Lectures on Geometric Kinetic Equations

Jacques Smulevici*

October 5, 2016

Contents

1 Preliminary remarks 2

2 Introduction: the basic equations 22.1 The relativistic setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Remarks on the regularity of the distribution functions . . . . . . . . . . 42.3 The non-relatistivic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 The free transport equations 53.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Decay estimates for transport equations: heuristics . . . . . . . . . . . . 63.4 Decay estimates from the method of characteristics . . . . . . . . . . . 7

4 The vector field approach to decay of velocity averages 84.1 Complete lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 The complete lifts of the Killing vector fields of the Minkowski space . 114.3 Restrictions of the complete lifts . . . . . . . . . . . . . . . . . . . . . . . 114.4 Velocity averages and the complete lift of the Killing fields . . . . . . . . 124.5 Vector field identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.6 Decay estimates for velocity averages of massless distribution via the

vector field method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 Decay estimates for velocity averages of massive distribution via the

vector field method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 Klainerman-Sobolev inequalities and decay estimates: massive case . 224.9 Distribution functions for massive particles with compact support in x 244.10 The classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Yet another proof of decay for velocity averages 265.1 Weights preserved by the flow . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Lp decay of velocity averages . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Interlude: a vector field approach to decay of solutions to Schrödiner

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Null condition and null decomposition of tensors for Vlasov fields 286.1 Null decomposition of the energy-momentum tensor and decay . . . . 316.2 A null form for a wave/particle interaction . . . . . . . . . . . . . . . . . 32

*Laboratoire de Mathématiques, Université Paris-Sud 11, bât. 425, 91405 Orsay, France.

1

7 Some standard systems of kinetic theory 407.1 The Vlasov-Poisson system . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2 The Vlasov Norström system. . . . . . . . . . . . . . . . . . . . . . . . . . 447.3 The Vlasov-Maxwell system. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1 Preliminary remarks

These lectures have been designed for the June 2016 summer school on "Geometricanalysis of wave and fluids", organized at MIT, Boston. In particular, I expect thestudents to have follow the lectures of the first week and thus, to be familiar withthe vector field method for the standard wave equation on Minkowski space and thebasics of Lorentzian geometry.

It is clear by now that the vector field method provides a robust framework forthe study of quasi-linear wave equations. One of the aims of these lectures is to showthat it has in fact a much wider range of applications, as we will develop a vector fieldtype method for kinetic transport equations. We will however pursue the analogiesbetween wave and kinetic equations a bit further. For instance, we know that fornon-linear wave equations in low dimensions, structure (think null-condition) playsa very important role and we can naturally wonder if there are analogues of thesestructural properties for non-linear systems of kinetic equations.

Apart from trying to understand kinetic equations using approaches initially de-veloped for wave equations, I will also present some standard techniques from ki-netic theory: we all need to learn the basics.

For simplicity, I will often fix the number of spatial dimensions below to 3, butthe interested reader can easily derive generalizations to arbitrary numbers of di-mensions.

These lectures contain many exercices as well as some open problems, the maindifference being that the author of these notes is only certain to know how to solvethe exercices. I strongly encourage the students to do at least part of the exercices.

Most of the results and new techniques (in particular the vector field methodfor Vlasov fields) presented below have been obtained through my joint work withJérémie Joudioux and David Fajman. We refer thus to our work [FJS15] for details.

These lectures are still in preparation. In particular, with probability one, theycontain typos/mistakes or inaccurate references. Feel free to email me for com-ments.

Signature conventions: −+++ for a Lorentzian metric.

2 Introduction: the basic equations

We are interested in the description of a large ensemble of particles, which for sim-plicity, we assume all have same rest mass m. The equations and the definitionsof the objects of study depend on the physical situation we are considering. In thenon-relativistic case, the equations are derived from Newtonian mechanics whilein the general relativistic case, the equations come from General Relativity. In allcases, the main unknown will be a counting or distribution function f , dependingon space-time location as well as velocities. Given a point x in space-time and a

2

velocity v , f (x, v) represents the density of particles at position x and velocity v , sothat if dµx,v is a volume form in phase space, then f (x, v)dµx,v should represent thenumber of particles in that infinitesimal volume.

2.1 The relativistic setting

An easy example is the flat Minkowski space. If (t , x) ∈R4, v ∈R3, then f (t , x, v)d x4d v3gives the number of particles at space-time location (t , x) with 4-velocity1 (vα) :=(v0 :=

√m2 +|v |2, v). In a general, time-oriented, Lorentzian manifold (M , g ), f

will be defined on a submanifold of the tangent-bundle, namely, the set of points(x, v) ∈ T M such that gx (v, v) = −m2, with v future directed2. This submanifold iscalled the mass shell and will be denoted P . We denote by π : T M → M the canon-ical projection. By a small abuse of notation, we will denote by the same letter theinduced projection on P .

In the absence of an external force field, the particles are free-falling and theirtrajectories are given by the geodesics of the Lorentzian manifold. In these lectures,we shall neglect all possible collisions between particles, which implies that the dis-tribution function f should be constant along geodesics.

To write down the equation satisfied by f , let (xα) be a coordinate system. Givenany vector V in Tx M , we know that can decompose V as V = V α∂xα . This impliesthat given a coordinate system (xα), there is a natural3 coordinate system on T M ,given by (xα, vα). The (vα) are called conjugate coordinates to the (xα). Let thus nowconsider a geodesic γ : s → γ(s). Recall that on the mass shell, we have at each point(x, v) ∈P , gx (v, v) =−m2, i.e.

g00v0v0 +2g0i v i v0 + gi j v i v j +m2 = 0,

Assume that ∂x0 is timelike (so that g00 < 0) and that the level set of x0 are space-like hypersurfaces (so that gi j is positive definite), then

v0 =g0i v i ±

√(g0i v i

)2 − g00 (gi j v i v j +m2)−g00

,

with only the + sign kept above if we want only future directed vectors and our man-ifold is time oriented by the vector field ∂x0 .

Thus, in this setting, we can used (xα, v i ) where 1 ≤ i ≤ 3 as a coordinate systemon the mass shell, the remaining v0 being obtained by the above formula.

Assume now that γ(s) = (γα(s)) in some coordinate system. Then, the geodesicequations reads

d 2γα

d s2+Γαβσ

dγβ

d s

dγσ

d s= 0, (1)

where the Γ := Γ(g ) are the Christoffel symbols of the Lorentzian metric. (Recall that

Γαβσ =1

2gαγ

(gγβ,σ+ gγσ,β− gσβ,γ

).

1Actually, this is only the 4 velocity when m = 1, otherwise, it corresponds to the 4-momentum. Inthese lectures, I will, by a small abuse of language use the word velocity often instead of momentum.

2Note that a future directed timelike vector is never zero. On the other hand, for null vectors, we adopthere the convention that a future directed null vector is never 0.

3Here, by a small abuse of notation, we denote by the same symbols the xα and their lifts to T M . Inthe remainder of this text, we will use freely these types of abuse of notation for simplicity.

3

Let f now be a distribution function which is constant along geodesics. Since fis defined on the mass shell, we can identify locally f with a function of (xα, v i ).

Now, f being constant along geodesics imply that f(γα(s), dγ

i

d s (s))

should be

constant in s. Differentiating, we obtain that f should solve

0 = dγα

d s

∂ f

∂xα+ d

2γi

d s2∂ f

∂v i,

which using the geodesic equation (1) can be rewritten as

0 = dγα

d s

∂ f

∂xα−Γiβσ

dγβ

d s

dγσ

d s

∂ f

∂p i.

Since this relation should be true for any future timelike (or null if m = 0) geodesic,we have obtained that f must solve

0 = vα ∂ f∂xα

−Γiβσvβvσ∂ f

∂v i, (2)

at each point (x, v) ∈ P . This is a linear transport equation, called the Vlasov equa-tion, whose characteristics are, by construction, the future timelike or null geodesicsof the space-time.

In the particular case of Minkowski space in Cartesian coordinates (so that theChristoffel symbols vanishes and t = x0), the Vlasov equation becomes

vα∂xα f = 0,where v0 =

√m2 +|v |2.

We will refers to this as the "free transport" (even though this might be physi-cally incorrect since (2) is also describing free falling particles). Let us define the freetransport operator as T = vα∂xα . Note that T actually depends on the value of m.In particular, there will be an important difference between the massless particlesm = 0 and the massive ones m > 0.

Exercices:

1. Write the Vlasov equation in a general Lorentzian manifold for charged parti-cles of charge q in presence of an external electromagnetic field described bya 2-form (Faraday tensor) F . (recall that the trajectory γ of a charged particleis given by

∇γ̇γ̇α = qFαβ γ̇β.)

2. Assume that f is a (sufficiently regular) distribution function defined on all ofT M . By restriction, f defines a regular distribution function on P , denotedhere g . Compute ∂xαg and ∂v i g in terms of ∂xα f and ∂vα f , where (x

α, vα)denotes a standard coordinate system on T M as introduced above.

2.2 Remarks on the regularity of the distribution functions

We have seen that f can be expressed in coordinates (at least locally) as a functionof (xα, v i ), with x0 to be thought of as a time function.

Recall also the definition of the mass-shell: P is the set of points (x, v) ∈ T Msuch that gx (v, v) =−m2, where v is future-directed (and non-zero if m = 0).

4

The distribution functions that we consider in this lecture notes can all be con-structed by solving an initial value problem. Assuming that the initial data is, say,smooth and compactly supported, then the resulting distribution function will besmooth and its restriction to a spatial section will have compact support.

As is classical, all that is actually needed below is a finite number of derivativestogether with polynomial type decay for the initial data depending only on the di-mension.

Note moreover that in physics, f represents a particle density, and is thus re-quired to be positive. This will not play a role in what we do (and in fact is not pre-served by differentiation) so I will ignore this from now on.

2.3 The non-relatistivic case

The classical equivalent of the above free transport operator is the operator ∂t +v i∂xi . If the particles are moving in a force field F so that their trajectories is givenby Newton’s equation

mẍi = F i

then we obtain the equation

vα∂xα f −F i∂v i f = 0,with v0 = 1.

3 The free transport equations

The equations that we want to consider first are

1. The classical (i.e. non-relativistic) free transport equation

∂t f + v i∂xi f = 0,

where f = f (t , x, v), (t , x, v) ∈Rt ×Rnx ×Rnv .2. The relativistic transport equations

vα∂xα f = 0,

where v0 =√

m2 +|v |2, m ≥ 0 is a constant, f = f (t , x, v), (t , x, v) ∈Rt ×Rnx ×Rnvfor m > 0 while (t , x, v) ∈Rt ×Rnx ×Rnv \ {0} if m = 0.

We will write any of the above equation of T ( f ) = 0 where T is one of the free trans-port operators.

Let us note the following lemma.

Lemma 3.1. Let f solve T ( f ) = 0. Let χ be a smooth function χ :R→R, thenT (χ( f )) = 0.Moreover, we have that T (| f |) = 0 in the sense of distribution.For the proof in the case of the absolute value, recall that if f is W 1,1 then | f | is

in W 1,1 and that ∂| f | = f| f |∂ f in the sense of distribution.

