∗=:mi

for m∗=R

R³ρ∗dx. The center of mass z_i^ω(t)of each component is such thatlimω→0+ω^2/3z_i^ω(t) =:zi(t)is a relative equilibrium of the N-body problem (Newton’s equations)

Some notations

Letx = (x^′,x3)∈R²×R, fixλ1, . . . λN andω >0, small: the problem is

−∆u=

N

X

i=1

ρi inR³, ρi:= u−λi+¹₂ω²|x^′|²p

+χi (3) where χi denotes the characteristic function ofKi

Boundary condition lim|x|→∞u(x) = 0

Mass and center of mass associated to each component by mi :=

Z

R³

ρi dx and xi := 1 mi

Z

R³

xρi dx

We shall say that two solutionsu1andu2are equivalent if there is a rotation R∈SO(2)× {Id},i.e. a rotation in the plane orthogonal to the direction of rotation, such thatu2(x) =u1(R x) for anyx ∈R³. We shall say that u1 andu2aredistinctif they are not equivalent

A second result: Multiplicity of polytropic relative equilibria

Theorem

Forωsmall enough, and for almost every positive

(λ1, . . . λN)∈(0,∞)^N, (3) has at least[2^N−1(N−2) + 1] (N−2) ! distinct solutions which continuoulsy depend on ω

If u^ω is such a solution, there are pointsξ^ω₁, . . . ξ^ω_N∈R³such that as ω →0,|ξ^ω_j −ξ_i^ω| → ∞ for any j 6=i and u^ω(·+ξ^ω_i )locally converges to the unique radial nonnegative solution of

−∆w = (w−λi)^p₊ inR³

Forω >0 small enough, the support ofρ^ω has N connected components

ω→0limm_i^ω=λ^(3−p)/2_i m∗=:mi and lim

ω→0ω^2/3ξ_i^ω:=ξi

and(ξ1, . . . ξN)is a relative equilibrium with masses(mi)1≤i≤N

The scaling invariance is recovered only in the limitω→0+

The basic cell problem and the ansatz

−∆w = (w−1)^p₊ inR³ (4)

Lemma

Under the conditionlim|x|→∞w(x) = 0, Equation (4) has a unique solution, up to translations, which is positive and radially symmetric. It is a non-degenerate critical point of ¹₂R

R³|∇w|²dx+_p+1¹ R

R³(w−1)^p+1₊ dx wi(x) :=w^λi(x−ξi), Wξ :=

N

X

i=1

wi

Compatibility condition: for a large, fixedµ >0, and all smallω >0,

|ξi|< µ ω⁻²³, |ξi−ξj|> µ⁻¹ω⁻²³ (5) Ansatz: we look for a solution of (3) of the form

u=Wξ+φ

withsupp(w^λi −λi)+⊂B(0,R) for alli= 1, . . .N forR>0 large

∆φ+

N

X

i=1

p Wξ−λi+¹₂ω²|x^′|²p−1

+ χiφ=−E−N[φ]

The nonlinear perturbation problem

We want to solve

∆φ+

N

X

i=1

p Wξ−λi+¹₂ω²|x^′|²p−1

+ χiφ=−E−N[φ]

where

E := ∆Wξ+

N

X

i=1

Wξ−λi+¹₂ω²|x^′|²p + χi

N[φ] :=

N

X

i=1

h Wξ−λi+¹₂ω²|x^′|²+φp

+− Wξ−λi+¹₂ω²|x^′|²p +

−p Wξ−λi+¹₂ω²|x^′|²p−1 + φi

χi

The variational scheme

◮ A linear theory

◮ The projected nonlinear problem (Lagrange multipliers)

◮ The variational reduction

◮ A variational approach in finite dimension [Floer-Weinstein 1986]+ many others...

