Dispersion and flux-limited diffusion in classicaland relativistic gravitational systems

Juan Soler

Departamento de Matemática AplicadaUniversidad de Granada

Kinetic Description of Multiscale Phenomena

Brown, May 2010

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 1

Outline

Analysis of dispersion properties to (self-gravitating) kineticsystems vs the existence of steady states

Preservation of structures: patterns, fronts, stationary states,special configurations, breathers, singularities, ...

How to introduce some stochasticity, either because an explicitmodel is unknown or because there are too many factors to betaken into account that make the model much more complex.

If we know that these uncontrollable variables have a smallinfluence on the real phenomena, we want it to be the same forour equations without destroying most of the structures we areinterested in. This problem is usually associated with the conceptof (linear) diffusion.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 2

Outline

Analysis of dispersion properties to (self-gravitating) kineticsystems vs the existence of steady states

Preservation of structures: patterns, fronts, stationary states,special configurations, breathers, singularities, ...

How to introduce some stochasticity, either because an explicitmodel is unknown or because there are too many factors to betaken into account that make the model much more complex.

If we know that these uncontrollable variables have a smallinfluence on the real phenomena, we want it to be the same forour equations without destroying most of the structures we areinterested in. This problem is usually associated with the conceptof (linear) diffusion.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 2

Outline

Analysis of dispersion properties to (self-gravitating) kineticsystems vs the existence of steady states

Preservation of structures: patterns, fronts, stationary states,special configurations, breathers, singularities, ...

How to introduce some stochasticity, either because an explicitmodel is unknown or because there are too many factors to betaken into account that make the model much more complex.

If we know that these uncontrollable variables have a smallinfluence on the real phenomena, we want it to be the same forour equations without destroying most of the structures we areinterested in. This problem is usually associated with the conceptof (linear) diffusion.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 2

Outline

Analysis of dispersion properties to (self-gravitating) kineticsystems vs the existence of steady states

Preservation of structures: patterns, fronts, stationary states,special configurations, breathers, singularities, ...

How introduce some stochasticity, either because an explicitmodel is unknown or because there are too many factors to betaken into account that make the model much more complex?

If we know that these uncontrollable variables have a smallinfluence on the real phenomena, we want it to be the same forour equations without destroy most of the structure we would beinterested. This problem is usually associated with the concept of(linear) diffusion. Use of nonlinear flux–limited diffusion?

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 3

Dispersion in Vlasov systems

Objectives

Our main goal is to be able to predict the dynamics of a 3-Dmany-particle self-gravitating system solely from some of itsmacroscopic parameters:

mass energy linear momentum

To make such theoretical predictions, the role of particle models isovertaken by kinetic theory. We study this feature in the followingsystems:

Vlasov–Poisson

Nordström–Vlasov (Relativistic Scalar Gravity)

Einstein–Vlasov.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 4

Dispersion in Vlasov systems

Objectives

Our main goal is to be able to predict the dynamics of a 3-Dmany-particle self-gravitating system solely from some of itsmacroscopic parameters:

mass energy linear momentum

To make such theoretical predictions, the role of particle models isovertaken by kinetic theory. We study this feature in the followingsystems:

Vlasov–Poisson

Nordström–Vlasov (Relativistic Scalar Gravity)

Einstein–Vlasov.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 4

Dispersion in Vlasov systems

Dynamical behavior

Some plausible dynamical behaviors are:

stationary equilibrium configurations

oscillatory behavior (periodic solutions, breathing modes)

dispersive behavior (to be defined...)

More complicated dynamics are possible (formation of singularities,splitting...).

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 5

Dispersion in Vlasov–Poisson systems

Classical kinetic description of galactic dynamics

Galaxies are modeled as large collections of stars which interact bythe mean gravitational potential that they generate collectively andwithout colliding with each other.

Each star is subject to the Newtonequations of motion

x = vv = −∇xφ

being x ∈ R3 the position, v ∈ R3 themomentum and φ(t , x) the mean gravi-tational potential generated by the starsaltogether.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 6

Dispersion in Vlasov–Poisson systems

Classical kinetic description of galactic dynamics

Galaxies are modeled as large collections of stars which interact bythe mean gravitational potential that they generate collectively andwithout colliding with each other.

Each star is subject to the Newtonequations of motion

x = vv = −∇xφ

being x ∈ R3 the position, v ∈ R3 themomentum and φ(t , x) the mean gravi-tational potential generated by the starsaltogether.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 6

Dispersion in Vlasov–Poisson systems

The Vlasov-Poisson system

Hence we shall model a self-gravitating system using theVlasov-Poisson system, which we write in the following way:

∂t f + v · ∇x f −∇xφ · ∇v f = 0

ρ(t , x) =

∫R3

v

f (t , x , v) dv ,

∆φ = ρ(t , x), lim|x |→∞

φ(t , x) = 0 ∀t ∈ R+.

f (0, x , v) = f0(x , v) ∈ L1(R6) initial distribution of particles.

Units were chosen so that 4πG = m = 1.

I Classical Theory (PFAFFELMOSER, SCHAEFFER, HORST,...)I Weak Theory (HORST, HUNZE, LIONS, PERTHAME,...)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 7

Dispersion in Vlasov–Poisson systems

The Vlasov-Poisson system

Hence we shall model a self-gravitating system using theVlasov-Poisson system, which we write in the following way:

∂t f + v · ∇x f −∇xφ · ∇v f = 0

ρ(t , x) =

∫R3

v

f (t , x , v) dv ,

∆φ = ρ(t , x), lim|x |→∞

φ(t , x) = 0 ∀t ∈ R+.

f (0, x , v) = f0(x , v) ∈ L1(R6) initial distribution of particles.

Units were chosen so that 4πG = m = 1.

I Classical Theory (PFAFFELMOSER, SCHAEFFER, HORST,...)I Weak Theory (HORST, HUNZE, LIONS, PERTHAME,...)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 7

Dispersion in Vlasov–Poisson systems

Macroscopic quantities

The following quantities are conserved during evolution:

the total massM =

∫R6

f dxdv

the linear momentum

Q =

∫R6

v f (t , x , v)dvdx

the energy

H = Ekin(f )− Epot(f )

=12

∫v2f (t , x , v) dvdx − 1

2||∇φ(t)||22 .

We can also define the center of mass:

cp(t) =1M

∫R3

x ρdx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 8

Dispersion in Vlasov–Poisson systems

Macroscopic quantities

The following quantities are conserved during evolution:

the total massM =

∫R6

f dxdv

the linear momentum

Q =

∫R6

v f (t , x , v)dvdx

the energy

H = Ekin(f )− Epot(f )

=12

∫v2f (t , x , v) dvdx − 1

2||∇φ(t)||22 .

We can also define the center of mass:

cp(t) =1M

∫R3

x ρdx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 8

Dispersion in Vlasov–Poisson systems

Macroscopic quantities

The following quantities are conserved during evolution:

the total massM =

∫R6

f dxdv

the linear momentum

Q =

∫R6

v f (t , x , v)dvdx

the energy

H = Ekin(f )− Epot(f )

=12

∫v2f (t , x , v) dvdx − 1

2||∇φ(t)||22 .