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3.1 Conservation laws

Lemma 3.2. Let f solves T ( f ) = 0. Then, for all t , and any 1 ≤ p 0, is computed on a ball centered at xt and of radius Rt .Thus, we have a bound

||ρ(t , x)||. || f (0, ., .)||L∞x,vR3

t 3

and we see that ρ(t , x) decays, the decay being due to the v support of f (t , x, .)shrinking.

Remark 3.1. • The above decay estimates is a bit crude, however, it has been usedin many situations.

• The v integrals such as ρ(t , x) are often called velocity averages in the littera-ture.

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3.4 Decay estimates from the method of characteristics

A variation of the above argument will first help us to remove the compact supportassumptions and then apply it to the relativistic case.

We start again with the non-relativistic case.Let f thus solves

T ( f ) = 0, T = ∂t + v i∂xi .Now, we have

ρ(t , x) =∫

v∈R3f (t , x, v)d v

=∫

v∈R3f (0, x − v t , v)d v

≤∫

v∈R3supw∈R3

f (0, x − v t , w)d v

For t > 0, we can apply the change of variable y = x − v t to get

ρ(t , x) ≤ 1t 3

∫y∈R3

supw∈R3

f (0, y, w)d v.

We have thus obtained the decay estimates

||ρ(t , x)||L∞ ≤ 1t 3

|| f (0, ., .)||L1x L∞v .

Consider now the relativistic case with m > 0. We have again

ρ(t , x) =∫

v∈R3f (t , x, v)d v (4)

=∫

v∈R3f (0, x − v

v0t , v)d v (5)

≤∫

v∈R3supw∈R3

f (0, x − vv0

t , w)d v (6)

We now want to apply the change of variable y = x − vv0

t . Note however that as

v →+∞,∣∣∣ vv0 ∣∣∣ → 1. This implies that for large v , we are trying to map a 3d space, to

a sphere, i.e. the change of variable degenerates for large v . The way this would bereflected in our estimates is that if you compute the Jacobian of the change of vari-able, then this Jacobian is not uniformly bounded in v . For simplicity, we thereforeassume that f has compact v support. Let us define

V = sup{|v |, v ∈R3 : ∃x ∈R3, f (0, x, v) 6= 0}(where the sup should be replaced by an essential supremum if f is, say, not contin-uous). Then, we can bound the Jacobian by a constant depending only on V .

Thus, we obtain a decay estimate of the form

ρ(t , x) ≤C (V ) 1t 3

∫y∈R3

supw∈R3

f (0, y, w)d v,

where C (V ) > 0 depends only V and C (V ) →+∞ as V →+∞.

Exercice:

7

1. Compute explicity the Jacobian in the above change of variable and give anestimate for C (V ).

2. in the massless case, show, using the method of characteristics as above, thatthe following decay estimate holds∫

v∈R3\{0}f (t , x, v)d v3 .

1

t 2V 3

∫ν∈S2t

supw∈R3

f (0, x −ν, w)dν,

.1

t 2V 3 sup

xsupr≥0

∫ν∈S2r

supw∈R3

f (0, x −ν, w)dν,

where Sr denotes the round sphere of radius r centered at 0, and V is againan upper bound on the size of the v support of the initial distribution.

The above decay estimates are very easy to prove. However,

1. one needs a good control of the characteristics for this method to work. Forinstance, in the Lorentzian setting, one would need to control the exponentialmap, as well as derivative of the exponential map for this to work.

2. without extra work, we need a compact support assumption in v for them towork and we do not obtain directly spatial decay, or improved decay outsideor near the light-cones in the massless or massive cases.

3. These decay estimates are not functional inequalities (that is to say they onlyapply to solutions of the equations, compare with the Klainerman-Sobolevinequality for instance).

The vector field approach that we shall now present will remediate to all of this.The estimates that we have obtained all required some L∞ bounds in v of the data.We will replace these by requiring that some (weighted) v derivatives are integrable.By Sobolev, the latter is of course a stronger requirement to the former. In otherwords, the aim of the vector field approach is not to obtain sharper estimates as faras regularity is concerned. The main advantage will be that in non-linear applica-tions, we can mostly forget the characteristics of the system and treat it in a way verysimilar to say, a system of non-linear waves. In particular, there will be no need tocontrol the v support of the solution and in fact, no need for any boundedness in vof the initial support.

4 The vector field approach to decay of velocity aver-ages

First, let us recall that there are two approaches to the vector field method for thewave equation: the original approach of Klainerman, which is based mostly on com-mutators, and the approach following Morawetz, where the key piece is to prove anintegrated decay estimate for a local or weighted energy.

We shall consider mostly the vector field method following the approach of Klain-erman, but let us mention immediately that there also exists integrated decay typeestimates for velocity averages.

Let us now recall what are the key ingredients to Klainerman’s vector field methodfor the wave equation

8

1. A coercive conservation law: in the case of the wave equation this is simplythe energy estimate.

2. A set of commuting vector fields: for the wave equation, they are given by theKilling vector fields, and the scaling vector fields (thus parts of the conformalKilling fields). The set of commuting vector fields is therefore tied to the sym-metries of the equation.

3. Weighted identities between the algebra of the commuting vector fields: forinstance the usual derivative ∂t can be rewritten as

∂t = tS −xiΩ0i

t 2 −|x|2 ,

where S denotes the scaling vector field and theΩ0i are the Lorentz boosts.

4. The Klainerman-Sobolev inequality, which is derived from the usual Sobolevinequalities and a careful analysis of the weights, some of them coming fromthe vector fields idendities, some of them from the volume forms.

5. Finally, using the conservation laws, one can relate the norms appearing onthe right-hand side of the Klainerman-Sobolev inequality to the initial normof the data.

To obtain a vector field method for kinetic equations, we shall therefore obtainan analogue for each of the above ingredients.

The conservation laws that we will use is simply the conservation of the L1x,vnorms4 of f .

The next ingredient is the construction of the commuting vector fields. We willdo it geometrically for the relativistic transport equations.

4.1 Complete lift

The commuting vector fields will be given by the complete lifts of the Killing vectorfields.

Let M be a smooth manifold and X be a vector field on M . Let φs denote theflow of X . For each s (for simplicity, we assume here that X is complete, but theconstructions below are local), φs is a map

φs : M → M

and by definition, since φs is the flow of X , we have φs (x) = γ(s) with γ the uniquecurve starting at x and which solves

γ̇= X ◦γ.

Given x ∈ M , recall that the differential of φs is a linear map:

dφs (x) : Tx M → Tφs (x)M .4Other Lp norms of f could be used, and would lead to sharp decay estimates of velocity averages of

f p . The L1 norm is needed if we want to obtain sharp estimates for velocity averages of f only.

9

Recall that by definition, given a vector in v ∈ Tx M , if we take a curve in M αwith α(0) = x, α̇(0) = v , then t → φs ◦α(t ) is a curve and dφs (x)(v) = d(φs◦α)d t (t = 0).Moreover, in coordinates

[dφs (x)(v)]α = ∂x

α ◦φs∂xβ

vβ.

If now v ∈ Tx M , we can consider the point (φs (x),dφs (x)(v)) ∈ T M . When smoves, this defines a curve in T M

s → (φs (x),dφs (x)(v)) ∈ T M .

Its tangent vector at s = 0 is a vector in the tangent space of T M at the point (x, v)i.e. a vector in T(x,v)T M , denoted here X̃(x,v).

Lemma 4.1. The map: (x, v) → X̃(x,v) is smooth and thus defines a vector field onT T M called the complete lift of the vector field X . In local canonical coordinates,

X̃ = Xα∂xα + vγ ∂Xα

∂xγ∂vα .

Proof. Exercice.

Remark 4.1. The map X → X̃ depends only the differentiable structure of M, and isthus independent of any metric structure on M.

Why do we care about complete lifts? Let X be a rotation in Minkowski space, forinstance X = y∂x −x∂y . Then, a quick computation shows that [T, X ] = v y∂x −v x∂y ,for T the free transport operator. Thus, the rotations do not commute in generalwith our equation. On the other hand, the complete lifts of the rotations will com-mute, as explained in the following lemma.

Lemma 4.2 (Commutation of the transport operator with the complete lifts). Let Xbe a vector field on Rt ×Rnx and X̃ its complete lift. Let T = vα∂xα be the relativistictransport operator (massless or massive), then

[T, X̃ ] = vβvσ ∂Xα

∂xβxσ∂vα .

In particular, if X is any of the Killing fields of the Minkowski space then [T, X̃ ] = 0.Things are more complicated for the conformal Killing fields because it will de-

pends on the value of the mass. However, we have in all cases

Lemma 4.3. Let S = xα∂xα be the scaling vector field. Then

[T,S] = T,

for T the relativistic massless or massive transport operator.

Note that here, we are not using its complete lift (but its vertical lift cf later in thelectures).

Exercice:

10

1. Let (M , g ) be a Lorenztian manifold and T = vα∂xα−vαvβΓσαβ∂vσ its geodesicsvector field. Prove that for any vector field X on M , we have

[T, X̃ ] = vαvβ[∇α∇βXσ−RσβαγX γ

]∂vσ f .

2. The equation ∇α∇βXσ−RσβαγX γ = 0 is called the Jacobi equation. Prove thatany Killing fields of g solve the Jacobi equation.

4.2 The complete lifts of the Killing vector fields of the Minkowskispace

Now we have a good receipe to construct commuting vector fields, let us write themdown.

The complete lift of the Killing fields of the Minkowski space are given by

• translations: ∂̂xα = ∂xα .• Rotations (including Lorentz boost): with Ωαβ = xα∂xβ − xβ∂xα , we get Ω̂αβ =

xα∂xβ −xβ∂xα + vα∂vβ − vβ∂vα .As for the usual Killing fields of Minkowski space, the commutator of two com-

plete lifts is linear combination of complete lifts with constant coefficient coeffi-cients.

4.3 Restrictions of the complete lifts

Now, recall that the distribution functions we want to consider are actually definedonly on the mass shell, not on the whole tangent bundle. Fortunately, complete liftsof Killing fields are always tangent to the mass shell, and hence their restrictions tothe mass shell are honest differential operators.

Lemma 4.4. Let (M , g ) be a Lorentzian manifold, and P its mass shell. Let X bea Killing field and X̂ its complete lift. Then, X̂ is tangent to P and thus, for any(sufficiently regular) distribution function f defined on P , X̂ ( f ) is well defined.

Proof. Let X be Killing. Without loss of generality, choose a coordinate system onM such that X coincide with one of the partial derivative: X = ∂xi . Then, X̂ = ∂xiwhile a 1-form normal to the mass shell is given by ν= gαβ,γvαvβd xγ+2gαβvβd vα.Since gαβ must be independent of x

i in the (xα) coordinate system, it follows that< ν, X̂ >= 0, i.e. X̂ is tangent to the mass shell.

