Document [original]

Cosmological solutions of the Einstein-Vlasov-scalar field system

vorgelegt von

Diplom-Mathematiker

David Tegankong

von der Fakult¨at II-Mathematik und Naturwissenschaften

der Technische Universit¨at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. R. Nabben

Berichter: Priv.-Doz. Dr. A. D. Rendall

Berichter: Prof. Dr. A. Unterreiter

Tag der wissenschaftlichen Aussprache: 19. Mai 2005

Berlin 2005

D 83

To my wife Blandine Pulcherie Tamatcho and our child Marl`ene, Audrey and

Brice Tegankong.

Acknowledgements

I am grateful to my supervisors Pr. Alan D. Rendall and Pr. Norbert Noutchegueme

for helpful suggestions that both gave me during this work. A major part of

this work was carried out during two three-months stays at the Albert Einstein

Institute where I received pleasure of collaboration of Pr. Alan D. Rendall. So

I would like to give him many thanks.

I gratefully acknowledge the financial support of the VolkswagenStiftung,

Federal Republic of Germany, that allows me to do this work.

Besides, I thank Frau Hauschild and Frau Lampe for a warm welcome they

brought me during each of my visit at the Albert Einstein Institute in G¨olm.

Contents

Abstract 1

Introduction 2

1 Equations and preliminary results 5

1.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Auxiliary system and preliminary results . . . . . . . . . . . . . . 10

1.3 The reduced system . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Local existence and continuation of solutions 26

2.1 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Continuation criteria . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Global existence and asymptotic behaviour of solutions in the

past 50

3.1 Global existence in the past . . . . . . . . . . . . . . . . . . . . . 50

3.2 On past asymptotic behaviour . . . . . . . . . . . . . . . . . . . . 54

4 Global existence and asymptotic behaviour of solutions in the

future 62

4.1 Global existence in the future . . . . . . . . . . . . . . . . . . . . 62

4.2 The future asymptotic behaviour . . . . . . . . . . . . . . . . . . 68

4.2.1 Integration of equations . . . . . . . . . . . . . . . . . . . 69

4.2.2 Geodesic completeness . . . . . . . . . . . . . . . . . . . . 70

Conclusion 75

Appendices 76

iii

Abstract

The aim of this thesis is to obtain as much information as possible, about global

solutions of the Cauchy problem for the Einstein-Vlasov-scalar field system with

spherical, plane and hyberbolic symmetries written in areal coordinates. The

sources of this system are generated by both a distribution function and a linear

scalar field subject to the Vlasov and wave equations respectively. This system

describes the evolution of self-gravitating collisionless matter and scalar waves

within the context of general relativity. We consider the cosmological case. That

is spacetimes possess a compact Cauchy hypersurface and then, data are given

on a compact 3-manifold.

We extend the local-in-time results obtained by G. Rein for the Einstein-

Vlasov system with collisionless matter alone. This extension concerns pointwise

estimates for hyperbolic equations by the method of characteristics. This means

that the system is transformed to a system of ordinary differential equations

which are integrated along characteristics. The constraint equation on the initial

data reduced to an ordinary differential equation of first order and is solved. In

the past direction, we show global existence results for general data. The proof

is based on a change of variables inspired by the work of M. Weaver. The nature

of singularity is analyzed. The curvature invariant called Kretschmann scalar

blows up as ttends to 0 so that there is a singularity at tequal zero.

We prove that there is no global solution in the future in the spherical

symmetry case. In the plane and hyperbolic symmetries, the area radius goes

to infinity and so we obtain global solutions in the expanding direction. In the

special case of plane symmetry without Vlasov contribution, we show that the

asymptotics are Kasner-like at early time. Moreover the spacetime obtained in

this case is future geodesically complete.

We conclude the work by showing that the spatially homogeneous solutions

of the plane and hyperbolic symmetric Einstein-Vlasov-scalar field system exist

globally in the future and the corresponding spacetimes are geodesically com-

plete. Future asymptotics are Kasner-like in the plane symmetric case.

Introduction

The Einstein-Vlasov system governs the time evolution of a self-gravitating col-

lisionless gas in the context of general relativity. In the mathematical study of

general relativity, one of the main problems is to establish the existence and

properties of global solutions of the Einstein equations coupled to various mat-

ter fields such as collisionless matter described by the Vlasov equation (see [1],

[21] for reviews) or a scalar field (see [19] for the cosmological case and [6], [7]

and references therein for the asymptotically flat case). The aim of the present

investigation is to establish in the cosmological case the global in time existence

of solutions and their behaviour near the singularity and in the future. In this

case the whole universe is modelled and the ”particles” in the kinetic description

are galaxies or even clusters of galaxies.