We can also define the center of mass:

cp(t) =1M

∫R3

x ρdx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 8

Dispersion in Vlasov–Poisson systems

Macroscopic quantities

The following quantities are conserved during evolution:

the total massM =

∫R6

f dxdv

the linear momentum

Q =

∫R6

v f (t , x , v)dvdx

the energy

H = Ekin(f )− Epot(f )

=12

∫v2f (t , x , v) dvdx − 1

2||∇φ(t)||22 .

We can also define the center of mass:

cp(t) =1M

∫R3

x ρdx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 8

Dispersion in Vlasov–Poisson systems

Examples in the VP setting

Kurth’s example:We can build up a family of f (t , x , v) such that

ρ(t , x) =3

4πR(t)−3χ|x |<R(t),

being R(t) the radius of the system. Behavior depends on theassociated energy spectrum of the family:

dispersive solutionsperiodic solutions−3/5 0

E

R. KURTH, A global particular solution to the initial value problem of the stellar dynamics, Quart. Appl. Math., 36(1978), 325–329.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 9

Dispersion in Vlasov–Poisson systems

Examples in the VP setting: Polytropes

Polytropes are stationary solutions of VP defined by

νµ,k (x , v) = C(

E0 −12|v |2− φνµ(x)

)µ+

|x ∧ v |k ,

where φνµ,k is a solution of the Poisson equation.

(G. WOLANSKY, Y. GUO, G. REIN, O. SÁNCHEZ, J.S., M. LEMOU, F. MEHATS, P. RAPHAEL)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 10

Dispersion in Vlasov–Poisson systems

Galilean invariance

The VP system satisfies the property of Galilean invariance, that is, theequivalence of all inertial observers.

In mathematical terms, given any V ∈ R3, if

f (t , x , v) is a solution with initial data f0(x , v),

then

f (t , x − tV , v − V ) is the solution with initial data f0(x , v − V ).

Traveling solutions can be constructed. This feature has to be takeninto account whenever dealing with stability or dispersion.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 11

Dispersion in Vlasov–Poisson systems

Galilean invariance

The VP system satisfies the property of Galilean invariance, that is, theequivalence of all inertial observers.

In mathematical terms, given any V ∈ R3, if

f (t , x , v) is a solution with initial data f0(x , v),

then

f (t , x − tV , v − V ) is the solution with initial data f0(x , v − V ).

Traveling solutions can be constructed. This feature has to be takeninto account whenever dealing with stability or dispersion.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 11

Dispersion in Vlasov–Poisson systems

The concept of strong dispersion

A solution of the V-P system is said to be strongly dispersive if thereexists p > 1 such that limt→+∞||ρ(t)||p exists and is zero.

Example: small data solutions.

For small smooth initial data we get the following long time behavior:

||ρ(t)||∞ ∼ t−3 (same rate as in free transport)

Epot (f ) = 12 ||∇φ||

22 ∼ t−1

The global behavior for large times is like that of free streamingparticles.

C. BARDOS, P. DEGOND, Global existence for the Vlasov-Poisson equation in 3 space variables with small initialdata, Ann. Inst. Henri Poincaré, Analyse non linéaire, 2 (1985), 101–118.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 12

Dispersion in Vlasov–Poisson systems

The concept of strong dispersion

A solution of the V-P system is said to be strongly dispersive if thereexists p > 1 such that limt→+∞||ρ(t)||p exists and is zero.

Example: small data solutions.

For small smooth initial data we get the following long time behavior:

||ρ(t)||∞ ∼ t−3 (same rate as in free transport)

Epot (f ) = 12 ||∇φ||

22 ∼ t−1

The global behavior for large times is like that of free streamingparticles.

C. BARDOS, P. DEGOND, Global existence for the Vlasov-Poisson equation in 3 space variables with small initialdata, Ann. Inst. Henri Poincaré, Analyse non linéaire, 2 (1985), 101–118.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 12

Dispersion in Vlasov–Poisson systems

The concept of strong dispersion

A solution of the V-P system is said to be strongly dispersive if thereexists p > 1 such that limt→+∞||ρ(t)||p exists and is zero.

Example: small data solutions.

For small smooth initial data we get the following long time behavior:

||ρ(t)||∞ ∼ t−3 (same rate as in free transport)

Epot (f ) = 12 ||∇φ||

22 ∼ t−1

The global behavior for large times is like that of free streamingparticles.

C. BARDOS, P. DEGOND, Global existence for the Vlasov-Poisson equation in 3 space variables with small initialdata, Ann. Inst. Henri Poincaré, Analyse non linéaire, 2 (1985), 101–118.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 12

Dispersion in Vlasov–Poisson systems

Concentration-compactness theory

The concentration-compactness lemma: Let ρn be a sequence inL1(RN) such that ∫

RNρn dx = M > 0.

Then there exists a subsequence ρnk satisfying one of the following:

Compactness: there exists yk ∈ RN such that

∀ε > 0, ∃R <∞ such that∫

yk +BR

ρnk (x) dx > M − ε.

( P.L. LIONS)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 13

Dispersion in Vlasov–Poisson systems

Concentration-compactness theory

The concentration-compactness lemma: Let ρn be a sequence inL1(RN) such that ∫

RNρn dx = M > 0.

Then there exists a subsequence ρnk satisfying one of the following:

Vanishing: for all R > 0 there holds that

limk→∞

supy∈RN

∫y+BR

ρnk (x) dx = 0.

( P.L. LIONS)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 13

Dispersion in Vlasov–Poisson systems

Concentration-compactness theory

The concentration-compactness lemma: Let ρn be a sequence inL1(RN) such that ∫

RNρn dx = M > 0.

Then there exists a subsequence ρnk satisfying one of the following:

Dichotomy: There exists α ∈]0,M[ such that ∀ε > 0 we can find0 6 ρout

k , ρink ∈ L1(RN) that for advanced k satisfy

dist (Supp ρoutk ,Supp ρin

k )→ +∞

‖ρnk − (ρoutk + ρin

k )‖1 6 ε,

∣∣∣∣∫RNρout

k dx − α∣∣∣∣ 6 ε,

∣∣∣∣∫RNρin

k dx − (M − α)

∣∣∣∣ 6 ε.

( P.L. LIONS)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 13

Dispersion in Vlasov–Poisson systems

Vanishing of mass

Define for a solution to the V-P system, whenever it is possible, thefunction

M(R) = limt→∞

supx0∈R3

∫|x−x0|<R

ρ(t , x) dx .

Then we give the name outgoing mass to the quantity

Mout = M − limR→∞

M(R).

Given a solution for the V-P system such that the above constructionhas a meaning, we say that such a solution is totally, respectivelypartially dispersive ifMout = M, respectivelyMout ∈]0,M[.

(Galilean invariance vs R. GLASSEY, W. STRAUSS, ...)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 14

Dispersion in Vlasov–Poisson systems

Vanishing of mass

Define for a solution to the V-P system, whenever it is possible, thefunction

M(R) = limt→∞

supx0∈R3

∫|x−x0|<R

ρ(t , x) dx .