A linear theory

kφk∗= sup

x∈R³ N

X

i=1

|x−ξi|+ 1

!

|φ(x)|, khk∗∗= sup

x∈R³ N

X

i=1

|x−ξi|⁴+ 1

!

|h(x)| Consider the projected problem

L[φ] =h+

N

X

i=1 3

X

j=1

cijZijχi

where Zij:=∂xjwi, subject to orthogonality conditions Z

R³

φZijχidx= 0 ∀i, j = 1,2. . .N

Lemma

Assume that (5) holds. Given h with khk∗∗<+∞, there is a unique solutionφ=: T[h]and there exists a positive constant C , which is independent ofξsuch that, forω >0 small enough,

kφk^∗≤Ckhk^∗∗

The projected nonlinear problem

Findφwithkφk∗<+∞, solution of L[φ] =−E−N[φ] +

N

X

i=1 3

X

j=1

cijZijχi

such thatφ(x)→0 as|x| →+∞and Z

R³

φZijχi dx= 0 for alli, j.

To do this analysis we have to measure the size of the error E. We recall that

E =

N

X

i=1

h(wi+X

j6=i

wj−λi+¹₂ω²|x^′|²)^p₊−(wi−λi)^p₊i χi

=

N

X

i=1

p(wi−λi+t(X

j6=i

wj+¹₂ω²|x^′|²))^p−1₊ (X

j6=i

wj+¹₂ω²|x^′|²)χi

for somet ∈(0,1)

|E| ≤C

N

X

i=1

h X

j6=i

1

|ξi−ξj|+1

2ω²|ξi|²i

χi ≤Cω²³

N

X

i=1

χi

Thus kEk^∗∗≤Cω²³. Moreover, forkφk^∗≤1,

|N[φ]| ≤C

N

X

i=1

|φ|^γχi , γ= min{p,2} kN[φ]k^∗∗≤Ckφk^γ∗ andkN(φ1)−N(φ2)k^∗∗≤o(1)kφ1−φ2k^γ∗ We look for a fixed pointφ=A[φ] :=−T

E + N[φ]

on the region B=n

φ : kφk∗≤Kω²³o

Lemma

∃!φξ =φ(ξ1, . . . ξk)which depends continuously on its parameters for thek k∗-norm andkφξk∗≤Cω²³,

φ_e^θAξ =φξ(e^−θA·) and ci·(e^θAξ) =e^−θAci·

The variational reduction

Lemma

We have that cij = 0for all i, j if and only if the k-tuple(ξ1, . . . ξN)is a critical point of the functional

(ξ1, . . . ξN)7→Λ(ξ1, . . . ξN) :=J(Wξ+φξ) A Taylor expansion

J(Wξ) =J(Wξ+φξ)−DJ(Wξ+φξ)[φξ]+1 2

Z 1 0

D²J(Wξ+(1−t)φξ)[φξ]²dt D²J(Wξ+ (1−t)φξ)[φξ]²=O(ω⁴³)

Λ(ξ) =

N

X

i=1

λ^5−p_i e∗ −h1 2

X

i6=j

mimj

|ξi−ξj|+¹₂ω²

N

X

i=1

mi|ξi|²i

+ O(ω⁴³)

In the region B=n

(ξ1, . . . ξk) : |ξi−ξj|> ρ ω⁻²³, |ξi|< ρ⁻¹ω⁻²³ for alli, jo where ρ >0 is chosen small enough and fixed, we have that

sup

B

Λ > sup

∂B

Λ

so that this functional has a local maximum somewhere inB. Hence a critical point of this functional does exist inB

Lemma

For any λi >0,ξi, we have found a critical point ofΛ, i.e. a solution of L[φ] =−E−N[φ]

A variational approach in finite dimension Um(q1, . . .qN) =− 1

8π

N

X

i6=j=1

mimj

|qj−qi| is a Morse function on Sm. Take a local system of coordinates

(η1, ..., η2N−4) on a neighborhood of a critical point ¯qand for α >0, let ξ(α,p, η) = (α(q1(η) +p), . . . α(qN(η) +p))

Φ(α,p, η) =ω⁻²³Λ(ξ(ω⁻²³α,p,q(η)))