Note that the restrictions of the xα translations and the usual (not hyperbolic)rotations are given by the same expressions in the (xα, v i ) coordinate system, whilefor the hyperbolic rotations, one obtain

Ω̂0i = t∂xi +xi∂t + v0∂v i ,

where v0 =√

m2 +|v |.Let P be the set of all Killing fields (the Poincaré algebra).We will denote by

P̂≡ {Ẑ |Z ∈P},

11

the set of all complete lifts and by

K= P̂∪{S},where S = xα∂xα is the scaling vector field (not its complete lift).

4.4 Velocity averages and the complete lift of the Killing fields

We have found our commuting vector fields, but we still need to understand how toexploit them. Recall that what decays are not the distribution functions themselves,but their velocity averages. Thus, we would like to estimate quantities of the form

Z

(∫v

f d v

)for Z a vector field. However, if we integrate in v first, then

∫v f d v are quantities

defined on M not T M (or rather P ), so where how are we going to make use or ourcomplete lift ? The answer is contained is the following lemma:

Vector fields and the operator of averaging in v essentially commute in the fol-lowing sense.

Lemma 4.5. Let f be a regular distribution function for the massless case. Then,

• for any translation ∂xα , we have

∂xα[ρ( f )

]= ρ (∂xα ( f ))= ρ (∂̂xα ( f )) .• for any rotationΩi j , 1 ≤ i , j ,≤ n, we have

Ωi j[ρ( f )

]= ρ (Ω̂i j ( f )) ,where Ω̂i j is the complete lift of the vector fieldΩi j .

• for any Lorentz boostΩ0i , 1 ≤ i ≤ n, we have

Ω0i[ρ( f )

]= ρ (Ω̂0i ( f ))+2ρ ( v i|v | f)

.

• for the scaling vector field S, we have

S[ρ( f )

]= ρ (Ŝ( f ))+ (n +1)ρ( f ).• finally, all the above equalities hold (almost everywhere) with f replaced by | f |,

for instanceS

[ρ(| f |)]= ρ (Ŝ(| f |))+ (n +1)ρ(| f |).

Proof. Let us consider, for instance, a Lorentz boostΩ0i = t∂xi +xi∂t , then

Ω0i[ρ( f )

] = ∫v

(t∂xi +xi∂t

)( f )|v |d v. (7)

12

On the other hand, note that∫v

(t∂xi +xi∂t

)( f )|v |d v =

∫v

(t∂xi +xi∂t +|v |∂v i

)( f )|v |d v −

∫v|v |2∂v i ( f )d v

=∫

vΩ̂0i ( f )|v |d v +2

∫v

v i

|v | ( f )|v |d v

= ρ (Ω̂0i ( f ))+2ρ ( v i|v | f)

,

using an integration by parts in v i . The other cases can all be treated similarly, thetranslations being trivial since ∂̂xα = ∂xα . That f can be replaced by | f | follows fromthe standard property of differentiation of the absolute value5.

In the massive case, we have the following lemma, whose proof is left to thereader since it is very similar to the above.

Lemma 4.6. Let f be a regular distribution function for the massive case. Then,

• for any translation ∂xα , we have

∂xα[ρ( f )

]= ρ (∂xα ( f ))= ρ (∂̂xα ( f )) .• for any rotationΩi j , 1 ≤ i , j ,≤ n, we have

Ωi j[ρ( f )

]= ρ (Ω̂i j ( f )) ,where Ω̂i j is the complete lift of the vector fieldΩi j .

• for any Lorentz boostΩ0i , 1 ≤ i ≤ n, we have

Ω0i[ρ( f )

]= ρ (Ω̂0i ( f ))+2ρm ( v iv0

f

).

• finally, all the above equalities holds with f replaced by | f |.Remark 4.2. Although we do not have for all commutation vector fields Zρ = ρẐ , wedo have that |Zρ(| f |)|. ρ (|Ẑ ( f )|)+ρ(| f |) and this is all we shall need from the above.Note also that if we were looking at other moments, then similar formulae wouldhold with different coefficients. For instance, we haveΩ0i

∫v f dµm =

∫v Ω̂0i f dµm for

sufficiently regular f .

Exercice:Let (M , g ) be Lorentzian manifold, X a Killing field of the metric g and Tµν[ f ] an

energy momentum tensor of some regular Vlasov field f :

Tµν[ f ] =−∫

vf vµvν

pg

d v3

v0.

Prove that LX Tµν[ f ] = Tµν[X̂ ( f )]. (note also that above, we did not always usethe Lie derivative...)

5Recall that f ∈ W 1,1 implies that | f | ∈ W 1,1 with ∂| f | = f| f |∂ f almost everywhere. See for instance[LL97], Chap 6.17.

13

4.5 Vector field identities

The following classical vector field identities will be used later.

Lemma 4.7. The following identities hold:

(t 2 − r 2)∂t = tS −xiΩ0i ,(t 2 − r 2)∂i = −x jΩi j + tΩ0i −xi S,

(t 2 − r 2)∂r = t xi

rΩ0i − r S.

Furthermore,

∂s ≡ 12

(∂t +∂r ) = S +ωiΩ0i

2(t + r ) , ∂i ≡ ∂i −ωi∂r =ω jΩi j

r= −ωiω

iΩ0 j +Ω0it

. (8)

4.6 Decay estimates for velocity averages of massless distributionvia the vector field method

We are now ready to prove decay estimates for velocity averages using the vectorfield method, at least for massless fields. For massive fields, we will need a little bitof extra knowledge, as we will use the so-called hyperboloidal foliation.

Theorem 4.1 (Klainerman-Sobolev inequalities for velocity averages of massless dis-tribution functions). Let f be a regular distribution function for the massless casedefined on [0,T ]×Rnx ×

(Rnv \ {0}

)for some T > 0. Then, for all (t , x) ∈ [0,T ]×Rnx ,

ρ(| f |)(t , x). 1(1+|t −|x| |)(1+|t +|x| |)n−1 ‖ f ‖K,n(t ), (9)

where

‖ f ‖K,k (t ) ≡∑

|α|≤k

∑Ẑα∈K̂|α|

∫Σt

ρ(∣∣Ẑα f ∣∣) (t , x)d x.

Proof. The proof is very similar to the proof of the usual Klainerman-Sobolev in-equality for the wave equation, apart from a technical problem due to the pres-ence of the absolute value. In the usual approach, one typically starts by applyingSobolev’s Lemma. Here, the absolute value present us from doing, so we will reproveSobolev’s Lemma as we prove our decay estimate.

Let (t , x) ∈ [0,T ]×Rnx and assume first that |x| ∉ [t/2,3/2t ] and t +|x| ≥ 1. Let ψbe defined as

ψ : y → ρ (| f |(t , x + (t +|x|)y)) ,where y = (y1, y2, .., yn). Note that

∂yiψ(y) = ∂y i[ρ

(| f |(t , x + (t +|x|)y)]= (t +|x|)∂xi (ρ [| f |]) (t , x + (t +|x|)y).Assume now that |y | ≤ 1/4. Using the fact that we are away from the light-cone andthe condition on |y |, it follows that

1/C ≤ |t +|x|||t −|x + (t +|x|)y | ≤C ,

for some C > 0. It then follows from the vector field identities of Lemma 4.7 that|∂y i ρ(| f |(t , x + (t +|x|)y)|.

∑Z∈K

∣∣Z (ρ [| f |])∣∣ (t , x + (t +|x|)y).14

From Lemma 4.5, we then obtain that∣∣∣∂y i ρ [| f |(t , x + (t +|x|)y)]∣∣∣ . ∑|α|≤1,Ẑα∈K̂|α|

∣∣ρ [Ẑ (| f |)]∣∣ (t , x + (t +|x|)y)+ρ(| f |)(t , x + (t +|x|)y),.

∑|α|≤1,Ẑα∈K̂|α|

∣∣ρ [Ẑα(| f |)]∣∣ (t , x + (t +|x|)y),.

∑|α|≤1,Ẑα∈K̂|α|

ρ[∣∣Ẑα(| f |)∣∣] (t , x + (t +|x|)y),

.∑

|α|≤1,Ẑα∈K̂|α|ρ

[∣∣Ẑα( f )∣∣] (t , x + (t +|x|)y),where we have used in the last line that for any vector field Ẑ ,

∣∣Ẑ (| f |)∣∣ = ∣∣Ẑ ( f )∣∣ (al-most everywhere and provided f is sufficiently regular), which essentially follows

from the fact that ∂| f | = f| f |∂ f almost everywhere if f ∈ W 1,1. Let now δ = 116n , sothat if |yi | ≤ δ1/2 for all 1 ≤ i ≤ n, we then have |y | ≤ 1/4. Applying now a 1 dimen-sional Sobolev inequality in the variable y1, we have

|ψ(0)| = ρ [| f |] (t , x) . ∫|y1|≤δ1/2

(∣∣∂y1ψ(y1,0, ..,0)∣∣+ ∣∣ψ(y1,0, ..,0)∣∣)d y1,.

∫|y1|≤δ1/2

( ∑|α|≤1,Ẑα∈K̂|α|

ρ[∣∣Ẑα( f )∣∣](t , x + (t +|x|)(y1,0, ..,0))

)d y1.

We can now apply a 1 dimensional Sobolev inequality in the variable y2 and repeatthe previous argument, with |Zα( f )| replacing | f |, to obtain

|ψ(0)|.∫|y1|≤δ1/2

∫|y2|≤δ1/2

( ∑|α|≤2,Ẑα∈K̂|α|

ρ0[∣∣Ẑα( f )∣∣](t , x + (t +|x|)(y1, y2, ..,0))

)d y1d y2.

Repeating the argument up to exhaustion of all variables, we obtain that

ρ0[| f |] (t , x).

∫|y1|≤δ1/2

∫|y2|≤δ1/2

..∫|yn |≤δ1/2

( ∑|α|≤n,Ẑα∈K̂|α|

ρ[∣∣Ẑα( f )∣∣](t , x + (t +|x|)(y1, y2, .., yn))

)d y1d y2..d yn .

Applying the change of variable z = (t +|x|)y gives us a (t +|x|)n factor which com-pletes the proof of the inequality in this particular case. The case where (t +|x|) ≤ 1follows from simpler considerations and is therefore left to the reader.

Let us thus turn to the case where x ∈ [t/2,3/2t ] and (t + |x|) ≥ 1 . Note thatit then follows that t > 2/5 and |x| > 1/3. Let us introduce spherical coordinates(r,ω) ∈ [0,+∞)×Sn−1, such that x = rω and denote by q the optical function q ≡ r−t .Let v(t , q,ω) ≡ ρ0( f )(t , (t +q)ω).

Note that ∂q v = ∂rρ, q∂q v = (r −t )∂r and that there exist constants Ci j such that

∂ωv = ∂ω(ρ0( f )(t , (q + t )ω)

)= ∑i< j

Ci jΩi jρ0( f ),

where theΩi j are the rotation vector fields.Let q0 = |x|− t . We need to prove that

t n−1(1+|q0|)|v(t , q0,ω)|. ‖ f ‖K,n(t ).