Then we give the name outgoing mass to the quantity

Mout = M − limR→∞

M(R).

Given a solution for the V-P system such that the above constructionhas a meaning, we say that such a solution is totally, respectivelypartially dispersive ifMout = M, respectivelyMout ∈]0,M[.

(Galilean invariance vs R. GLASSEY, W. STRAUSS, ...)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 14

Dispersion in Vlasov–Poisson systems

Vanishing of mass

Define for a solution to the V-P system, whenever it is possible, thefunction

M(R) = limt→∞

supx0∈R3

∫|x−x0|<R

ρ(t , x) dx .

Then we give the name outgoing mass to the quantity

Mout = M − limR→∞

M(R).

Given a solution for the V-P system such that the above constructionhas a meaning, we say that such a solution is totally, respectivelypartially dispersive ifMout = M, respectivelyMout ∈]0,M[.

(Galilean invariance vs R. GLASSEY, W. STRAUSS, ...)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 14

Dispersion in Vlasov–Poisson systems

Examples in the VP setting

Outgoing shells:Let f0 describe a spherically symmetric shell of (smoothly distributed)matter of total mass M, internal radius R1(0) and external radiusR2(0). Let w = x · v/r -radial velocity variable-. If

W := infsupp f0

w2 − M2πR1(0)

> 0

thenR2(t) > R1(t) > R1(0) + Wt .

(ANDREASSON, KUNZE & REIN)

The same dynamics can be reproduced if we place a ball-shapedsteady state inside the initial shell.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 15

Dispersion in Vlasov–Poisson systems

Examples in the VP setting

Outgoing shells:Let f0 describe a spherically symmetric shell of (smoothly distributed)matter of total mass M, internal radius R1(0) and external radiusR2(0). Let w = x · v/r -radial velocity variable-. If

W := infsupp f0

w2 − M2πR1(0)

> 0

thenR2(t) > R1(t) > R1(0) + Wt .

(ANDREASSON, KUNZE & REIN)

The same dynamics can be reproduced if we place a ball-shapedsteady state inside the initial shell.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 15

Dispersion in Vlasov–Poisson systems

The concept of statistical dispersion

For any solution of the Vlasov-Poisson system we can define its spatialvariance as

〈(∆x)2〉 =1M

∫R3|x − cp(t)|2 ρ(t , x) dx .

This serves as a “mean radius” for the system. The unlimited growth ofthis quantity can be seen as some kind of weak dispersion. We claimstatistical dispersion to happen whenever supt>0〈(∆x)2〉 = +∞.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 16

Dispersion in Vlasov–Poisson systems

The concept of statistical dispersion

For any solution of the Vlasov-Poisson system we can define its spatialvariance as

〈(∆x)2〉 =1M

∫R3|x − cp(t)|2 ρ(t , x) dx .

This serves as a “mean radius” for the system. The unlimited growth ofthis quantity can be seen as some kind of weak dispersion. We claimstatistical dispersion to happen whenever supt>0〈(∆x)2〉 = +∞.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 16

Dispersion in Vlasov–Poisson systems

Strong dispersion: characterizations

Result:Let f be a regular solution of the Vlasov–Poisson system. Then thefollowing assertions are equivalent:

1 f is strongly dispersive2 f is totally dispersive3 the potential energy vanishes as t →∞.

Moreover, if any of the above holds then f satisfies the inequalities

H ≥ Q2

2M

‖ρ(t)‖p > C(1 + t)−3(p−1)

p for t 1, p ∈]1,∞].

(S. CALOGERO, J. CALVO, O. SANCHEZ, J.S., DCDS 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 17

Dispersion in Vlasov–Poisson systems

Statistical dispersion: sufficient conditions

Result

Let f be a regular solution of the V-P system. Then H > Q2

2M impliesthat 〈(∆x)2〉 = O(t2) for large times.

This condition is verified for all the family of Galilean transforms of agiven solution if and only if it is so for the chosen solution.

Noteworthy, in the spherically symmetric setting the condition H > 0implies statistical dispersion.

In the opposite case H < Q2

2M ⇒

EPOT (f ) > C ,

‖ρ(t , ·)‖L

53 (R3)

> C′ .

J. DOLBEAULT , O. SÁNCHEZ, J. SOLER, Asymptotic behaviour for the Vlasov-Poisson system in the stellar dynamics case, Arch.Rat. Mech. Anal., 171 (2004), 301–327.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 18

Dispersion in Vlasov–Poisson systems

Statistical dispersion: sufficient conditions

Result

Let f be a regular solution of the V-P system. Then H > Q2

2M impliesthat 〈(∆x)2〉 = O(t2) for large times.

This condition is verified for all the family of Galilean transforms of agiven solution if and only if it is so for the chosen solution.

Noteworthy, in the spherically symmetric setting the condition H > 0implies statistical dispersion.

In the opposite case H < Q2

2M ⇒

EPOT (f ) > C ,

‖ρ(t , ·)‖L

53 (R3)

> C′ .

J. DOLBEAULT , O. SÁNCHEZ, J. SOLER, Asymptotic behaviour for the Vlasov-Poisson system in the stellar dynamics case, Arch.Rat. Mech. Anal., 171 (2004), 301–327.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 18

Dispersion in Vlasov–Poisson systems

The threshold for statistical dispersion

Result

Regular solutions of the Vlasov-Poisson system which satisfy H = Q2

2Mare statistically dispersive. The variance grows at least linearly in time.

( J. CALVO, 2010)

Kurth solutions with zero energy are statistically dispersive and〈(∆x)2〉 grows like t4/3.

R. KURTH, A global particular solution to the initial value problem of the stellar dynamics, Quart. Appl. Math., 36(1978), 325–329.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 19

Dispersion in Vlasov–Poisson systems

Other results

Results:Time periodic solutions of the Vlasov–Poisson system satisfyH < −Q2

2M .

Any steady state of the Vlasov–Poisson system has negative energy.

Any traveling steady state verifies H < Q2

2M .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 20

Dispersion in Vlasov–Poisson systems

Some (equivalent) useful relations

The pseudoconformal law:

ddt

∫R6

(x − tv)2f dxdv =ddt

(2t2Epot

)− 2tEpot.

( ILLNER & REIN, PERTHAME)

The dilation identity:

ddt

∫R6

x · v f dxdv = H + Ekin.

An equation for the spatial variance:

Md2

dt2 〈(∆x)2〉 = 2H + 2Ekin − 2Q2

M.

( ARRIOLA & SOLER, JSP 2001, DOLBEAULT, SÁNCHEZ & SOLER, ARMA 2004)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 21

Dispersion in Vlasov–Poisson systems

Some (equivalent) useful relations

The pseudoconformal law:

ddt

∫R6

(x − tv)2f dxdv =ddt

(2t2Epot

)− 2tEpot.

( ILLNER & REIN, PERTHAME)

The dilation identity:

ddt

∫R6

x · v f dxdv = H + Ekin.

An equation for the spatial variance:

Md2

dt2 〈(∆x)2〉 = 2H + 2Ekin − 2Q2

M.