In [15] and [16] G. Rein obtained cosmological solutions of the Einstein-

Vlasov system with surface symmetry written in areal coordinates. In [27] and

[30], these results were generalized to the case of non-vanishing cosmological

constant. In the present work, we extend the results of [15] to the case where

the source terms of the Einstein equations are generated by both a distribution

function fof particles, which is subject to the Vlasov equation, and a massless

scalar field φ, which is subject to the wave equation. The first result we establish

is a local in time existence theorem together with a continuation criterion. With

this, we prove global existence in time and study the asymptotic behaviour of

solutions when the time coordinate ttends to its limiting values, which might

correspond to the approach to the singularity or a phase of unending expansion.

There are several reasons why it is of interest to look at the case of a scalar

field. The first is that it is the simplest situation in which wave phenomena can

be examined in the context of the Einstein-Vlasov system. In surface symmetry

all wave propagation can be eliminated from the Einstein equations by the use of

suitable coordinate conditions. This is an analogue of the well-known statement

that there are no gravitational waves in spherical symmetry. Mathematically

it means that controlling solutions of the Einstein equations can be reduced to

controlling solutions of ordinary differential equations in time and in space. The

Vlasov equation, being a scalar hyperbolic equation of first order, can also easily

be solved in terms of its characteristics. This was the strategy used in [15] and

[16]. In the presence of a cosmological constant it is possible to follow the same

route.

The inclusion of a scalar field introduces waves into the system which cannot

be eliminated. Mathematically this means that it introduces a non-trivial hy-

perbolic equation, the wave equation. This work is concerned with symmetric

situations where there is a symmetry group acting on two-dimensional spacelike

orbits. This means that the wave equation reduces to an effective equation in

one space dimension. As a consequence part of the strategy used previously

can be carried over. That was based on pointwise estimates and not on integral

estimates (energy estimates) as is usual in the theory of hyperbolic equations.

Pointwise estimates for solutions of wave equations in terms of data can be ob-

tained in one space dimension but not in higher space dimensions (See e.g. the

discussion in [12], p. 14.)

Pointwise estimates for hyperbolic equations in one space dimension can be

obtained using the method of characteristics. This means that in fact ordinary

differential equations appear once again but this time they are integrated not at

a constant value of the spatial or time variable but along characteristics. This

method will be applied in the following, the characteristics in this case being

null curves of the spacetime geometry.

It should be mentioned that there are global existence results in the literature

where the Einstein-Vlasov system is considered in a context where hyperbolic

equations play an important role. In fact if we relax the assumptions from sur-

face symmetry, where there are three local Killing vectors, to the case where

there are only two local spacelike Killing vectors, then hyperbolic equations nec-

essarily occur. Some relevant papers are [20], [4], [32]. In those references no

direct local existence proof was given. Instead an indirect argument was used.

First a known local existence theorem for the Einstein-Vlasov equation without

symmetry was quoted. Then it was shown that coordinates could be introduced

which are well-adapted to making use of the symmetry when proceeding to ob-

tain global results. Apart from the methodological interest of having a direct

local existence proof, the direct proof gives stronger results concerning the dif-

ferentiability required of the initial data and obtained for the solutions. This is

very difficult to control in the indirect method and for this reason the latter has

only been applied to the case where everything is of infinite differentiability.

The inclusion of a scalar field can be seen as a step towards certain ques-

tions of physical interest. In recent years cosmological models with accelerated

expansion have become a very active research topic in response to new astro-

nomical observations [25]. The easiest way to obtain models with accelerated

expansion is to introduce a positive cosmological constant, a possibility studied

mathematically in [28], [30] and [13]. A more sophisticated way is to introduce

a scalar field with potential (see [22], section 4.3., [23], [14]). We treat only

the case of a linear scalar field but it is likely that the approach developed here

will be useful in the nonlinear case. Scalar fields also play a role in theories of

gravity generalizing Einstein’s theory, such as the Jordan-Brans-Dicke theory.

In that case, in contrast to the one considered here, there is a direct coupling be-

tween the scalar field and the distribution function. The techniques developed

here could serve as a first step towards the study of these more complicated

situations, which have hardly been looked at mathematically yet. (See however

[2] and [5] where the coupling of the Vlasov equation to a scalar field of the

Jordan-Brans-Dicke type in the absence of Einstein gravity is considered).