15

Using a one dimensional Sobolev inequality, we have for any ω ∈Sn−1.

|v(t , q0,ω)|.∫|q | 1. Let χ ∈ C∞0 (−1/2,1/2) be a smooth cut-off function

such that χ(0) = 1 and define Vq0 (t , q,ω) ≡ χ((q − q0)/q0)v(t , q,ω). To get the extrafactor of |q0|, we apply the method used above replacing the function v by the func-tion (s,η) → Vq0 (t , q0 + q0s,η) and applying first a 1−d Sobolev inequality in s on|s| < 1/2. The extra power of q0 appearing are then absorbed since |q0 + q0s| ∼ |q0|in the region of integration and since (r − t )∂r can be expressed as a linear combina-tion of commutation vector fields from Lemma 4.7 (with coefficients homogeneousof degree 0). The rest of the proof is similar to the one just given when |q0| ≤ 1 andtherefore omitted.

16

Since the norm on the right-hand side is conserved for solutions of the homoge-neous massless transport equations, we obtain in particular,

Theorem 4.2 (Decay estimates for velocity averages of massless distribution func-tions [FJS15]). Let f be a regular distribution function for the massless case, a solu-tion to T ( f ) = 0 on Rt ×Rnx ×

(Rnv \ {0}

). Then, for all (t , x) ∈Rt ×Rnx ,

ρ(| f |)(t , x). 1(1+|t −|x| |)(1+|t +|x| |)n−1 ‖ f ‖K,n(0). (10)

4.7 Decay estimates for velocity averages of massive distributionvia the vector field method

The analogy between the wave equation and massless particles is replaced in thatcase by the analogy between the Klein Gordon equation and massive particles, wherethe Klein Gordon equation (with mass m = 1) is

(ä−1)ψ= 0, ψ :=ψ(t , x).Now, how is the vector field method working for Klein Gordon fields ? We could

try to repeat the same estimates as the one obtained for the wave equation but thereis one caveat: the scaling vector field does not commute with ä− 1. Without thescaling vector field, we can no longer use the vector field idendities of Lemma 4.7.

Instead, the traditional techniques (see [Kla93]) use

1. The hyperboloidal foliation.

2. Vector field identities on each hyperboloidal slice.

3. the extra control of the L2 norm of ψ in the energy estimates.

Let us explain each ingredient, starting with the foliation.

4.7.1 The hyperboloidal foliation

Let us fix global Cartesian coordinates (t , xi ), 1 ≤ i ≤ n on Rn+1 and denote by Σtthe hypersurface of constant t . The hypersurfaces Σt , t ∈ R then give a completefoliation of Rn+1.

The hyperboloidal foliation is defined as follows. For any ρ > 0, define Hρ by

Hρ ={(t , x)

∣∣ t ≥ |x| and t 2 −|x|2 = ρ2} .For any ρ > 0, Hρ is thus only one sheet of a two sheeted hyperboloid6

Note that ⋃ρ≥1

Hρ ={(t , x) ∈Rn+1 ∣∣ t ≥ (1+|x|2)1/2} .

The above subset of Rn+1 will be referred to as the future of the unit hyperboloid. Onthis set, we will use as an alternative to the Cartesian coordinates (t , x) the followingtwo other sets of coordinates.

6The hyperboloidal foliation was originally introduced in [Kla85] in the context of the non-linear Klein-Gordon equation. For more recent applications, see [WW15] and [LM15] which concern the stability ofthe Minkowski space for the Einstein-Klein-Gordon system.

17

t

r

Hρ

H1

Σt

Figure 1: The Hρ foliations in the (t ,r ) plane, ρ > 1

Hρ1

H1

Hρ2

r=

0

t =r

I+

Figure 2: The Hρ foliations in a Penrose diagram of Minkowski space, ρ2 > ρ1 > 1

Spherical coordinatesWe first consider spherical coordinates (r,ω) on Rnx , where ω denotes spherical co-ordinates on the n −1 dimensional spheres and r = |x|. (ρ,r,ω) then defines a co-ordinate system on the future of the unit hyperboloid. These new coordinates aredefined globally on the future of the unit hyperboloid apart from the usual degener-ation of spherical coordinates and at r = 0.

Pseudo-Cartesian coordinatesThese are the coordinates (y0, y j ) ≡ (ρ, x j ). These new coordinates are also definedglobally on the future of the unit hyperboloid.

18

For any function defined on (some part of) the future of the unit hyperboloid,we will move freely between these three sets of coordinates.

4.7.2 Geometry of the hyperboloids

The Minkowski metric η is given in (ρ,r,ω) coordinates by

η=−ρ2

t 2(dρ2 −dr 2)− 2ρr

t 2dρdr + r 2σSn−1 ,

where σSn−1 is the standard round metric on the n −1 dimensional unit sphere, sothat for instance

σS2 = sinθ2dθ2 +dφ2,in standard (θ,φ) spherical coordinates for the 2-sphere. The 4 dimensional volumeform is thus given by

ρ

tr n−1dρdr dσSn−1 ,

where dσSn−1 is the standard volume form of the n −1 dimensional unit sphere.The Minkowski metric induces on each of the Hρ a Riemannian metric given by

d s2Hρ =ρ2

t 2dr 2 + r 2σSn−1 .

A normal differential form to Hρ is given by td t −r dr while t∂t +r∂r is a normalvector field. Since

η (t∂t + r∂r , t∂t + r∂r ) =−ρ2,the future unit normal vector field to Hρ is given by the vector field

νρ ≡ 1ρ

(t∂t + r∂r ) . (11)

Finally, the induced volume form on Hρ , denoted dµHρ , is given by

dµHρ =ρ

tr n−1dr dσSn−1 .

4.7.3 The particle vector field and the stress energy tensor of Vlasov fields

For the massless case, we used the conservation in t of || f (t , ., .)||L1x,v . For the massivecase, we want to use the hyperboloidal foliation, so we are looking for a conservationof some norm of f of the form

∫Hρ

∫v f where Hρ is the hypersurface of constant ρ.

Unfortunately, a naive attempt such as∫Hρ

∫v

f d vdµρ

does not lead to a conserved quantity in ρ.We could try to look for a conservation law by hand, but we can also just remem-

ber that the Vlasov equation, like the wave equation, has an energy-momentum ten-sor.

Let us first define a volume form in v

dµm ≡ d v1 ∧ . . .∧d vn

v0= d v√

m2 +|v |2, (12)

where as usual m = 0 in the massless case.

19

Remark 4.3. In the massless case, the volume form d v|v | is singular near v = 0. Inthe remainder of this article, we will however study mostly energy densities, whichintroduce an additional factor of |v |2 in the relevant integrals and thus remove thissingular behaviour near v = 0. Note also that, in dimension 2 or greater, f ∈ L∞loc issufficient for f|v | ∈ L1loc .

We now define the particle vector field in the case of massive particles as

Nµm ≡∫Rn

f vµdµm ,

and in the case of massless particles as

Nµ0 ≡∫Rn \{0}

f vµdµ0,

as well as the energy momentum tensors

T µνm ≡∫Rn

f vµvνdµm ,

and

T µν0 ≡∫Rn \{0}

f vµvνdµ0,

where dµm and dµ0 are the volume forms defined in (12). More generally, we candefine the higher moments

Mα1...αpm ≡

∫Rn

f vα1 . . . vαp dµm ,

and similarly for the massless system.The interest in any of the above quantities is that if f is a solution to the associ-

ated massless or massive transport equations, then these quantities are divergencefree. Indeed, we have

∂µTµν0 =

∫Rn \{0}

T0( f )vνdµ0, (13)

∂µTµνm =

∫Rn

Tm( f )vνdµm . (14)

We will be interested in particular in the energy densities

ρ0( f ) ≡ T0(∂t ,∂t ) =∫Rn \{0}

f |v |d v, (15)

for the massless case, while for the massive case we define

ρm( f ) ≡ Tm(∂t ,∂t ) =∫Rn

f v0d v. (16)

In the following, we will denote by ρ( f ) any of the quantities ρm( f ) or ρ0( f ) de-pending on whether we are looking at the massive or the massless relativistic oper-ator.

In the massive case, we will also make use of the following energy density

χm( f ) ≡ Tm(∂t ,νρ), (17)

20

where νρ is the future unit normal to Hρ introduced in Section 4.7.2. We compute

χm( f ) =∫

v∈Rnf v0

(t

ρv0 + r

ρv r

)dµm ,

=∫

v∈Rnf

(t

ρv0 − x

i

ρvi

)d v.

The following lemma will be used later.

Lemma 4.8 (Coercivity of the energy density normal to the hyperboloids). Assumingthat t ≥ r , we have

χm( f ) ≥ t2ρ

∫v∈Rn

f[(

1− rt

)((v0)2 + v2r

)+ r 2σAB v A vB +m2] d vv0

. (18)

Proof. Using that(v0)2 = v2r + r 2σAB v A vB +m2

where σAB denotes the components of the metric σSn and v A , vB are the angularvelocities, we have

(v0)2 = (v0)2

2+ 1

2

(v2r + r 2σAB v A vB +m2

)and thus

v0(

t

ρv0 − x

i

ρvi

)= t

2ρ

((v0)2 + v2r + r 2σAB v A vB +m2 −2

xi

tvi v

0)

.

The lemma now follows from

(v0)2 + v2r −2xi

tvi v

0 ≥(1− r

t

)((v0)2 + v2r

),

assuming t ≥ r .Remark 4.4. • Since (v0)2 ≥ v2r , we will use (18) in the form

χm( f ) ≥ t2ρ

∫v∈Rn

f[(

1− rt

)(v0)2 + r 2σAB v A vB +m2

]dµm .

• We also remark that

χm(| f |) ≥ m2

2

∫v| f |d v

v0= m

2

2ρm

( | f |(v0)2

),

since t2ρ ≥ 12 , and, furthermore,

χm(| f |) ≥ t − r2ρ

ρm(| f |) = ρ2(t + r )ρm(| f |).

• Finally, independently of Lemma 4.8, since by the Cauchy-Schwarz inequalityfor Lorentzian metrics, as the vectorsνρ , and v are both timelike future directed,∣∣∣∣ t v0 −xi viρ

∣∣∣∣= |〈v,νρ〉| ≥ |v ||νρ | = m, where |v | = |g (v, v)| 12 ,we get immediately∫

v| f |d v ≤

∫v

1

m

∣∣∣∣ t v0 −xi viρ∣∣∣∣ | f |d v = 1mχm(| f |).