∇Φ(α,p, η) = ∇

−1 2

N

X

i6=j=1

mimj

α|qj(η)−qi(η)| −1 2α²

N

X

i=1

mi|qi(η)|²

+1 2α²|p|²

N

X

i=1

mi

+O(ω²³)

(¯λ¹³,0,η) is nondegenerate, so the local degree¯ deg(∇Φ,˜ U,0) is well defined and nonzero: there exists a critical point (α^∗,p^∗, η^∗) asω→0

A flat model: theory and

numerical results

A two-dimensionalflatmodel

[JD-Fern´andez]Written in cartesian coordinates, the equation is

∂tf +v· ∇xf +ω²x· ∇vf + 2ωv∧ ∇vf − ∇xφ· ∇vf = 0 φ=− 1

4π|x|∗ Z

R²

f dv

where a∧b:=a^⊥·b=a1b2−a2b1 andx,v∈R²

Definition

Alocalized minimizeris a critical point ρofGω which is compactly supported in a ballB(0,R−ε) for some R>0 and ε∈(0,R), and which is a minimizer ofG^ω restricted to the set

{ρ∈L¹₊(R²) : supp(ρ)⊂B(0,R)and Z

R²

ρdx=M}

Results

Theorem

For any M >0, there existsω∗(M) =ω∗>0andω^∗(M) =ω^∗>0 with ω∗≤ω^∗such that

(i) Ifω∈[−ω∗, ω∗], thereduced free energy functional Gω[ρ] := κ

m−1 Z

R²

ρ^mdx−^ω2²

Z

R²

|x|²ρdx− 1 8π

Z Z

R²×R²

ρ(x)ρ(y)

|x−y| dx dy admits a localized minimizer

(ii) If|ω|>ω^∗,G^ω[ρ]admits no localized minimizer

More detailed results in the radial case. How does symmetry breaking occur ?

Numerical schemes

Goal: investigate the energy landscape[JD-Fern´andez-Salomon]

◮ Local minimizers (under appropriate constraints) have a very small basin of attraction

◮ Compact support has to be enforced at each step

◮ Iteration method inspired by mean-field models in quantum

mechanics work, with similar difficulties: the Canc`es-LeBris method of relaxation has to be introduced to achieve convergence

Trying to understand the symmetry breaking

Polytropic case: β(f) =κf^q, without angular velocity: minimizing the free energy

F[f] :=

Z Z

R³×R³

β(f)dx dv+1 2

Z Z

R³×R³|v|²f dx dv−1 2 Z

R³|∇φ|²dx amounts to control the potential energy and find the optimal function in Z

R³|∇φ|²dx≤Ckfk^α1

RR

R³×R³β(f)dx dvβ(2−α) RR

R³×R³|v|²f dx dvβ(2−α)

Explanation: Withρ(x) =R

R³f(x,v)dv,HLS inequality Z

R³

|∇φ|²dx= Z

R³×R³

ρ(x)ρ(y)

4π|x−y| dx dy ≤CHLSkρk^α1kρk^2−αq

+interpolationkρkq≤Cp,q

RR

R³×R³β(f)dx dvβRR

R³×R³|v|²f dx dvβ

Relative equilibria: The constraintRR

R³×R³|x^′|²f dx dv=Const (+ localization inx-space) may break the symmetry

Conclusion: symmetry breaking and stability

◮ In nonlinear PDEs,symmetry breakingusually occurs because of a competition between the nonlinearity and an external potential

◮ A classical example is the (PDE) H´enon problem

◮ The case covered by the theorem of[Gidas-Ni-Nirenberg]is the trivial one: the nonlinearity and an external potential cooperate

◮ Symmetry breaking is usually achieved by eigenvalue considerations

◮ Here we have an example based on multiscale analysis: this is new !

◮ Main issue (especially in gravitation) isdynamical / orbital stability Constrained (localized) minimization and mass transport methods [McCann]but new ideas are required

◮ Building other examples of periodic solutions (choregraphies ?) would be an intermediate step