A large part of our investigation will focus on the initial value problem

for the Einstein-Vlasov-scalar field system with surface symmetry. In the first

chapter, we split the wave equation in φinto a system of two partial differential

equations of first order. This permits us to bound the derivatives of φby the

solutions of the field equations and to introduce an auxiliary system. Next, we

solve each equation of the auxiliary system when the other unknowns are fixed.

Using the results of [15], and under some constraints on the initial data, the full

system is equivalent to the auxiliary system and is reduced to a subsystem. This

chapter ends with the solvability of the constraint equation on data. By iterating

the solutions of the auxiliary system, local in time existence and uniqueness of

solutions in both time directions and continuation criteria are established in the

second chapter.

The third chapter concentrated on the contracting direction. Solutions of the

Einstein-Vlasov-scalar field system with spherical, plane and hyperbolic sym-

metry exist on the whole interval ]0,1] for general initial data. The proof is

based on a change of variables inspired by [32] where the existence up to t= 0

for a certain class of T2-symmetric solutions of the Einstein-Vlasov system with

vanishing cosmological constant was studied. The structure of the initial sin-

gularity is analyzed as in [15]. We show that the spacetime has a curvature

singularity. Moreover the singularity is crushing for any solution in ]0,1]. An

idea from [24] which studied the singularity for solutions of the Einstein equa-

tions in Gowdy spacetimes, allows us to prove in the special case of the spherical,

plane and hyperbolic symmetric Einstein-scalar field system that the singularity

is velocity-dominated.

In the expanding direction in chapter four, a global existence result in the

cases of plane and hyperbolic symmetry is obtained. But this fails in the spher-

ical case. The spacetime is future geodesically complete in the special case of

plane symmetry without Vlasov contribution. The same result holds for spa-

tially homogeneous spacetimes which are solutions of the full system.

The present work is oganized as follows : chapter 1 is concerned with the

formulation of the surface symmetric Einstein-Vlasov-scalar field system written

in areal coordinates and the proof of some preliminary results. The results of

chapter 1 are used to obtain in chapter 2, local existence theorems with contin-

uation criteria in both time directions. Chapter 3 focuses on the existence of

solutions up to t= 0 and their behaviour near the initial singularity. In chapter

4, we prove a global existence result in the future and geodesic completeness.

Chapter 1

Equations and preliminary

results

1.1 Equations

Let us recall the formulation of the Einstein-Vlasov-scalar field system ; for the

moment we do not assume any symmetry of the spacetime.

We consider a four-dimensional spacetime manifold M, with local coordi-

nates (xα) = (t, xi) on which x0=tdenotes the time and (xi) the space

coordinates. Unless otherwise specified in what follows Greek indices always

run from 0 to 3, and Latin ones from 1 to 3. On M, a Lorentzian metric gis

given with signature (−,+,+,+). The metric is assumed to be time-orientable,

i.e. that the two halves of the light cone at each point of Mcan be labelled

past and future in a way which varies continuously from point to point. With

this global direction of time, it is possible to distinguish between future-pointing

and past-pointing timelike vectors. The worldline of a particle of non-zero rest

mass mis a timelike curve in spacetime. The unit future-pointing tangent vec-

tor to this curve is the four-velocity vαof the particle. Its four-momentum pα

is given by mvα. Here we assume that all particles have the same mass m,

normalized to unity and no distinction need be made between four-velocity and

four-momentum. There is also the possibility of considering massless particles,

whose worldlines are null curves. In the case m= 1 the possible values of the

four-momentum are precisely all future-pointing unit timelike vectors. These

form a hypersurface (seven-dimensional submanifold)

PM := {gαβpαpβ=−1, p0>0},

in the tangent bundle TM called the mass shell and coordinatized by (t, xi, pi).

If the coordinates are such that the components g0ivanish then the component

p0is expressed by the other coordinates via

p0=p−g00q1 + gijpipj.

The distribution function f, which represents the density of particles with given

spacetime position and four-momentum, is a non-negative real-valued function

on PM. A basic postulate in general relativity is that a free particle travels

along a geodesic. Consider a future-directed timelike geodesic parameterized

by proper time. Then its tangent vector at any time is future-pointing unit

timelike. Thus this geodesic has a natural lift to a curve on PM, by taking its

position and tangent vector together. This defines a flow on PM. Denote the

vector field which generates this flow by X. The condition that frepresents

the distribution of a collection of particles moving freely in the given spacetime

is that it should be constant along the flow, i.e. that Xf = 0. This is the

Vlasov equation. In addition we consider a scalar field φwhich is a real-valued

function on M. The Vlasov equation can be coupled to the Einstein-scalar field

equations, giving rise to the Einstein-Vlasov-scalar field system. The unknowns

are a 4-manifold, a (time orientable) Lorentz metric gon M, a non-negative

real-valued function fon the mass shell defined by gand a real-valued function

φon M. The Einstein-Vlasov-scalar field system now reads:

∂tf+pi

p0∂xif−1

p0Γi

βγpβpγ∂pif= 0

Gαβ = 8πTαβ

Tαβ =−ZR3

fpαpβ|g|1

2dp1dp2dp3

+ (∇αφ∇βφ−1

2gαβ∇νφ∇νφ)

where pα=gαβpβ,|g|denotes the modulus of determinant of the metric gαβ,

Γλ

αβ the Christoffel symbols,

Gαβ := Rαβ −1

2gαβR

the Einstein tensor, and Tαβ the energy-momentum tensor. Ris the scalar

curvature of gand

Rαβ =Rνα,νβ =∂νΓν

αβ −∂βΓν

αν + Γν

νρΓρ

αβ −Γν

ρβΓρ

αν

the Ricci tensor.

Lemma 1.1 For any scalar field φof class C2defined on M, we have :

∇α(∇αφ∇βφ−1

2gαβ∇νφ∇νφ) = gφ∇βφ

Proof: We have, using the properties of g:

∇α(∇αφ∇βφ) = ∇α(∇αφ)∇βφ+∇αφ∇α∇βφ

=gφ∇βφ+∇αφ∇α∇βφ

gαβ∇α(∇νφ∇νφ) = gαβ∇α(∇νφ)∇νφ+gαβ∇νφ(∇α∇νφ)

= (∇β∇νφ)∇νφ+∇νφ(∇β∇νφ)

=∇νφ∇β∇νφ+gλν∇λφ(∇βgνη∇ηφ)

=∇νφ∇β∇νφ+δη

λ∇λφ∇β∇ηφ

=∇νφ∇β∇νφ+∇λφ∇β∇λφ

= 2∇νφ∇β∇νφ

Therefore

∇α(∇αφ∇βφ−1

2gαβ∇νφ∇νφ) = gφ∇βφ+∇αφ∇α∇βφ−∇νφ∇β∇νφ=gφ∇βφ.

Remark 1.2 Due to the Bianchi Identities, the Einstein equations imply the

conservation law ∇αTαβ = 0. Now since the contribution of fto the energy-

momentum tensor is divergence-free [8], we deduce that:

∇α(∇αφ∇βφ−1

2gαβ∇νφ∇νφ) = 0

which is equivalent to

gφ∇βφ= 0

i.e gφ= 0 or ∇βφ= 0. Consider the open set S={(t, r)|gφ(t, r)6= 0}.

Suppose that Sis non- empty. On S,∇βφ= 0 then ∇β∇βφ= 0 i.e gφ= 0,

which is a contradiction. Therefore Sis empty. This means that

gφ= 0,

which is the wave equation for φ.

Remark 1.3 The Vlasov equation in a fixed spacetime is a linear hyperbolic

equation for a scalar function and hence solving it is equivalent to solving the

equations for its characteristics. In coordinate components these are :

(dXi

ds =Pi

dP i

ds =−Γi

βγPβPγ

Let Xi(s, xα, pi),Pi(s, xα, pi)be the unique solution of the previous system with

initial conditions Xi(t0, xα, pi) = xiand Pi(t0, xα, pi) = pi. Then the solution

of the Vlasov equation can be written as :

f(xα, pi) = f0(Xi(t0, xα, pi), Pi(t0, xα, pi))

where f0is the restriction of fto the hypersurface t=t0. This function f0

serves as initial datum for the Vlasov equation.

In [18], a definition of spacetimes with spherical, plane and hyperbolic sym-

metry was given. The spacetime (M, g) is topologically of the form ]0,∞[×S1×

S, where Sis a 2-sphere, a 2-torus, or a hyperbolic plane, in the case of spherical,

plane or hyperbolic symmetry respectively. We now consider a solution of the

Einstein-Vlasov-scalar field system where all unknowns are invariant under one

of these symmetries and write the Einstein-Vlasov system in areal coordinates.