21

As in the massless case, let us define the following norms for massive fields

Definition 4.1. Let f be a regular distribution function for the massive case definedon

⋃1≤ρ≤P

Hρ ×Rnv . For k ∈N, we define, for all ρ ∈ [1,P ],

‖ f ‖P,k (ρ) ≡∑

|α|≤k

∑Ẑα∈P̂|α|

∫Hρχm(|Ẑα f |)dµHρ . (19)

4.8 Klainerman-Sobolev inequalities and decay estimates: massivecase

In the massive case m > 0, we will proveTheorem 4.3 (Klainerman-Sobolev inequalities for velocity averages of massive dis-tribution functions[FJS15]). Let f be a regular distribution function for the massivecase defined on

⋃1≤ρ

The remainder of the proof is then similar to the massless case. We have∣∣∣∣∫vΩ01(| f |)(y0, x1 + t y1, x2, .., xn , v)dµm

∣∣∣∣. ∑|α|≤1

∫v|Ẑα f |(y0, x1+t y1, x2, .., xn , v)dµm .

Inserting in the Sobolev inequality and repeating up to exhaustion of all the variables(the fact that for all j , |y j | ≤ 1

(8n1/2), guarantees that |y | = (∑nj=1 |y j |2)1/2 ≤ 1/8 so that

we still have tt (y0,x j +t y j ) ∼ 1), we obtain∫

v| f |(y0, x1, x2, .., xn , v)dµm .

∑|α|≤n

∫|y |≤1/8

∫v|Ẑα f |(y0, x j + t y j , v)dµmd y.

Recall that the volume form on each of the Hρ is given in spherical coordinates byρt r

n−1dr dσ, or in yα coordinates by y0

t d y . Thus, we have∫v| f |(y0, x1, x2, .., xn , v)dµm .

∑|α|≤n

∫|y |≤1/8

∫v|Ẑα f |(y0, x j + t y j , v)dµm t (y

0, x j + t y j )y0

dµHρ ,

.t (y0, x j )

y0∑

|α|≤n

∫|y |≤1/8

∫v|Ẑα f |(y0, x j + t y j , v)dµmdµHρ ,

where we have used again that t (y0, x j + t y j ) ∼ t (y0, x j ) in the region of integration.Applying the change of coordinates z j = t y j and noticing that the quantities on theright-hand side are controlled by the estimate (22) applied to Ẑα( f ) completes theproof.

Since the norm on the right-hand side of (20) is conserved if f is a solution to themassive transport equation, we obtain, as a corollary, the following pointwise decayestimate.

Theorem 4.4 (Pointwise decay estimates for velocity averages of massive distribu-tion functions [FJS15]). Let f be a regular distribution function for the massive casesatisfying the massive transport equation Tm( f ) = 0 on

⋃1≤ρ

Proof. We have νρ = Sρ with S the scaling vector field. On the other hand, recallthat S essentially commutes with the massive transport operator, so that in par-

ticular Tm(S( f )) = 0 if Tm( f ) = 0. Thus,∫

v S( f )d vv0

= S(∫

v fd vv0

)satisfies the same

decay estimates as∫

v fd vv0

, which shows the improved decay for νρ(∫

v fd vv0

). Thehigher order derivatives follow similarly. Indeed, using that S(ρ) = ρ, we have forinstance S2( f ) = ρ2ν2ρ( f )+ S( f ). Applying the decay estimates for the velocity av-erages of S2( f ) and S( f ) gives the correct improved decay for velocity averages ofν2ρ( f ). Higher normal derivatives can be treated similarly. Finally, the improved de-cay for tangential derivatives of velocity averages is an easy consequence of the factthat ∂yk = 1t Ω0k .

4.9 Distribution functions for massive particles with compact sup-port in x

The decay estimates for massive fields require that the initial data be given on theinitial hyperboloid H1 instead of a more traditional t = const hypersurface. We willexplain here how we can go from the t = 0 hypersurface to H1 provided the initialdata on t = 0 has compact support in x. For simplicity, consider the homogeneousmassive transport equation with initial data f0 given at t = 0. Assume that the sup-port of f0 is contained in the ball of radius R. Without loss of generality, we maytranslate the problem in time, so that we now consider the problem with data attime t =

pR2 +1.

Tm( f ) = 0, (23)f (t =

√R2 +1) = f0. (24)

Now, by the finite speed of propagation, the solution to this problem vanishes out-side of the cone

C (R) ≡{

(t ,r,ω)/ t − r =√

R2 +1−R,ω ∈Sn−1, t ≥√

R2 +1}

∪{

(t ,r,ω)/ t + r =√

R2 +1+R,ω ∈Sn−1, t ≤√

R2 +1}

depicted below. Thus, the trace of f on H1 is compactly supported and as a conse-quence, the norm appearing on the right-hand side of the decay estimate is finite.Recall also that the decay estimates hold for t ≥

√1+|x|2. On the other hand, the

region t <√

1+|x|2 would coresponds here to the exterior of C (R). Thus, for com-pactly supported initial data given on some t = const hypersurfaces, we can applyour decay estimates and obtain a 1/t n decay uniformly in x.

24

t

rH1

r = Rr = R

C(R) C(R)

t =√R2 + 1

t =r

Figure 3: The trace of a distribution function with compact support on H1.

4.10 The classical case

Finally, let us show how the same type of techniques can be used to derive decayestilmates for the classical transport equation

∂t f + v i∂xi f = 0.Exercice:

1. Show that the following set of vector fields commute with the classical trans-port operator

• The translations ∂xα .

• The complete lift of the usual rotationsΩxi j +Ωvi j .• Vector fields associated with the Galilean invariance7 t∂xi +∂v i .• Spatial scaling xi∂xi + v i∂v i• Space-time scaling xα∂xα .

2. Prove the following decay estimates∫v| f |(t , x, v)d v . 1

1+|x|+ |t |∑

|α|≤n||Ẑα f (0, x, v)||L1x,v ,

where the Ẑα are compositions of |α| commuting vector fields.7Another point of view is to consider them (formally) as degenerate versions (as c →+∞) of Lorentz

boosts.

25

5 Yet another proof of decay for velocity averages

5.1 Weights preserved by the flow

Recall that in a general Lorentzian manifold with metric g , if γ is a geodesic withtangent vector γ̇ and K denotes a Killing field, then g (γ̇,K ) is preserved along γ.

Exercice: prove indeed that is K is Killing and s → γ(s) is a geodesic, thend

d sg (K , γ̇) = 0.

In this section, we explain how to transpose and exploit this fact for the freetransport operators.

We define the sets of weights

km ≡ {vαxβ−xαvβ, vα}, m > 0 (25)and

k0 ≡ {xαvα, vαxβ−xαvβ, vα}. (26)The following lemma can be easily checked.

Lemma 5.1. Let Tm be the relativistic transport operator of mass m. For all z ∈ km ,[Tm ,z] = 0.

The weights in km or k0 also have good commutation properties with the vectorfields in P̂ and K̂.

Lemma 5.2. For any z ∈km and any Ẑ ∈ P̂,

[Ẑ ,z] = ∑z′∈km

cz′z′

where the cz′ are constant coefficients.Similarly for any z ∈k0 and any Ẑ ∈ K̂,

[Ẑ ,z] = ∑z′∈k0

dz′z′,

for some constant coefficients dz′ .

Proof. This follows from straightforward computations.

In the classical case, the weights that are preserved by the flow are the one asso-ciated with the Galilean symmetries.

Lemma 5.3. Let z= v i , xi − v i t , xi v j −x j v i , then (∂t + v i∂xi )z= 0.Again, these z weights have good commutation properties with the commuting

vector fields.

26

5.2 Lp decay of velocity averages

These extra weights can be used to prove decay estimates of velocity averages.We start with the easy classical case:

Proposition 5.1. Let f be a regular distribution function. Then, for all t > 0,∣∣∣∣∣∣∣∣∫v

f (t , ., v)d v

∣∣∣∣∣∣∣∣L2x

.1

t 3/2

∣∣∣∣ f < x − v t >n/2+∣∣∣∣L2x,v .In particular, if f solves the free transport equation ∂t + v i∂xi f = 0, then

||∫

vf (t , ., v)d v ||L2x .

1

t 3/2|| f (0, ., .) < x >n/2+ ||L2x,v .

(Here n/2+ is any number of the form n/2+ ² with ² > 0 and the constant in the in-equalities a priori degenerate as ²→ 0. Moreover, < . > is the Japanese bracket, mean-ing < x >=

√1+|x|2.)

Proof. We have∫v

f d v =∫

vf< x − v t >n+/2< x − v t >n+/2 d v, (27)

≤ || f < x − v t >n+/2 ||L2v || < x − v t >−n+/2 ||L2v , (28)

using the Cauchy-Schwarz inequality. This leads to

||∫

vf d v ||L2x ≤ || f < x − v t >

n+/2 ||L2x,v || < x − v t >−n+/2 ||L∞x L2v .

Let us thus compute

|| < x − v t >−n/2+ ||2L2v

=∫

v∈Rn< x − v t >−n+ d v.

Aplying again the change of variable w = x − v t we get the results.Exercice:

• Write decay estimates for Lp norms of∫

v f d v using the same type of argu-ments.

• We consider the relativistic massive transport operator. Again, we would liketo use the Cauchy-Schwarz inequality (or Hölder) as follows∫

vf d v =

∫v

f < v0x − v t >n+/2< v0x − v t >−n+/2 d v, (29)

≤ || f < v0x − v t >n+/2 ||L2 || < v0x − v t >−n+/2 ||L2v . (30)

The difficulty comes from the second term on the right-hand side, as it doesnot have uniform L∞ decay. A possible solution is to add extra v weights:∫

vf d v ≤ || f < v0 >q< v0x − v t >n+/2 ||L2 || < v0 >−q< v0x − v t >−n

+/2 ||L2v .

Prove an L2x decay estimate for q sufficiently large. (Start with n = 1 for sim-plicity).

• Write an Lp analogue of these estimates.

27

5.3 Interlude: a vector field approach to decay of solutions to Schrödinerequations

Consider the 1d Schrödinger equation8,

(i∂t +∂xx ) (u) = 0.This equation admits several conservations, but in particular, you can check easilymass conservation

||u(t )||L2x = ||u(0)||L2x .Using, say, the fundamental solution, it is easy to see that |u(t )|2. 1/t d . Is there

a vector field way to capture this ?Again, we need to find the commuting vector fields, and they typically need to

contains weights in t or x to get decay.After trial and error, one can derive that (formally) if u solves the Schrödinger

equation, so does

i t∂x u + x2

u.

Unfortunately, because of the second term, the operator u → i t∂x u + x2 u is not apure differential operator, i.e. it does not correspond to a vector field.

The idea is to rewrite it by adding an phase function, as follows

i t∂x u + x2

u = e i x2

4t i t∂x

(e−

i x24t u

).

Exercice: use the above formula to prove decay estimates for solutions to theSchródinger equation.

(Small) Open problem: The Airy equation reads

∂t +∂3x .It is a typical linear dispersive pde, that arises naturally in some model of fluid dy-namics (for instance, it is the linear part of the so called Korteweg-de-Vries (KdV)equation). Using a Fourier decomposition of the solutions, one can easily see thatsolutions to the Airy equation will decay like t−1/3.

Does there exists a vector field approach to this decay, perhaps using a similarphase conjugation ?