( ARRIOLA & SOLER, JSP 2001, DOLBEAULT, SÁNCHEZ & SOLER, ARMA 2004)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 21

Dispersion in Vlasov–Poisson systems

Some (equivalent) useful relations

The pseudoconformal law:

ddt

∫R6

(x − tv)2f dxdv =ddt

(2t2Epot

)− 2tEpot.

( ILLNER & REIN, PERTHAME)

The dilation identity:

ddt

∫R6

x · v f dxdv = H + Ekin.

An equation for the spatial variance:

Md2

dt2 〈(∆x)2〉 = 2H + 2Ekin − 2Q2

M.

( ARRIOLA & SOLER, JSP 2001, DOLBEAULT, SÁNCHEZ & SOLER, ARMA 2004)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 21

Dispersion in Vlasov–Poisson systems

Md2

dt2 〈(∆x)2〉 = 2H + 2Ekin − 2Q2

M.

We prove the optimal range for the kinetic energy

Ekin(f ) ∈ [E−kin,E+kin],

where

E±kin = −2IM(

1− H2IM±√

1− HIM

),

IM = inf

H = Ekin(g)− Epot(g); g≥0, ‖g‖L1(R6) =M, ‖g‖L∞(R6)≤1.

ResultFor all M>0, there exits a minimum of IM .

The minimum is reached inν0(· − x ′, ·); x ′ ∈ R3 .

( DOLBEAULT, SÁNCHEZ, SOLER, ARMA, 2004)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 22

Orbital Stability of the Vlasov–Poisson Polytropes

Orbital Stability for Spherical Polytropic (H < 0)

Let µ ∈ [0,7/2) and ε > 0. Also, let νµ be a spherical polytropic. Then,there exists δ = δ(ε) > 0 such that for every initial condition f 0

satisfying1 H(f0)− H(νµ) ≤ δ ,2 f0 ∈ f ∈ L1 ∩ L1+1/µ, f ≥ 0, ‖f‖L1 = M, ‖f‖L1+1/µ ≤ J ∩ C1

0 ,the associated solution f to the VP system verifies

infk∈R3

‖f (t , ·, ·)− νµ(· − k , ·)‖L1(R6) ≤ ε , ∀t ∈ (0,∞).

If µ 6= 0 we also have

infk∈R3

‖f (t , ·, ·)− νµ(· − k , ·)‖L1+1/µ(R6) ≤ ε , ∀t ∈ (0,∞) .

O. SÁNCHEZ, J. SOLER, Orbital stability for polytropic galaxies, Annales de l’institut Henri Poincaré, 23 (2006), 781–802.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 23

Dispersion in Norsdröm–Vlasov

The Norsdröm–Vlasov system

∂t f + p√

e2φ+|p|2· ∇x f −∇x (

√e2φ + |p|2) · ∇pf = 0 ,

∂2t φ−∆xφ = −e2φ

∫R3

p

f (t , x ,p)√e2φ + |p|2

dp .

f : [0,T [×R3x×R3

p→ [0,∞[ particle distribution

φ : [0,T [×R3→ [0,∞[ potential

Supplied with initial data f (t = 0, x ,p) = f0(x ,p),φ(t = 0, x) = φ0(x) and ∂tφ(t = 0, x) = φ1(x).

Written in units such that 4πG = c = m = 1.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 24

Dispersion in Norsdröm–Vlasov

The Norsdröm–Vlasov system

∂t f + p√

e2φ+|p|2· ∇x f −∇x (

√e2φ + |p|2) · ∇pf = 0 ,

∂2t φ−∆xφ = −e2φ

∫R3

p

f (t , x ,p)√e2φ + |p|2

dp .

The spacetime is the Lorentzian manifold (R4,e2φη).

Particles are moving along the geodesic curves of the metric.

This model satisfies the fundamental property of Lorentzinvariance.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 24

Dispersion in Norsdröm–Vlasov

Some associated quantities

The density and total mass:

ρ(t , x) =

∫R3

f (t , x ,p) dp, M =

∫R3ρ(t , x) dx .

The local and total energy:

h(t , x) =

∫R3

√e2φ + |p|2 f dp +

12

(∂tφ)2 +12|∇xφ|2,

H =

∫R3

h(t , x) dx .

The local and total momentum:

q(t , x) =

∫R3

pf dp − ∂tφ∇xφ , Q =

∫R3

q(t , x) dx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 25

Dispersion in Norsdröm–Vlasov

Some associated quantities

The density and total mass:

ρ(t , x) =

∫R3

f (t , x ,p) dp, M =

∫R3ρ(t , x) dx .

The local and total energy:

h(t , x) =

∫R3

√e2φ + |p|2 f dp +

12

(∂tφ)2 +12|∇xφ|2,

H =

∫R3

h(t , x) dx .

The local and total momentum:

q(t , x) =

∫R3

pf dp − ∂tφ∇xφ , Q =

∫R3

q(t , x) dx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 25

Dispersion in Norsdröm–Vlasov

Some associated quantities

The density and total mass:

ρ(t , x) =

∫R3

f (t , x ,p) dp, M =

∫R3ρ(t , x) dx .

The local and total energy:

h(t , x) =

∫R3

√e2φ + |p|2 f dp +

12

(∂tφ)2 +12|∇xφ|2,

H =

∫R3

h(t , x) dx .

The local and total momentum:

q(t , x) =

∫R3

pf dp − ∂tφ∇xφ , Q =

∫R3

q(t , x) dx .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 25

Dispersion in Norsdröm–Vlasov

Dynamical behavior

Preliminary result:

Let (f , φ) be a regular solution of the Norsdtröm–Vlasov system.Assume that

H2 − HM − |Q|2 > 0.

Then, there exist a time instant t0 and positive constants 0 < C1 < C2such that the spatial variance ∆x (t) of the unitary energy densityfunction,

∆x (t) =

∫R3|x − h(t)|2 h(t , x)

Hdx , where h(t) =

∫R3

xh(t , x)

Hdx ,

satisfiesC1t2 ≤ ∆x (t) ≤ C2t2 ∀t > t0 .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 26

Dispersion in Norsdröm–Vlasov

Dynamical behavior

Preliminary result:

Let (f , φ) be a regular solution of the Norsdtröm–Vlasov system.Assume that

H2 − HM − |Q|2 > 0.

Then, there exist a time instant t0 and positive constants 0 < C1 < C2such that the spatial variance ∆x (t) of the unitary energy densityfunction,

∆x (t) =

∫R3|x − h(t)|2 h(t , x)

Hdx , where h(t) =

∫R3

xh(t , x)

Hdx ,

satisfiesC1t2 ≤ ∆x (t) ≤ C2t2 ∀t > t0 .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 27

Dispersion in Norsdröm–Vlasov

Dynamical behavior

Conjecture:

Let (f , φ) be a regular solution of the Norsdtröm–Vlasov system.Assume that

H2 −M2 − |Q|2 > 0.