The circumstances under which coordinates of this type exist are discussed in

[3]. The metric gtakes the form

ds2=−e2µ(t,r)dt2+e2λ(t,r)dr2+t2(dθ2+ sin2

kθdϕ2) (1.1)

where

sinkθ=









sin θfor k= 1 (spherical symmetry);

1 for k= 0 (plane symmetry);

sinh θfor k=−1 (hyperbolic symmetry)

t > 0 denotes a time-like coordinate, r∈Rand (θ, ϕ) range respectively in the

domains [0, π]×[0,2π], [0,2π]×[0,2π], [0,∞[×[0,2π] respectively, and stand for

angular coordinates. The functions λand µare periodic in rwith period 1. It

has been shown in [15] and [16] that due to the symmetry, fcan be written as

a function of

t, r, w := eλp1and F:= t4[(p2)2+ sin2

kθ(p3)2],

i.e. f=f(t, r, w, F). In these variables, we have p0=e−µp1 + w2+F/t2.

The scalar field is a function of tand rwhich is periodic in rwith period 1.

We denote by a dot and by a prime the derivatives of the metric components

and of the scalar field with respect to tand rrespectively. We specify the

regularity properties which we require.

Definition 1.4 Let Ij]0,∞[be an interval and (t, r)∈I×R.

a) f∈C1(I×R2×[0,∞[) is regular if f(t, r + 1, w, F)=f(t, r, w, F)for

(t, r, w, F)∈I×R2×[0,∞[,f≥0and suppf(t, r, ., .)is compact uniformly in

rand locally uniformly in t.

b) µ∈C1(I×R)is regular, if µ0∈C1(I×R)and µ(t, r + 1) =µ(t, r).

c) λ∈C1(I×R)is regular, if ˙

λ∈C1(I×R)and λ(t, r + 1) =λ(t, r).

d) eµ(or φ1,φ2)∈C1(I×R)is regular, if eµ(t, r + 1) =eµ(t, r).

e) ρ(or p,j,q)∈C1(I×R)is regular, if ρ(t, r + 1) =ρ(t, r).

f) φ∈C2(I×R)is regular, if φ(t, r + 1) =φ(t, r).

Lemma 1.5 Let f=f(t, r, w, F )be regular on I×R2×[0,∞[,Ij]0,∞[an

interval; λand µregular on I×R. Then the nontrivial components of the

energy-momentum tensor are:

T00(t, r) = e2µπ

t2Z+∞

−∞ Z+∞

0p1 + w2+F/t2f(t, r, w, F)dFdw+1

2(˙

φ2+e2(µ−λ)φ02)

T11(t, r) = e2λπ

t2Z+∞

−∞ Z+∞

p1 + w2+F/t2f(t, r, w, F)dFdw+1

2(e2(λ−µ)˙

φ2+φ02)

T01(t, r) = −eλ+µπ

t2Z+∞

−∞ Z+∞

wf(t, r, w, F)dFdw +˙

φφ0

T22(t, r) = π

2t2Z∞

−∞ Z∞

p1 + w2+F/t2f(t, r, w, F)dFdw+1

2t2(e−2µ˙

φ2−e−2λφ02)

T33(t, r, θ) = T22(t, r) sin2

kθ.

Proof: Set Tαβ =Tf

αβ +Tφ

αβ where Tf

αβ and Tφ

αβ are respectively the contri-

bution of fand φto the energy-momentum tensor. Concerning the calculation

of components Tf

αβ, we refer to [15]. We calculate only the components

Tφ

αβ := ∇αφ∇βφ−1

2gαβ∇νφ∇νφ. We have

∇νφ∇νφ=gνα∇νφ∇αφ=g00(∇0φ)2+g11(∇1φ)2=−e−2µ˙

φ2+e−2λφ02. Thus

Tφ

00 =˙

φ2−1

2g00∇νφ∇νφ=1

2(˙

φ2+e2(µ−λ)φ02)

Tφ

01 =∇0φ∇1φ=˙

φφ0

Tφ

11 =φ02−1

2g11∇νφ∇νφ=1

2(e2(λ−µ)˙

φ2+φ02)

Tφ

22 =−1

2g22∇νφ∇νφ=1

2t2(e−2µ˙

φ2−e−2λφ02)

Tφ

33 =Tφ

22 sin2

kθ.

The remaining components being zero.