6 Null condition and null decomposition of tensors forVlasov fields

In the case of wave equations, we know that due to the slow decay (1/t ) of wavesin 3d , we need structural conditions on the non-linearities for global solutions toexist9. The standard structures are the null forms

Q0(φ,ψ) = g (dφ,dψ),Qi j (φ,ψ) = (∂xiψ∂x jφ−∂x jψ∂xi )

8I would like to thanks Sun Jin Oh for explaining me this example.9See the seminal work of F. John [Joh79].

28

Where is the improvement coming ?

Improvement for decay:

In terms of decay, recall that the tangential derivative to the light cone, such as∂t +∂r , can be rewritten as

L = ∑|α|=1

1

tcαZ

α.

Thus, they generate a 1/t extra decay compared to a random vector field.So the idea is that in the null forms Q we want to check that each product always

contains a tangential derivative.This leads to formulae such as

Q0(φ,ψ) = 1t

∑|α|,|β|=1

cαβZα(φ)Zβ(ψ)

and the extra 1/t is the reason for the gain.

Improvement for regularity:

Imagine that we want to prove an energy estimate for an equation such as

�ψ= ∂ψ.∂ψ.After multiplying the equation by ∂tψ and running the usual arguments for the

energy estimate, we need to control the error term∫t ,x

|∂tψ.∂ψ.∂ψ|(s, x)d s.∫

t|∂tψ|L2x (s)×|∂ψ.∂ψ|L2x (s)d s.

The first factor on the right hand side can be absorbed using a Gronwall type argu-ment but we still need a bound on the second. This can be done using the Sobolevinequality, but implies that we need to control ψ in H n/2+1+².

However, for null forms, we can do much better:

Theorem 6.1 (Klainerman-Machedon estimates (in dimension 3) [KM93]). For Q anull form, if

�ψ = Q(ψ,φ),�φ = Q(ψ,φ)

then, locally in time

||Q(∂ψ.∂φ)||L2t ,x . E(∂ψ)+E(ψ)+E(φ)+E(∂φ).

where E is the usual energy for the wave equation.

So there is a gain of 1/2+ ². (without the null forms, we can actually also im-prove to just H 2+² regularity, using the so-called Strichartz estimates, see for in-stance [PS93]).

Moreover, we are also familiar with the idea of decomposing tensors into tensorcomponents using a null frame. A typical null frame in Minkowski space is given by

29

L̄ := ∂t −∂r ,L := ∂t +∂r ,e1,e2, where e1,e2 are a local orthonormal frame of vectorfields tangent to the 2-spheres of constant t and r .

If we consider say a Maxwell 2-form F , one can prove (see [CK90]), using thestandard vector field method (including the Morawetz vector field as a multiplier)

|FAL |. (1+ t +|x|)−5/2, |FAL̄ |. (1+ t +|x|)−1(1+ t −|x|)−3/2, ...and these types of estimates can easily be saturated.

We will present here some analogue of these estimates in the case of masslessVlasov fields. (Note that the dynamics of massless fields are also relevant to the mas-sive fields, in the sense that many special properties of massive fields degenerate aseither |v |→+∞ or m → 0.)

One of the main motivations comes from the non-linear problems.

Let us illustrate this with the so-called Einstein-Vlasov system10

Ri c(g )− 12

g R(g ) = Twhere Ri c(g ) and R(g ) denotes respectively the Ricci tensor and Ricci scalar of someLorentzian metric g and where T = T [ f ] is the energy momentum tensor of a Vlasovfield f which solves the Vlasov equation

vα∂xα f − vγvβΓiγβ(g )∂v i f = 0,where Γi

γβ(g ) are the Christoffel symbols of the metric g .

We recall that, schematically, the first equation takes the form (in so called wavecoordinate)

äg gµν =Qµν(∂g ,∂g )+Tµν[ f ],where äg is the wave operator associated with the metric g and Q is some bilinearform.

Consider massless particles. In 3d, the decay that we have obtained so far giveswould imply that for g close to the Minkowskis metric, |Tµν[ f ]| decays like 1/t . Thiswill lead to logarithmic growth in the energy estimates.

This type of logarithmic obstacle for small data global existence is classic fornon-linear waves with quadratic non-linearities in low dimension and without anyspecial structure on the equation, the best we can hope is an almost global existence(say existence of solutions up to time exp(1/²) where ² measure the "size" of the so-lution initiallly).

Thus, we also need a form of the null condition here for the terms coming fromT [ f ].

Moreover, we will also need a form of the null condition in the Vlasov equation.We will give below several illustrations of null conditions for Vlasov fields, either

at the linear level or for model problems .

10See [Rin13] for a thorough presentation of this system.

30

6.1 Null decomposition of the energy-momentum tensor and de-cay

Consider again the energy-momentum tensor of massless Vlasov fields in the flatMinkowski space.

Tµν[ f ] =∫

v∈R3f vµvν

d v

v0.

Let L = ∂t +∂r . If we want to consider couple systems of wave and particles, weneed to understand why TLL should decay better than say TL̄L̄ .

One way is, as above, to exploit the weights propagated by the flow.Indeed, in the same sense that one can write vector identities such as

(t + r )L = ∑|α|=1

cαZα

where cα are homogeneous of degree zero, we have the weights identities

Lemma 6.1. Let vα = mα be the mulplier associated to any of the translation vectorfields. Let ωi = xi /|x| and let ∂r = ωi∂i and mr = ωi mi , i = 1, ..,3. Finally let mS bethe multiplier associated with the scaling vector field. Then, we have

vt = tmS −xiω0i

t 2 − r 2 (31)

vr = tωiω0i − r mst 2 − r 2 (32)

vi =−x jωi j + tω0i −xi mS

t 2 − r 2 =−xi ms

t 2 − r 2 +xi x jω0 jt (t 2 − r 2) +

ω0i

t. (33)

In particular,

vL = 12

(vt + vr ) = ms +ωiω0i

2(t + r ) , (34)

v i := vi −ωi mr =ω jωi j

r=−ωiω

jω0 j +ω0it

. (35)

Here v i is the projection to the cone of the vi vector. If v is a covector, we shallwe shall denote by v the tangential11 part of v (i.e. < L, v >= 0, with L = ∂t − ∂r ).Thus, |v | = |vL |+ |vi |.

We then have the estimates

Lemma 6.2. Let q = r − t . Then,

(1+ t +|q |)|v |+ (1+|q |)|v |.∑I|m I (v)|.

where the sum is over all possible multipliers corresponding to translation, angularmomentum and scaling.

11Note than we have vL = − 12 v q , vq = − 12 vL , thus, the tangential part of the velocity (index down)corresponds to the transvertial part of the momentum, index up.

31

Proof. The proof is very similar to the one of the standard vector field method. Notefirst that if |t | + |r | ≤ 1, then the estimate holds since the usual translation are in-cluded in the sum. The idendity for |v | following directly from the previous lemma.

The conclusion is that we get improved decay for TLL[ f ] but the norm on theright-hand side will have extra weights corresponding to the z weights.

6.2 A null form for a wave/particle interaction

Consider the partially non-linear problem

T ( f ) = v0Q(∂φ, f ),äφ = 0,

where T is the massless relativistic transport operator.Here Q(∂φ, f ) is a linear combination of terms of the form ∂φ. f with constant (or

homogeneous) coefficients and the extra v0 is here only because of homogeneity.

Exercice:

1. Write a commutator formula for the above equation after N commutations.

2. Write the energy estimates.

3. Show that solutions to the above problem have the same decay as in the linearcase in dimension n ≥ 4.

In dimension 3, since ∂φ decays a priori no faster than 1/t , standard energy es-timates for the above transport equation would lead to a log growth.

As in the standard application of the theory of null forms, we want to understandwhether there might exist special non-linearities, for which this log growth is absent.

We will consider the following special non-linearity:

Q(∂φ, f ) = 1v0

T (φ) f ,

where T (φ) = vα∂xα (φ).To understand why this non-linearity is better, we first show that, like Q0, this

special non-linearity can be eliminated using a simple transformation.(recall that ifäφ=Q0(φ,φ) andφ is small then, settingφ= ln(1+v), then v solves

the homogeneous wave equation. This is the so-called "Niremberg example". )Indeed, with the above choice of Q, the equation becomes

T ( f ) = T (φ) f ,which can be rewritten as

T ( f e−φ) = 0and using g = f e−φ as our unknown, we can apply all the linear estimates to g andthen go back to our original variable.

32

In the actual non-linear physical systems, such as the Einstein-Vlasov, Vlasov-Norström or Vlasov-Maxwell, the non-linearities typically have a different structure,closer to

T ( f )+Q(∂xφ,∂v f ) = 0In the (massless) Vlasov-Norström case, one has for instance

T ( f )+T (φ)v i∂v i f = 0. (36)The extra v derivatives acting on f is a true annoyance, as ∂v i does not belongs

to our algebra of commuting fields. Note however that v i∂v i does commute with T ,in the sense that [T, v i∂v i ] =−T .

So we will enhance our algebra of commuting fields to include v i∂v i .However, with (36), we can no longer eliminate the non-linearity by our trick. So

we need to estimate products of the form T (φ)v i∂v i f . We therefore need an esti-mate for products of the form T (φ)g , where g solves a transport equation.

Why is this product better ? For wave equations, we need to see some sort ofderivatives tangential to the light-cone for a product to be better. We therefore forcethe apparition of ∂t +∂r as follows

T (φ)g = vα∂xαφ.g ,

= v0(∂t + v

i

v0∂xi

)(φ)g ,

= v0(∂t +∂r −∂r + v

i

v0∂xi

)(φ)g ,

= v0 (∂t +∂r ) (φ)g

+ v0(−∂r + v

i

v0∂xi

)(φ)g .

Now the first term on the RHS is good, (it has extra decay) so we focus on the second.

v0(−∂r + v

i

v0∂xi

)(φ)g = v0

(− x

i

|x|∂xi +v i

v0∂xi

)(φ)g

= v0(− x

i

|x| +v i

v0

)∂xiφ.

Now, away from the light-cone, we know that we can get extra decay and thereforeclose the estimates. Near the light-cone on the other hand |x| ∼ t , so let us pretendthat we can rewrite the above factor as

1

t

(−xi + t v

i

v0

)= 1

tzi

where zi are part of the weights we can propagated along the characteristic flow.Thus, we can incorporate these weights in the energy estimates for g .

6.2.1 Applications to the massless Vlasov-Nordström system

We consider in this section, the so-called Vlasov-Norström system. The Norströmtheory is a scalar theory of gravity that still contains gravitational wave. Formally,

33

one can obtain it from the usual Einstein equations by considering only conformaldeformation of Minkowski space and by removing all non-linear wave interactions.

We will denote by Tφ the transport operator defined by

Tφ ≡ vα ∂∂xα

− vα∇αφ · v i ∂∂v i

,

i.e.Tφ = T0 −T0(φ) · v i∂v i .