Then, there exist a time instant t0 and positive constants 0 < C1 < C2such that the spatial variance ∆x (t) of the unitary energy densityfunction,

∆x (t) =

∫R3|x − h(t)|2 h(t , x)

Hdx , where h(t) =

∫R3

xh(t , x)

Hdx ,

satisfiesC1t2 ≤ ∆x (t) ≤ C2t2 ∀t > t0 .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 28

Steady States and Stability in Norsdrön–Vlasov

About steady states

Result:Let f be a static regular asymptotically flat solution of the NV system.Then

H ≤ M.

Moreover, equality above implies that the support of the static solutionis unbounded.

Idea of the proof:

ddt

∫R3

q · x − φ∂tφdx =

∫R3

h dx +

∫R3

e2φ(φ− 1)

∫R3

f dp√e2φ + |p|2

dx

>∫

R3h − ρdx = H −M.

Math formalization: KLAINERMAN’s vector field multipliers technique.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 29

Steady States and Stability in Norsdrön–Vlasov

About steady states

Result:Let f be a static regular asymptotically flat solution of the NV system.Then

H ≤ M.

Moreover, equality above implies that the support of the static solutionis unbounded.

Idea of the proof:

ddt

∫R3

q · x − φ∂tφdx =

∫R3

h dx +

∫R3

e2φ(φ− 1)

∫R3

f dp√e2φ + |p|2

dx

>∫

R3h − ρdx = H −M.

Math formalization: KLAINERMAN’s vector field multipliers technique.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 29

Steady States and Stability in Norsdrön–Vlasov

About steady states

Result:Let f be a static regular asymptotically flat solution of the NV system.Then

H ≤ M.

Moreover, equality above implies that the support of the static solutionis unbounded.

Idea of the proof:

ddt

∫R3

q · x − φ∂tφdx =

∫R3

h dx +

∫R3

e2φ(φ− 1)

∫R3

f dp√e2φ + |p|2

dx

>∫

R3h − ρdx = H −M.

Math formalization: KLAINERMAN’s vector field multipliers technique.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 29

Steady States and Stability in Norsdrön–Vlasov

The vector fields multipliers method

Local conservation laws:

∂µTµν = 0, µ, ν = 0, . . . ,3.

Integral identity: ∫∂Ω

Tµνξνnµdσ =

∫Ω

Tµν∂µξ

νdtdx .

Choose the region Ω = [0,T ]× x ∈ R3 : |x | ≤ R.

Choose the vector field ξ0 = 0, ξi = χ(r)ωi for suitable χ.

S. CALOGERO, J. CALVO, O. SÁNCHEZ, J. SOLER, Virial inequalities for steady states in relativistic galactic dynamics,Nonlinearity 2010.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 30

Steady States and Stability in Norsdrön–Vlasov

NV system: Orbital stability

The stability for the isotropic polytropes

f0(x ,p) =

(E0 − E

c

)k

+

, E =√

e2φ0 + |p|2.

Here k > −1, c > 0 and E0 > 0 are constants, E is the particle energy.

They are associated to the existence of a minimizer to the variationalproblem

infH(f , φ, ψ), f ∈ ΓkM,J , φ ∈ D1, ψ ∈ L2, Ekin(f ,0) <∞,

provided the mass M is sufficiently large (depending on J and k ),where H is the energy functional and k ∈ (0,2).

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 31

Steady States and Stability in Norsdrön–Vlasov

Orbital Stability (H ≤ M)

Let (f0, φ0) be the minimizer associated to 0 < k < 2, J > 0 andM > M0. For every ε > 0, there exists δ = δ(ε) such that, for all initialdata (f in, φin

0 , φin1 ) = (f , φ, ∂tφ)|t=0 of the NV system in the class

0 ≤ f in ∈ ΓkM,J ∩ C1

c , φin0 ∈ C3 ∩ D1, φin

1 ∈ C2 ∩ L2

and ∣∣H(f in, φin0 , φ

in1 )− H(f0, φ0,0)

∣∣ ≤ δ,the associated solution (f , φ) ∈ C1 × C2 satisfies, for all t > 0,

infy∈R3

‖f − Ty f0‖L1 + infy∈R3

‖f − Ty f0‖L1+1/k ≤ ε,

infy∈R3

‖∇φ− Ty∇φ0‖L2 + ‖∂tφ‖L2 ≤ ε.

S. CALOGERO, O. SÁNCHEZ, J. SOLER, Orbital stability for polytropic galaxies, Arch. Rat. Mech. Anal, 194 (2009), 743–773.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 32

Steady States and Stability in Norsdrön–Vlasov

Advantages of the Poisson coupling

The Emden–Folder equation for the VP equation

(r2ψ′(r))′ = −r2(ψ(r))µ+3/2+ .

The strongly nonlinear and nonlocal character of the ODE

(r2ψ′(r))′ = r2e2ψ(r)

∫R3

f0√e2ψ(r) + |p|2

dp,

which is the equivalent counterpart of the Emden–Fowler equationin the NV case.

=⇒ Regularity (compactness), uniqueness, ...

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 33

Steady States and Stability in Norsdrön–Vlasov

Advantages of the Poisson coupling

The Emden–Folder equation for the VP equation

(r2ψ′(r))′ = −r2(ψ(r))µ+3/2+ .

The strongly nonlinear and nonlocal character of the ODE

(r2ψ′(r))′ = r2e2ψ(r)

∫R3

f0√e2ψ(r) + |p|2

dp,

which is the equivalent counterpart of the Emden–Fowler equationin the NV case.

=⇒ Regularity (compactness), uniqueness, ...

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 33

Steady States and Stability in Norsdrön–Vlasov

Advantages of the Poisson coupling

The Emden–Folder equation for the VP equation

(r2ψ′(r))′ = −r2(ψ(r))µ+3/2+ .

The strongly nonlinear and nonlocal character of the ODE

(r2ψ′(r))′ = r2e2ψ(r)

∫R3

f0√e2ψ(r) + |p|2

dp,

which is the equivalent counterpart of the Emden–Fowler equationin the NV case.

=⇒ Regularity (compactness), uniqueness, ...

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 33

Steady States in Einstein–Vlasov

The Einstein–Vlasov system

The fundamental equations of GR are the Einstein equations

Gµν := Rµν −12

Rgµν = 8πTµν + Λgµν ,

Gµν is the Einstein tensor, Λ the cosmological constant, Rµν theRicci tensor, R the scalar curvature and g the space-time metric.

We want to study the case when Λ = 0 and Tµν is determined byVlasov matter (kinetic description)

pα∂xα f − Γαβγpβpγ∂pα f = 0,

Gµν = 8πTµν = 8π∫

pµpν f |g|1/2 dp1dp2dp3

−p0 ,

with geodesics equations: dXαdτ = Pα, dPα

dτ = −ΓαβγPβPγ .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 34

Steady States in Einstein–Vlasov

The Einstein–Vlasov system

The fundamental equations of GR are the Einstein equations

Gµν := Rµν −12

Rgµν = 8πTµν + Λgµν ,

Gµν is the Einstein tensor, Λ the cosmological constant, Rµν theRicci tensor, R the scalar curvature and g the space-time metric.