Following [15], we can now write the complete Einstein-Vlasov-scalar field

system as :

∂tf+eµ−λw

p1 + w2+F/t2∂rf−(˙

λw +eµ−λµ0p1 + w2+F/t2)∂wf= 0 (1.2)

e−2µ(2t˙

λ+ 1) + k= 8πt2ρ(1.3)

e−2µ(2t˙µ−1) −k= 8πt2p(1.4)

µ0=−4πteλ+µj(1.5)

e−2λ(µ00 +µ0(µ0−λ0)) −e−2µ(¨

λ+ ( ˙

λ+1

t)( ˙

λ−˙µ)) = 4πq (1.6)

e−2λφ00 −e−2µ¨

φ−e−2µ(˙

λ−˙µ+2

t)˙

φ−e−2λ(λ0−µ0)φ0= 0 (1.7)

where (1.7) is the wave equation in φand :

ρ(t, r) = e−2µT00(t, r) = π

t2Z+∞

−∞ Z+∞

0p1 + w2+F/t2f(t, r, w, F)dFdw

2(e−2µ˙

φ2+e−2λφ02)

(1.8)

p(t, r) = e−2λT11(t, r) = π

t2Z+∞

−∞ Z+∞

p1 + w2+F/t2f(t, r, w, F)dFdw

2(e−2µ˙

φ2+e−2λφ02)

(1.9)

j(t, r) = −e−(λ+µ)T01(t, r) = π

t2Z+∞

−∞ Z+∞

wf(t, r, w, F)dFdw −e−(λ+µ)˙

φφ0

(1.10)

q(t, r) = 2

t2T22(t, r) = 2

t2sin2

kθT33(t, r, θ)

=π

t4Z∞

−∞ Z∞

p1 + w2+F/t2f(t, r, w, F)dFdw +e−2µ˙

φ2−e−2λφ02

(1.11)

We are going to study the initial value problem corresponding to this system

with unknowns f,λ,µ,φand prescribe initial data at time t= 1:

f(1, r, w, F) = ◦

f(r, w, F), λ(1, r) = ◦

λ(r), µ(1, r) = ◦

µ(r),

φ(1, r) = ◦

φ(r),˙

φ(1, r) = ψ(r)

The choice t= 1 is made only for convenience. Analogous results hold in the

case that data are prescribed on any hypersurface t=t0>0.

1.2 Auxiliary system and preliminary results

Using characteristic derivatives, we show in this section that the first and second

derivatives of φcan be bounded in terms of λand µ.

Lemma 1.6 Let D+=e−µ∂t+e−λ∂r;D−=e−µ∂t−e−λ∂r;

X=˙

φe−µ−φ0e−λ;Y=˙

φe−µ+φ0e−λ;

a= (−˙

λ−1

t)e−µ−µ0e−λ;b=−e−µ

t;c= (−˙

λ−1

t)e−µ+µ0e−λ

then X,Yare solutions of the system

D+X=aX +bY (1.12)

D−Y=bX +cY (1.13)

Proof: We have

D+X= (e−µ∂t+e−λ∂r)( ˙

φe−µ−φ0e−λ)

=e−µ(−˙µ˙

φe−µ+¨

φe−µ)−e−µ(−˙

λφ0e−λ+˙

φ0e−λ)

+e−λ(−µ0˙

φe−µ+˙

φ0e−µ)−e−λ(−λ0φ0e−λ+φ00e−λ)

=¨

φe−2µ−φ00e−2λ−˙µ˙

φe−2µ+λ0φ0e−2λ+ ( ˙

λφ0−µ0˙

φ)e−µ−λ

¿From (1.7), we deduce that

D+X= (µ0φ0−λ0φ0)e−2λ+ ( ˙µ˙

φ−˙

λ˙

φ)e−2µ

−2

t˙

φe−2µ−˙µ˙

φe−2µ+λ0φ0e−2λ+ ( ˙

λφ0−µ0˙

φ)e−µ−λ

=−˙

λ˙

φe−2µ+ ( ˙

λφ0−µ0˙

φ)e−µ−λ−2

t˙

φe−2µ

= [(−˙

λ−1

t)e−µ−µ0e−λ]( ˙

φe−µ−φ0e−λ)−1

t(˙

φe−µ+φ0e−λ)

=aX +bY

If we replace, rby −r(i.e ∂rby −∂r), the wave equation is invariant, Xand Y,

aand c,D+and D−interchange respectively; and we can write the equation

with D−to obtain (1.13)..

The full system above is overdetermined and we will show that a solution

(f, λ, µ, φ) of the subsystem consisting of equations (1.2), (1.3), (1.4) and (1.7)

solves the remaining equations (1.5) and (1.6). Notice that such a solution

determines the right hand side of (1.5) which is then a given function say ˜µ.