The massless Vlasov-Nordström system can then be rewritten as

äφ = 0, (37)Tφ( f ) = 0, (38)

which we complement by the initial conditions

φ(t = 0) =φ0, ∂tφ(t = 0) =φ1, (39)f (t = 0) = f0, (40)

where (φ0,φ1) are sufficiently regular functions defined on Rnx and f0 is a sufficientlyregular function defined on Rnx ×

(Rnv \ {0}

).

By sufficiently regular, we mean that all the computations below make sense. Wewill eventually require that EN [φ0,φ1] < +∞ where EN is the energy norm definedby (44) and similarly, we will also require below that || f0||K,N

Iterating the above, one obtains

Lemma 6.4. Let f be a solution to (38). For any multi-index α, we have the commu-tator estimate ∣∣[Tφ, Ẑα] f ∣∣≤C ∑

|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

|T (Zγφ)| · |Ẑβ f |, (41)

where the Z γ ∈K|γ| and the Ẑβ ∈ K̂|β|0 and C > 0 is some constant depending only on|α|.

6.2.3 Approximate conservation law

Similar to the conservation of the L1 norm for the free transport operator, we have

Lemma 6.5. Let h be a regular distribution function. Let g be a regular solution toTφ(g ) = v0h, with v0 = |v |, defined on [0,T ]×Rnx ×

(Rnv \ {0}

)for some T > 0. Then, for

all t ∈ [0,T ],∫Σt

ρ0(|g |)(t , x)d x

≤∫Σ0

ρ0(|g |)(0, x)d x +∫ t

0

∫Σs

ρ0(|h|)(s, x)d xd s + (n +1)∫ t

0

∫Σs

∫v∈Rnv \{0}

|T (φ) f |d vd xd s.(42)

Proof. As before, this follows, after regularization of the absolute value, from inte-gration by parts or an application of Stoke’s theorem. The term T (φ)v i∂v i | f |, whichappears in the computation, gives rise after integration by parts in v to the last termin (42) since ∂v i

(v i T (φ)

)= (n +1)T (φ).6.2.4 Massless case in dimension n ≥ 4In this section, we first consider the case n ≥ 4, the 3d case requiring the use of thenull condition as explained above.

If φ is a solution to the wave equation, let us consider the energy at time t = 0EN [φ](t = 0) ≡

∑|α|≤N ,|α|∈K|α|

∣∣∣∣Zα(∂φ)(t = 0)∣∣∣∣2L2(Rnx ) . (43)Now if φ(t = 0) = φ0 and ∂tφ(t = 0) = φ1, for pairs of sufficiently regular functions(φ0,φ1) defined on Rnx , then the above quantity can be computed purely in terms ofφ0, φ1, so we define12

EN [φ0,φ1] ≡ EN [φ](t = 0). (44)Similarly, if f is a solution to (38) arising from initial data f0 at t = 0, then we

define

EN [ f ](t = 0) ≡ || f ||K,N (t = 0)= ∑

|α|≤N ,Ẑα∈K̂α0

∣∣∣∣ρ0 (Ẑα( f )(t = 0))∣∣∣∣L1(Rnx ) (45)

12The alternative to the approach we use here is to assume that (φ0,φ1) are regular initial data withdecay fast enough in x, for instance by assuming compact support, so that the resulting EN [φ(t = 0)] isfinite. What we want to emphasize here is that the quantity EN [φ(t = 0)] can in fact be computed purelyin terms of the initial data (using the equation to rewrite second and higher time derivatives ofφ in termsof spatial derivatives), and that this is all that is needed in terms of decay in x.

35

and we remark that this quantity can be computed purely in terms of f0 so we willset

EN [ f0] ≡ EN [ f ](t = 0).We will prove

Theorem 6.2. [FJS15]Let n ≥ 4 and let N ≥ 3n2 +1. Let (φ0,φ1, f0) be an initial dataset for the massless Vlasov-Nordström system such that EN [φ0,φ1] + EN [ f0] < +∞.Then, the unique solution ( f ,φ) to (37)-(38) satisfying the initial conditions (39)-(40)verifies the estimates

1. Global bounds: for all t ≥ 0,

EN [ f ](t ) ≤ eCE1/2N [φ0,φ1]EN [ f0],

where C > 0 is a constant depending only on N ,n.2. Pointwise estimates for velocity averages: for all (t , x) ∈ [0,+∞)×Rnx and all

multi-indices α satisfying |α| ≤ N −n,

ρ0(|Ẑα f |)(t , x). eCE 1/2N [φ0,φ1]EN [ f0]

(1+|t −|x| |)(1+|t +|x| |)n−1 .

Proof. Let N ,n,φ0,φ1, f0 be as in the statement of the theorem. From the conser-vation of energy and the commutation properties of the Zα with the wave operator,we have, for all t ,

EN [φ](t ) = EN [φ0,φ1] ≡ EN .Applying the standard decay estimates obtained via the vector field method toφ, wehave for all multi-indices α satisfying |α| ≤ N − (n +2)/2 and for all (t , x) ∈Rt ×Rnx

|∂Zαφ(t , x)|2. EN [φ](t )(1+|t −|x| |)(1+|t +|x| |)n−1 . (46)

Note that it follows from a standard existence theory for regular data that for all t ,EN [ f (t )]

|hα| . 1v0

∑|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

|T (Zγφ)| · |Ẑβ f |

.∑

|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

|∂(Zγφ)| · |Ẑβ f |,

so thatρ0(|hα|).

∑|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

|∂(Zγφ)|ρ0(|Ẑβ f |

),

since φ is independent of v . Integrating over x, we obtain, for all s ∈ [0, t ],∫Σs

ρ0(|hα|)(s, x)d x .∑

|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

∫Σs

|∂(Zγφ)|ρ0(|Ẑβ f |

)(s, x)d x.

We now estimate each term in the above sum depending on the values of |γ| and|β|.

If |β| ≤ N −n, we then apply the pointwise estimates on ρ0(Ẑβ( f )) to obtain

∫Σs

|∂(Zγφ)|ρ0(|Ẑβ f |

)(s, x)d x .

∫Σs

|∂(Zγφ)| EN [ f ](s)(1+|s −|x| |)(1+|s +|x| |)n−1 d x.

Applying the Cauchy-Schwarz inequality and using that∣∣∣∣∣∣∣∣ 1(1+|s −|x| |)(1+|s +|x| |)n−1∣∣∣∣∣∣∣∣

L2x

.1

(1+ s)(n−1)/2 , (48)

we obtain ∫Σs

|∂(Zγφ)|ρ0(|Ẑβ f |

)(s, x)d x . E 1/2N [φ](s)

EN [ f ](s)

(1+ s)(n−1)/2 . (49)

If now |β| > N −n, then |γ| ≤ |α|+1−|β| ≤ N − (n +2)/2 and thus, we also have (49),using this time the pointwise estimates on ∂(Zγφ) given by (46). Applying Grönwall’sinequality finishes the proof of the theorem.

6.2.5 Massless case in dimension n = 3We now turn to the case of the dimension 3, where the slower pointwise decay of so-lutions to the wave equations leads to a slightly harder analysis. First, let us strengthenour norms for the Vlasov field.

For this, recall the algebra of weights k0 introduced in Section 5.1 and define arescaled version κ0 by

κ0 ≡ (v0)−1k0 ={ z

v0/z ∈k0

},

where we recall that v0 = |v | in the massless case. If α is a multi-index, we will write[z

v0

]α ∈ κ|α|0 to denote a product |α| elements of κ0 and [ |z|v0 ]α in case we take theproduct of the absolute values of these elements.

37

Let us now define, for any regular distribution function f , the weighted norm

EN ,q [ f ] ≡∑

|α|≤N ,|β|≤q

∑Ẑα∈K̂|α|0

∫Σt

ρ0

(|Ẑα f |

[ |z|v0

]β)(x)d x (50)

= ∑|α|≤N ,|β|≤q

∑Ẑα∈K̂|α|

∫Σt

∫v∈Rn \{0}

(|Ẑα f |(x, v)

[ |z|v0

]β)v0d vd x

,where the weights z

v0lie in κ0.

Theorem 6.3 (Asymptotic behaviour in dimension n = 3[FJS15]). Consider the di-mension n = 3. Let N ≥ 7 and q ≥ 1. Let (φ0,φ1, f0) be an initial data set for themassless Vlasov-Nordström system such that EN [φ0,φ1]+EN [ f0]N ,q 0 is a constant depending only on N ,n and q.2. Small data improvement for the low order norms: there exists an ε0 (depending

only on n, N , q) such that if EN [φ0,φ1] ≤ ε0, then for all t ∈Rt ,EN−(n+4)/2,q−1[ f ](t ) ≤ eCE

1/2N [φ0,φ1]EN ,q [ f0]. (52)

3. Under the above smallness assumption, we also have the optimal pointwise es-timates for velocity averages: for all (t , x) ∈Rt ×Rnx and all multi-indices α sat-isfying |α| ≤ N − (3n +4)/2 and all |β| ≤ q −1,

ρ0

(∣∣∣∣Ẑα( f )[ zv0]β∣∣∣∣) (t , x). eCE

1/2N [φ0,φ1]EN ,q [ f0]

(1+|t −|x| |)(1+|t +|x| |)n−1 .

Proof. First, let us note that for all z ∈k0, we havev i∂v i

( zv0

f)= z

v0v i∂v i f ,

from which it follows that for all regular distribution functions g ,[

Tφ,z

v0

]g = 0.

Thus, we can upgrade Lemma 6.4 to∣∣∣[Tφ,[ zv0

]σẐα

]f∣∣∣≤C ∑

|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

|T (Zγφ)| ·[ |z|

v0

]σ|Ẑβ f |, (53)

where the Zγ ∈K|γ|0 , the Ẑβ ∈ K̂|β|0 ,

[z

v0

]σ ∈ κ|σ|0 and C > 0 is some constant depend-ing only on |α|. Applying arguments similar to those used in the n ≥ 4 case yield

EN ,q [ f ](t ) ≤ EN ,q [ f0]+C∫ t

0

∑|β|≤|α|,|γ|≤|α|,|β|+|γ|≤|α|+1

∑|σ|≤q

∫Σs

|∂(Zγφ)|ρ0([ |z|

v0

]σ|Ẑβ f |

)(s, x)d xd s

≤ EN ,q [ f0]+C∫ t

0E 1/2N

EN ,q [ f ](s)

(1+ s) d s. (54)

38

Applying Grönwall inequality, we obtain (51).Now assume that EN ≤ ε0 with ε0 small enough so that,

EN ,q [ f ](t ) ≤ (1+ t )δEN ,q [ f0],

with δ=CE 1/2N < 1/2.The key to the improved estimates is the following decomposition of the trans-

port operator T :

T = v0∂t + v i∂xi = v0(∂t + x

i

|x|∂xi)− v0 x

i

|x|∂xi + vi∂xi

= v0(∂t + x

i

|x|∂xi)+ v

0xi

t |x| (|x|− t )∂xi +v i t −xi v0

t∂xi

= v0(∂t + x

i

|x|∂xi)

︸ ︷︷ ︸outgoing derivatives

−v0

t

xi

|x|︸︷︷︸bounded

(−x jΩi j + tΩ0i −xi S

t + r

)︸ ︷︷ ︸

≤C(|Ωi j |+|Ω0i |+|S|)+v0 z

v0t∂xi ,

where the weight z in the last term is v i t − xi v0 ∈ k0. Recall14 now the followingimproved decay for outgoing derivatives of solutions to the wave equations: for allmulti-indices α such that |α| ≤ N − (n +2)/2−1,∣∣∣∣(∂t + xir ∂xi

)Zα(φ)

∣∣∣∣. EN(1+ t )3/2 .To estimate the second term, we need to obtain decay for Zφ as solution to the waveequation. This is done by integrating the decay of ∂Zφ coming from the Klainerman-Sobolev inequality along ingoing null rays. One then obtains:∣∣∣∣∣

(−x jΩi j + tΩ0i −xi S

t + r

)Zαφ

∣∣∣∣∣. EN(1+ t )1/2 .As a consequence, it follows that for all multi-indices |α| ≤ N − (n +2)/2−1,

|T (Zαφ)|. EN v0(

1

(1+ t )3/2 +∑z∈k0

|z|v0

1

t (1+ t )

).