We want to study the case when Λ = 0 and Tµν is determined byVlasov matter (kinetic description)

pα∂xα f − Γαβγpβpγ∂pα f = 0,

Gµν = 8πTµν = 8π∫

pµpν f |g|1/2 dp1dp2dp3

−p0 ,

with geodesics equations: dXαdτ = Pα, dPα

dτ = −ΓαβγPβPγ .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 34

Steady States in Einstein–Vlasov

Einstein-Vlasov system in spherical symmetry

The spherically symmetric Einstein-Vlasov system in Schwarzschildcoordinates is given by the following set of equations:

∂t f + eµ−λv√

1 + v2· ∇x f −

(λt

x · vr

+ eµ−λµr

√1 + v2

) xr· ∇v f = 0,

e−2λ(2rλr − 1) + 1 = 8πr2h,e−2λ(2rµr + 1)− 1 = 8πr2prad,

where

h(t , r) =

∫ √1 + v2fdv , prad(t , r) =

∫ (x · vr

)2f

dv√1 + v2

.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 35

Steady States in Einstein–Vlasov

Einstein-Vlasov system in spherical symmetry

The spherically symmetric Einstein-Vlasov system in Schwarzschildcoordinates is given by the following set of equations:

∂t f + eµ−λv√

1 + v2· ∇x f −

(λt

x · vr

+ eµ−λµr

√1 + v2

) xr· ∇v f = 0,

e−2λ(2rλr − 1) + 1 = 8πr2h,e−2λ(2rµr + 1)− 1 = 8πr2prad,

where

h(t , r) =

∫ √1 + v2fdv , prad(t , r) =

∫ (x · vr

)2f

dv√1 + v2

.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 35

Steady States in Einstein–Vlasov

Einstein-Vlasov system in spherical symmetry

The functions λ, µ determine the metric of the space-time according to

ds2 = −e2µdt2 + e2λdr2 + r2dω2.

Initial condition:

0 ≤ f (0, x , v) = f0(x , v), f0(Ax ,Av) = f0(x , v), ∀A ∈ SO(3).

Boundary conditions:

limr→∞

λ(t , r) = limr→∞

µ(t , r) = λ(t ,0) = 0.

(CHOQUET–BRUHAT, REIN, RENDALL, SCHAEFFER, ANDREASSON, WOLANSKY,...)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 36

Steady States in Einstein–Vlasov

Some associated quantities

The ADM mass (or energy):

H =

∫R3

∫R3

√1 + |v |2 f dvdx .

The total rest mass:M =

∫R3

∫R3

eλf dvdx .

The central redshift:Zc(t) = e−µ(0,t) − 1.

The tangential pressure:

ptan(t , r) =

∫R3

∣∣∣x ∧ vr

∣∣∣2 fdv√

1 + |v |2.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 37

Steady States in Einstein–Vlasov

Some associated quantities

The ADM mass (or energy):

H =

∫R3

∫R3

√1 + |v |2 f dvdx .

The total rest mass:M =

∫R3

∫R3

eλf dvdx .

The central redshift:Zc(t) = e−µ(0,t) − 1.

The tangential pressure:

ptan(t , r) =

∫R3

∣∣∣x ∧ vr

∣∣∣2 fdv√

1 + |v |2.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 37

Steady States in Einstein–Vlasov

Some associated quantities

The ADM mass (or energy):

H =

∫R3

∫R3

√1 + |v |2 f dvdx .

The total rest mass:M =

∫R3

∫R3

eλf dvdx .

The central redshift:Zc(t) = e−µ(0,t) − 1.

The tangential pressure:

ptan(t , r) =

∫R3

∣∣∣x ∧ vr

∣∣∣2 fdv√

1 + |v |2.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 37

Steady States in Einstein–Vlasov

Some associated quantities

The ADM mass (or energy):

H =

∫R3

∫R3

√1 + |v |2 f dvdx .

The total rest mass:M =

∫R3

∫R3

eλf dvdx .

The central redshift:Zc(t) = e−µ(0,t) − 1.

The tangential pressure:

ptan(t , r) =

∫R3

∣∣∣x ∧ vr

∣∣∣2 fdv√

1 + |v |2.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 37

Steady States in Einstein–Vlasov

About steady statesResult:Let f be a static, compactly supported solution of the sphericallysymmetric Einstein–Vlasov system. Then the following inequality holdstrue

Zc ≥∣∣∣∣MH − 1

∣∣∣∣ .

This result is a consequence of the identity

H =

∫R3

eλ+µ(ptan + prad + h) dx

which can be proved using either:

the Tolman-Oppenheimer-Volkoff (TOV) equation.more general: the vector fields multipliers technique.

S. CALOGERO, J. CALVO, O. SÁNCHEZ, J. SOLER, Virial inequalities for steady states in relativistic galactic dynamics,Nonlinearity 2010.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 38

Steady States in Einstein–Vlasov

About steady statesResult:Let f be a static, compactly supported solution of the sphericallysymmetric Einstein–Vlasov system. Then the following inequality holdstrue

Zc ≥∣∣∣∣MH − 1

∣∣∣∣ .This result is a consequence of the identity

H =

∫R3

eλ+µ(ptan + prad + h) dx

which can be proved using either:

the Tolman-Oppenheimer-Volkoff (TOV) equation.more general: the vector fields multipliers technique.

S. CALOGERO, J. CALVO, O. SÁNCHEZ, J. SOLER, Virial inequalities for steady states in relativistic galactic dynamics,Nonlinearity 2010.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 38

Steady States in Einstein–Vlasov

About steady statesResult:

Any Jeans-type steady state with radius R satisfies

eµ(0) ≤ min

1,MH

√1− 2H

R.

Any static shell with inner radius Rin satisfies the inequality

Rin ≤18H

ln(∣∣M

H − 1∣∣+ 1

) .

These results are complementary to the Buchdahl inequality

supr>0

1− e−2λ(r) 689.

(. . . , ANDREASSON, STALKER, ANDREASSON AND REIN)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 39

Steady States in Einstein–Vlasov

About steady statesResult:

Any Jeans-type steady state with radius R satisfies

eµ(0) ≤ min

1,MH

√1− 2H

R.

Any static shell with inner radius Rin satisfies the inequality

Rin ≤18H

ln(∣∣M

H − 1∣∣+ 1

) .These results are complementary to the Buchdahl inequality

supr>0

1− e−2λ(r) 689.

(. . . , ANDREASSON, STALKER, ANDREASSON AND REIN)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 39

Steady States in Einstein–Vlasov

Properties of steady states

Vlasov–Poisson H < 0

Nordström–Vlasov H 6 M

Einstein–Vlasov∣∣M

H − 1∣∣ ≤ Zc

The first two have proven crucial for the stability theory of steadystates.

Polytropic models: ϕ(E ,F ) = EµF k .

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 40

Steady States in Einstein–Vlasov

Interplay between dynamics and staticity

Vlasov–Poisson H < 0 H > 0

Nordström–Vlasov H 6 M H > M

Einstein–Vlasov∣∣M

H − 1∣∣ ≤ Zc ? ?

Conjecture:

the third inequality could also play an important role in the stabilityanalysis or in the study of dispersive properties.