Then, since (1.4) already provides µ, an idea introduced in [15], and that we

will follow here, is to replace µ0in (1.2) and (1.7) by an auxiliary function ˜µ,

which is not assumed a priori to be a derivative, and prove later that, under

certain conditions, ˜µis nothing else than µ0. We then introduce the following

auxiliary system obtained by coupling (1.3)-(1.4) to the equations obtained by

replacing µ0by ˜µin (1.2),(1.5) and (1.12)-(1.13), i.e

∂tf+eµ−λw

p1 + w2+F/t2∂rf−(˙

λw +eµ−λ˜µp1 + w2+F/t2)∂wf= 0 (1.14)

˜µ=−4πteλ+µj(1.15)

and

D+X= ˜aX +bY (1.16)

D−Y=bX + ˜cY (1.17)

Where µ0is substituted by eµin aand cto obtain eaand ecrespectively.

Let us estimate the first and second order derivatives of the scalar field φ,

using the characteristic curves of (1.7) and system (1.16)-(1.17).

Proposition 1.7 Let

K0= 2 sup{| ψ(r)|e−◦

µ(r)+|◦

φ0(r)|e−

◦

λ(r);r∈R}

m(t) = sup{| ˙

λ(t, r)|+2

t+|˜µ(t, r)|e(µ−λ)(t,r);r∈R}

K(t) = sup{(|X|2+|Y|2)1

2(t, r) ; r∈R}.

If (X, Y )is a solution of (1.16)-(1.17) with

X(1) = e−◦

µ(r)ψ(r)−e−

◦

λ(r)◦

φ0(r)

and

Y(1) = e−◦

µ(r)ψ(r) + e−

◦

λ(r)◦

φ0(r)

then:

1) If t∈]T, 1],T≥0, we have

K(t)≤K0+ 2 Z1

m(s)K(s)ds (1.18)

2) If t≥1the analogous estimate holds with the limits tand 1exchanged in the

integral in (1.18).

Proof: The characteristic curves (t, γi), i= 1,2 of the second order partial

differential equation (1.7) satisfy g(u, u) = 0 with u= (1,˙γi,0,0). Then,

−e2µ+e2λ˙γ2

i= 0 and ˙γi=±eµ−λ. Therefore, for any function f,

dt (t, γi(t)) = ∂tf+ ˙γi∂rf=∂tf±eµ−λ∂rf

eµ(e−µ∂tf±e−λ∂rf) = eµD+for eµD−f

and we have D+=D−=e−µd

dt on the corresponding characteristic. Then

(1.16)-(1.17) become

dt X(t, γ1(t)) = eµ(˜aX +bY )(t, γ1(t))

dt Y(t, γ2(t)) = eµ(bX + ˜cY )(t, γ2(t))

Integrate this system on [t, 1], thus

(X(t, γ1(t)) = X(1, γ1(1)) −R1

teµ(s,γ2(s))(˜aX +bY )(s, γ1(s))ds

Y(t, γ2(t)) = Y(1, γ2(1)) −R1

teµ(s,γ2(s))(bX + ˜cY )(s, γ2(s))ds

Now take the absolute value in each equation and add the two inequalities to

obtain:

|X(t, γ1(t))|+|Y(t, γ2(t))| ≤ |X(1, γ1(1))|+|Y(1, γ2(1))|

+Z1

eµ(s,γ1(s))(|˜a||X|+|b||Y|)(s, γ1(s))ds +Z1

eµ(s,γ2(s))(|b||X|+|˜c||Y|)(s, γ2(s))ds

≤ |X(1, γ1(1))|+|Y(1, γ2(1))|+Z1

eµ(s,γ1(s))[(|˜a|+|b|)(|X|+|Y|)](s, γ1(s))ds

+Z1

eµ(s,γ2(s))[(|b|+|˜c|)(|X|+|Y|)](s, γ2(s))ds

Take the supremum in space of each term of the above inequality. Now, we have

|X(1, γ1(1))|+|Y(1, γ2(1))| ≤ K0,

(X+Y= 2 ˙

φe−µ

Y−X= 2φ0e−λ

thus (2|˙

φ|e−µ≤ |X|+|Y|

2|φ0|e−λ≤ |X|+|Y|

which implies

4(e−2µ˙

φ2+e−2λφ02)≤2(|X|+|Y|)2, i.e K(s)≤√2

2sup{(|X|+|Y|)(s, r); r∈R}.

Since, (|X|+|Y|)2≤2(|X|2+|Y|2), we have

sup{(|X|+|Y|)(s, r); r∈R} ≤ 2K(s).

And (1.18) follows. 