Repeating the previous ingredients then gives (52). The pointwise estimates thenfollow from the Klainerman-Sobolev inequality (20).

6.2.6 Improved regularity

Assume that φ solves the wave equations and f solves the free transport masslessequation. Then,

∫R3+1

∫v

T (φ)2 fd v

|v | d xd t .(||φ|t=0||2H 2 +||∂tφ|t=0||2H 1)∫

R3x

f (0, x, v)|v |d x. (55)14This can be obtained from the usual Klainerman-Sobolev inequality and the formula for ∂s in (8) by

integration along the constant t = |x| null lines.

39

7 Some standard systems of kinetic theory

7.1 The Vlasov-Poisson system

This is the system

∂t f + v i∂xi f ±∇xφ.∇v f = 0, ∆φ= ρ( f ), (56)

where∆ is the Laplacian15. The variable x can be inRn , a bounded domain (in whichcase, extra boundary conditions must be specified) or some manifold. A typicallystudied case is that of x ∈ T n (corresponding to periodic boundary conditions).

There are many standard results on the VP system so here is an extremely non-exhaustive list:

• Global existence in dimension less than 3 (including) for arbitrarily large data.There are essentially two distinct approaches, the Pfaffelmoser approach [Pfa92](see also [Sch91] for easier presentation following the same type of ideas),and the one via propagation of moments of Lions-Perthame [LP91] (see also[Pal14] for an recent improvement).

• Small data global existence and asymptotics in dimension 3 and larger. Recallthat ρ decays like 1/t n . This leads to the decay rate for the force field ∇φ.

|∇φ|. 1t n−1

.

Up to dimension 3, one can use these decay rates to prove global existenceand derive some asymptotics of the solutions. Compared to a say, a standardquasilinear wave equations in dimension 3, one can study these equationswith very little regularity on the initial data, so the original proof of [BD85]contained decay estimates for ρ but not for derivatives of ρ. To get sharp de-cay estimates for derivatives of ρ, you need to do more work, first results by[HRV11] and then [Smu15] using the vector field approach. Up to dimension4, you can use the standard vector fields, but in dimension 3, you need a re-fined version, namely modified vector fields. Dimension 2 is an interesting (buthard) open problem.Open problem: consider the Vlasov Poisson system with two spatial dimen-sions. What are the asymptotics of the solutions for small initial data ?

• There are plenty of stationary solutions to the Vlasov Poisson system (in the at-tractive case) and a natural question is that of stability. Again, the scaling andnon-linearities in these equations is such that, contrary to say the Einsteinequations, you can prove results about orbital stability without understand-ing asymptotic stability. In fact, there are very few asymptotic stability resultsknown for VP, even for quite simple stationnary state. The stronger orbital sta-bility results are in [LMR12]. See also [Mou13] for a nice introduction to thesequestions and a presentation of the orbital results and methods of [LMR12].

• Landau damping. This is a classical phenomena in plasma theory. It was orig-inally discover by Landau at the linearized level. A full proof in the non-linear

15Sometimes, extra boundary/decay/mean value conditions can be imposed on ρ to ensure uniquesolvability in the Poisson equation.

40

case was given by Villani-Mouhot [MV11] and there has been some slight im-provements and simplified proofs since [BMM16]. Landau damping concernsthe Vlasov-Poisson system for x ∈ T n . The Poisson equation is slightly differ-ent to account for the periodicity:

∆φ= ρ[ f ]−∫

S1ρ[ f ].

Because of the periodic boundary conditions, the velocity averages do notneed to decay anymore, instead they homogenize, that is to say they approachtheir mean value: ∣∣∣∣ρ( f )(t , x)−∫

yρ( f )(t , y)

∣∣∣∣→ 0as t goes to ∞. How fast is the convergence ? This depends on the regular-ity. Let illustrate this for the free transport equation, using the typical methodused first, and then using our vector field approach.

We thus consider (in 1d for simplicity)

(∂t + v∂x ) ( f ) = 0,for f := f (t , x, v) with (t , x, v) ∈R×S1 ×R.The typical techniques are Fourier based. Let ρ( f ) = ∫v f d v and let ρk beingits kth Fourier modes (in x):

ρ( f )(t ) = ∑k∈Z

ρk (t )ei kx .

Let also fk (t , v) denotes the kth Fourier modes of f .

Applying the Fourier transform to this equation leads to

∂t fk + i vk fk = 0

which can be easily integrated as

fk (t , v) = e−i vk fk (0, v).

Note that ρk ( f )(v) =∫

v fk (t , v)d v. One can then compute

ρk ( f )(t ) =∫

vfk (t , v)d v =

∫v

e−i vk fk (0, vd v = f̃k (0,kt ),

where f̃k (0,η) is the Fourier transform in v of fk (0, .).

Now assume that fk is initially very regular in v . Then, its Fourier transform inv decays fast as |η|→+∞. Thus, as t →+∞, we have that ρk ( f )(t ) → 0, unlessk = 0.It is then easy to see that ρ( f ) must converge (exercice: in which topology ? )to ρ0( f ), which is exactly the mean value convergence mentioned above.

How would you capture this using the vector field method ?

41

Let Z = t∂x +∂v . As before, we can commute the free transport N times andthus control ∫

x

∫v|Z N f |(t , x, v)d xd v

in terms of the initial data.

Now, recall the Wirtinger inequality. For a periodic function ψ in W 1,1,∣∣∣∣ψ(x)− 12π∫

S1ψ

∣∣∣∣. ∫S1|∂ψ|(y)d y.

Apply it to ρ( f ). This gives

|ρ( f )−ρ0( f )|.∫

S1|∂xρ( f )|d x.

Rewrite ∂xρ[ f ] as

∂xρ[ f ] = 1tρ[Z [ f ]].

Then, we get

|ρ( f )−ρ0( f )|. 1t

∫S1|ρ[Z [ f ]]|d x,

and we have obtain 1/t decay. To get further decay, note that we can reap-ply what we just did to ρ[Z [ f ]], expect that now this quantity is mean free:∫

x ρ[Z [ f ]]d x = 0. Thus, we get that

|ρ[Z [ f ]]| ≤ 1t

∫S1|ρ[Z 2[ f ]]|d x.

Iterating, we obtain that

|ρ( f )−ρ0( f )|. 1t N

∫S1|ρ[Z N [ f ]]|d x.

Using the conservation of the right-hand side in time, this implies that

|ρ( f )−ρ0( f )|. 1t N

∫S1|∂Nv f (0, x, v)|d xd v.

Now, in physics, the distribution f could represent the number of chargedparticles. This distribution of charged particles then will generate an electro-magnetic force E , which can be computed by solving the Poisson equation

∆φ= ρ[ f ]−∫

S1ρ[ f ], E =∇φ,φ=φ(t , x).

(here I have neglected all physical constants for simplicity)

To get uniqueness in the Poisson equation, we look forφwhich are mean free :∫S1 φd x = 0 (note that the physical quantity, the one that appears in the Vlasov

equation, is actually E , not φ).

42

According to what we explained above, ρ[ f ] goes to a constant, so the sourceterm in the Poisson equation decays.

Exercice: Prove that the electric field E decays polynomially in t , with a rate ofdecay depending on the initial regularity of the data. (Harder: assuming theinitial data to be real analytic, prove that the electric field decays exponentiallyin time.)

What happens is the the distribution of charges homogenize, and hence, thereare no more "potential difference", and hence no electric field.

Homogeneous distributions: These are by definitions distributions functionswhich depends only on v . Any such distribution is actually a stationnary so-lutions to the Vlasov-Poisson system. Moreover the corresponding electricfield vanishes. Landau damping is then the claim that if one perturbs some ofthese homogeneous solutions, the perturbed solutions should then convergequite rapidly towards a nearby homogeneous solutions. These special homo-geneous solutions must verify a linear stability condition (typicaly called thePenrose stability condition). The work of Villani and Mouhot [MV11] can thenbeen seen as a non-linear stability result for homogeneous linear stable distri-butions (but their analysis also contained improvement at the linear level).

The non-linear analysis is hard, so I will only here describe briefly the linearone.

Let us first write the linearized equation around a homogeneous solution f0(v).We obtain

∂t f + v∂x f +∇[φ].∂v f0 = 0.We first Fourier in x:

∂t fk + i vk fk + i kφk∂v f0 = 0

We also Fourier in x the Poisson equation:

For k 6= 0,

−k2φk = ρkwhich leads to

∂t fk + i vk fk − ik

k2ρk∂v f0 = 0

We solve this ode :

fk (t ) = e−i vkt fk (0, v)−∫ t

0e−i vk(t−s)

k

k2ρk (s)∂v f0(v)d s

and then integrate in v to obtain

ρk (t ) = f̃k (0, tk)+ i∫ t

0ρk (s)

k

k2f̃0(k(t − s))k(t − s)d s.

We are left with an equation for ρk of the form

43

a(t ) = b(t )+∫ t

0a(t )W (t − s)d s.

This is a well known integral equation known at the Voltera equation. Typi-cally, it is solved using the Laplace transform (a complex variant of the Fouriertransform).

aL(λ) =∫ +∞

0e2πλt a(t )d t .

(well defined if a decay exponentially andλ small enough, otherwise use com-plex deformation)

Then, the above equation can be rewritten as

aL = bL +aLW L

which gives

aL = bL

1−W L .

provided that 1−W L 6= 0.The stability condition is then typically written as |1 −W L(λ)| > κ > 0 (thisshould hold uniformly in λ in some complex strip).

Problem: this analysis allows to control ρk and proves decay of the electricfield. However, compared with the linear case, it does not allow a control onnorms such as

∫x,v |Z N ( f )|d xd v . Open problem: could we prove stronger sta-

bility bounds for the linear stabil