(ANDREASSON AND REIN, ZELDOVICH, PODURETS AND NOVIKOV)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 41

Steady States in Einstein–Vlasov

Interplay between dynamics and staticity

Vlasov–Poisson H < 0 H > 0

Nordström–Vlasov H 6 M H > M

Einstein–Vlasov∣∣M

H − 1∣∣ ≤ Zc ? ?

Conjecture:

the third inequality could also play an important role in the stabilityanalysis or in the study of dispersive properties.

(ANDREASSON AND REIN, ZELDOVICH, PODURETS AND NOVIKOV)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 41

Flux–Limitation Mechanisms

Dispersion vs diffusion in Gravitation

The Vlasov–Poisson system allows for a very rich dynamics andplenty of steady states.

But its description neglects close encounters.Modeling influences out of control: Fokker–Planck (Brownianmotion / linear diffusion) terms introduce stochasticity.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 42

Flux–Limitation Mechanisms

Dispersion vs diffusion in Gravitation

The Vlasov–Poisson system allows for a very rich dynamics andplenty of steady states.But its description neglects close encounters.

Modeling influences out of control: Fokker–Planck (Brownianmotion / linear diffusion) terms introduce stochasticity.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 42

Flux–Limitation Mechanisms

Dispersion vs diffusion in Gravitation

The Vlasov–Poisson system allows for a very rich dynamics andplenty of steady states.But its description neglects close encounters.Modeling influences out of control: Fokker–Planck (Brownianmotion / linear diffusion) terms introduce stochasticity.

∂t f + v · ∇x f −∇xφ · ∇v f = ν∆v f ,

ρ(t , x) =

∫R3

v

f (t , x , v) dv ,

∆φ = ρ(t , x), lim|x |→∞

φ(t , x) = 0 ∀t ∈ R+.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 42

Flux–Limitation Mechanisms

Dispersion vs diffusion in Gravitation

The Vlasov–Poisson system allows for a very rich dynamics andplenty of steady states.But its description neglects close encounters.Modeling influences out of control: Fokker–Planck (Brownianmotion / linear diffusion) terms introduce stochasticity.

The choice of Fokker–Planck terms to introduce stochasticityleads to very poor dynamics.

(CARRILLO, SOLER & VÁZQUEZ, JFA 1996)

Can we complement the Vlasov–Poisson system with some formof stochasticity that is able to respect some of the already existingstructures/dynamics?

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 42

Flux–Limitation Mechanisms

The problem of infinite speed of propagation

Fourier’s theory on heat conduction, based on the relation

heat flux = −k∇u(t , x)

and the subsequent equation

∂tu = k∆xu

predicts an infinite speed of propagation for the heat flux.

Flux-limitation mechanisms modify the law defining the heat flow tomake it saturate when gradients become unbounded.

Prototypical macroscopic equation: the relativistic heat equation.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 43

Flux–Limitation Mechanisms

The problem of infinite speed of propagation

Fourier’s theory on heat conduction, based on the relation

heat flux = −k∇u(t , x)

and the subsequent equation

∂tu = k∆xu

predicts an infinite speed of propagation for the heat flux.

Flux-limitation mechanisms modify the law defining the heat flow tomake it saturate when gradients become unbounded.

Prototypical macroscopic equation: the relativistic heat equation.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 43

Flux–Limitation Mechanisms

The problem of infinite speed of propagation

Fourier’s theory on heat conduction, based on the relation

heat flux = −k∇u(t , x)

and the subsequent equation

∂tu = k∆xu

predicts an infinite speed of propagation for the heat flux.

Flux-limitation mechanisms modify the law defining the heat flow tomake it saturate when gradients become unbounded.

Prototypical macroscopic equation: the relativistic heat equation.

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 43

Flux–Limitation Mechanisms

Monge-Kantorovich optimal mass transportation

The problem of optimal transport: how to move mass from a giveninitial configuration to a prescribed final configuration with minimal cost.

The relativistic heat equation is derived as a gradient flow of theBoltzmann entropy for the metric corresponding to the following costfunction:

k(z) =

c2(

1−√

1− |z|2

c2

)if |z| 6 c

+∞ if |z| > c

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 44

Flux–Limitation Mechanisms

Monge-Kantorovich optimal mass transportation

The problem of optimal transport: how to move mass from a giveninitial configuration to a prescribed final configuration with minimal cost.

∂u∂t

= ν div

|u|∇xu√u2 + ν2

c2 |∇xu|2

c maximum speed of propagation allowed.ν kinematic viscosity.singular front propagation, BV theory.

(ROSENAU, BRENNIER, ANDREU, CASELLES, MAZÓN, MOLL, CALVO, SOLER... 2004-2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 44

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 0.075

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER, VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 0.15

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER, VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 0.3

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER, VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 0.6

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER, VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 0.9

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER, VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 1.2

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER, VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Numerical comparison

classical heat eq.relativistic heat eq.

t = 1.5

10.80.60.40.20

2

1.5

1

0.5

0

(CALVO, MAZÓN, SOLER & VERBENI, 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 45

Flux–Limitation Mechanisms

Other flux limiters in gravitation

WILSON (MIHALAS & MIHALAS, 1984):

∂tu = ν div(

u∇uu + ν

c |∇u|

)

LEVERMORE & POMRANING, AJ 1981:

∂tu = c div (uλ(R)R) , λ(R) =1R

(1R− coth R

), R = − ∇u

σωu

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 46

Flux–Limitation Mechanisms

Nonlinear Hilbert expansion (F. Golse)

Consider the linear Boltzmann equation with parabolic scaling

ε2∂f ε

∂t+ εv · ∇x f ε + σ (f ε − 〈f ε〉) = 0, f ε0 (x , v) = f0(x , v),

where 〈f ε〉 = 14π

∫S2 f ε(t , x , v)dv , and σ > 0.

The standard diffusion approximation states that

〈f ε〉 −→ ρ in Lp, for each t ≥ 0

being ρ the solution of the linear diffusion equation

∂ρ

∂t=

13σ

∆ρ

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 47

Flux–Limitation Mechanisms

Nonlinear Hilbert expansion (F. Golse)

Consider the linear Boltzmann equation with parabolic scaling

ε2∂f ε

∂t+ εv · ∇x f ε + σ (f ε − 〈f ε〉) = 0, f ε0 (x , v) = f0(x , v),

where 〈f ε〉 = 14π

∫S2 f ε(t , x , v)dv , and σ > 0.

The standard diffusion approximation states that

〈f ε〉 −→ ρ in Lp, for each t ≥ 0

being ρ the solution of the linear diffusion equation

∂ρ

∂t=

13σ

∆ρ

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 47

Flux–Limitation Mechanisms

Nonlinear Hilbert expansion (F. Golse)

Consider the linear Boltzmann equation with parabolic scaling

ε2∂f ε

∂t+ εv · ∇x f ε + σ (f ε − 〈f ε〉) = 0, f ε0 (x , v) = f0(x , v).

Instead of the usual Hilbert expansion, seek f ε as

f ε = exp

∑k≥0

εk Φk

Order ε0: eΦ0 − 〈eΦ0〉 = 0 =⇒ Φ0 = φ0(t , x)

Order ε1: v · ∇xeφ0 + σeφ0 (Φ1 − 〈Φ1〉) = 0

=⇒ Φ1 = φ1(t , x)− 1σv · ∇xφ0

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 48

Flux–Limitation Mechanisms

Nonlinear Hilbert expansion (F. Golse)

Consider the linear Boltzmann equation with parabolic scaling

ε2∂f ε

∂t+ εv · ∇x f ε + σ (f ε − 〈f ε〉) = 0, f ε0 (x , v) = f0(x , v).

Set φ = φ0 + εφ1 and consider the truncate expansion

f ε = exp(φ− ε1

σv · ∇xφ

)Inserting ρε = 〈f ε〉 in the continuity equation for 〈f ε〉

∂〈f ε〉∂t

+1ε

divx〈v f ε〉 = 0

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 49

Flux–Limitation Mechanisms

Nonlinear Hilbert expansion (F. Golse)

Consider the linear Boltzmann equation with parabolic scaling

ε2∂f ε

∂t+ εv · ∇x f ε + σ (f ε − 〈f ε〉) = 0, f ε0 (x , v) = f0(x , v).

one finds the following flux-limited diffusion equation

∂ρε

∂t− 1ε

divx

(A(ε

σ

|∇ρε|ρε

)∇ρε

|∇ρε|ρε)

= 0

whereA(z) = coth(z)− 1

z

(LEVERMORE & POMRANING, 1981)

(Flux-limited and nonlinear Hilbert expansion, J.S et al., work in progress)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 50

Flux–Limitation Mechanisms

Hyperbolic-parabolic limit: Chemotaxis

Consider a system of two populations:(∂t + v · ∇x

)f1 = ν1 L1(f1) + η1 G1[f , f ] + µ1 I1[f , f ], (ε2t , εx)

(∂t + v · ∇x

)f2 = ν2L2(f2) + η2 G2[f , f ] + µ2 I2[f , f ], (εt , εx)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 51

Flux–Limitation Mechanisms

Hyperbolic-parabolic limit: Chemotaxis

Consider a system of two populations:(ε∂t + v · ∇x

)f ε1 =

1εpL1(f ε1 ) + εqG1[f ε, f ε] + εq+r1I1[f ε, f ε],

ε(∂t + v · ∇x

)f ε2 = L2[f ε1 ](f ε2 ) + εqG2[f ε, f ε] + εq+r2I2[f ε, f ε],

=⇒ Optimal transport theory implies that

the population n = 〈f2〉 and the chemical signal S = 〈f1〉 verify

∂tn = divx

Dnn∇xn√

n2 + D2n

c2 |∇xn|2− nχ

∇xS√1 + |∇xS|2

+ H2(n,S),

∂tS = divx(DS · ∇xS) + H1(n,S),

(BELLOMO, BELLOUQUID, NIETO, J.S., M3AS 2010, MAZÓN, CALVO, J.S. 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 52

Flux–Limitation Mechanisms

Hyperbolic-parabolic limit: Chemotaxis

Consider a system of two populations:(ε∂t + v · ∇x

)f ε1 =

1εpL1(f ε1 ) + εqG1[f ε, f ε] + εq+r1I1[f ε, f ε],

ε(∂t + v · ∇x

)f ε2 = L2[f ε1 ](f ε2 ) + εqG2[f ε, f ε] + εq+r2I2[f ε, f ε],

=⇒ Optimal transport theory implies that

the population n = 〈f2〉 and the chemical signal S = 〈f1〉 verify

∂tn = divx

Dnn∇xn√

n2 + D2n

c2 |∇xn|2− nχ

∇xS√1 + |∇xS|2

+ H2(n,S),

∂tS = divx(DS · ∇xS) + H1(n,S),

(BELLOMO, BELLOUQUID, NIETO, J.S., M3AS 2010, MAZÓN, CALVO, J.S. 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 52

Flux–Limitation Mechanisms

Hyperbolic-parabolic limit: Chemotaxis

Consider a system of two populations:(ε∂t + v · ∇x

)f ε1 =

1εpL1(f ε1 ) + εqG1[f ε, f ε] + εq+r1I1[f ε, f ε],

ε(∂t + v · ∇x

)f ε2 = L2[f ε1 ](f ε2 ) + εqG2[f ε, f ε] + εq+r2I2[f ε, f ε],

=⇒ Optimal transport theory implies that

the population n = 〈f2〉 and the chemical signal S = 〈f1〉 verify

∂tn = divx

Dnn∇xn√

n2 + D2n

c2 |∇xn|2− nχ

∇xS√1 + |∇xS|2

+ H2(n,S),

∂tS = divx(DS · ∇xS) + H1(n,S),

(BELLOMO, BELLOUQUID, NIETO, J.S., M3AS 2010, MAZÓN, CALVO, J.S. 2010)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 52

Flux–Limitation Mechanisms

Flux-limited models in gravitation

Our proposal is then:

∂t f + v · ∇x f −∇xφ · ∇v f = ν FL(f ,∇f ),

where FL(f ,∇f ) is a flux-limited term.

We need to combine and extend the applicability of differenttechniques coming from parabolic and kinetic contexts:

I Crandall-Liggett’s theorem, Stampacchia’s methodI Minty-Browder’s techniqueI BV-theory vs moments estimates. Front propagationI the concept of entropy solution, and the method of doubling

variables due to S. Kruzhkov

To summarize, the system behaves more in an hyperbolic-kineticthan in a parabolic way

(J. CALVO, V. CASELLES, J. SOLER, work in progress)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 53

Flux–Limitation Mechanisms

Flux-limited models in gravitation

Our proposal is then:

∂t f + v · ∇x f −∇xφ · ∇v f = ν FL(f ,∇f ),

where FL(f ,∇f ) is a flux-limited term.

We need to combine and extend the applicability of differenttechniques coming from parabolic and kinetic contexts:

I Crandall-Liggett’s theorem, Stampacchia’s methodI Minty-Browder’s techniqueI BV-theory vs moments estimates. Front propagationI the concept of entropy solution, and the method of doubling

variables due to S. Kruzhkov

To summarize, the system behaves more in an hyperbolic-kineticthan in a parabolic way

(J. CALVO, V. CASELLES, J. SOLER, work in progress)

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 53

Flux–Limitation Mechanisms

Future?

Extension of flux limitation to relativistic settings

Weaker concepts of dispersion in the classical setting

Suitable concepts of dispersion in the relativistic setting

These works (present and future) are possible thanks to thecollaboration of:

J. Calvo, S. Calogero, J. Dolbeault, P.E. Jabin, J. Mazón,V. Caselles, O. Sánchez, M. Verbeni, ...

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 54

Flux–Limitation Mechanisms

Future?

Extension of flux limitation to relativistic settings

Weaker concepts of dispersion in the classical setting

Suitable concepts of dispersion in the relativistic setting

These works (present and future) are possible thanks to thecollaboration of:

J. Calvo, S. Calogero, J. Dolbeault, P.E. Jabin, J. Mazón,V. Caselles, O. Sánchez, M. Verbeni, ...

J. Soler (U. Granada) Dispersion vs flux-limited diffusion FRG Meeting 54