Corollary 1.8 If gφ=Fwhere Fis a continuous function of variables t

and r, then we have the inequality

K(t)≤K0+ 2 Z1

[m(s)K(s) + sup{eµ(s,r)|F(s, r)|;r∈R}]ds

Lemma 1.9 Let D+and D−be defined as in Lemma 1.6 and define

X1=e−λ∂rX,Y1=e−λ∂rY;b1= (−2˙

λ−1

t)e−µ−(˜µ+µ0)e−λ;

b2=−e−µ

t;b3=−˙

λ0e−µ−λ+ (λ0˜µ−˜µµ0−˜µ0)e−2λ;

b4= (−2˙

λ−1

t)e−µ+ (˜µ+µ0)e−λ;b5=−˙

λ0e−µ−λ−(λ0˜µ−˜µµ0−˜µ0)e−2λ;

If Xand Ysatisfy (1.16) and (1.17) then X1and Y1satisfy

D+X1=b1X1+b2Y1+b3X(1.19)

D−Y1=b2X1+b4Y1+b5Y(1.20)

Proof: we have by definition,

D+X1= (∂rX)D+e−λ+e−λD+(∂rX) (1.21)

(∂rX)D+e−λ= (∂rX)(e−µ∂t+e−λ∂r)e−λ= (e−λ∂rX)(−˙

λe−µ−λ0e−λ)

= (−˙

λe−µ−λ0e−λ)X1

(1.22)

∂rD+=∂r(e−µ∂t+e−λ∂r) = D+∂r−µ0e−µ∂t−λ0e−λ∂r,

i.e

∂r(D+X) = D+(∂rX)−(µ0e−µ∂t+λ0e−λ∂r)X,

and then

D+(∂rX) = ∂r(D+X)+(µ0e−µ∂t+λ0e−λ∂r)X(1.23)

Now we have firstly,

(µ0e−µ∂t+λ0e−λ∂r)X=1

2(µ0+λ0)D+X+1

2(µ0−λ0)D−X

2(µ0+λ0)D+X+1

2(µ0−λ0)(D+X−2X1)

=µ0D+X−(µ0−λ0)X1

i.e

(µ0e−µ∂t+λ0e−λ∂r)X=µ0(˜aX +bY )−(µ0−λ0)X1(1.24)

and secondly,

∂r(D+X) = (∂r˜a)X+ ˜a∂rX+ (∂rb)Y+b∂rY

= (∂r˜a)X+ ˜aeλX1+ (∂rb)Y+beλY1

i.e

∂r(D+)X= ˜aeλX1+beλY1+[−˙

λ0e−µ+µ0(˙

λ+1

t)e−µ−˜µ0e−λ+˜µλ0e−λ]X+µ0

te−µY

(1.25)

Substituting (1.25) and (1.24) in (1.23) gives

D+(∂rX) = ˜aeλX1+beλY1+ [−˙

λ0e−µ+µ0(˙

λ+1

t)e−µ−˜µ0e−λ+ ˜µλ0e−λ]X

+µ0

te−µY+µ0(˜aX +bY )−(µ0−λ0)X1

= ˜aeλX1+beλY1+ [−˙

λ0e−µ+µ0(˙

λ+1

t)e−µ−˜µ0e−λ+ ˜µλ0e−λ]X

+µ0

te−µY+ [(−µ0˙

λ−µ0

t)e−µ−˜µµ0e−λ]X−µ0

te−µY−(µ0−λ0)X1

i.e

D+(∂rX) = (˜aeλ−µ0+λ0)X1+beλY1+ (−˙

λ0e−µ−˜µµ0e−λ−˜µ0e−λ+ ˜µλ0e−λ)X

(1.26)

Substituting (1.26) and (1.22) in (1.21) gives

D+X1= (−˙

λe−µ−λ0e−λ)X1+ [˜a+ (−µ0+λ0)e−λ]X1+bY1+ (−˙

λ0e−µ−˜µµ0e−λ

−˜µ0e−λ+ ˜µλ0e−λ)e−λX

= (−˙

λe−µ+ ˜a−µ0e−λ)X1+bY1+ [−˙

λ0e−µ−λ+ (−˜µµ0−˜µ0+ ˜µλ0)e−2λ]X

and equation (1.19) follows. If we replace e−λby −e−λ, the wave equation

is invariant, D+and D−,X1and −Y1,b1and b4,−b3and b5interchange

respectively; and we can write equation (1.20).

Proposition 1.10 Let K(t)be defined as in Proposition 1.7 and set:

A0= 2 sup{[(|ψ0|+|◦

µ0|| ψ|)e−◦

µ−

◦

λ+ (|◦

φ00 |+|◦

λ0|| ◦

φ0|)e−2

◦

λ](r) ; r∈R}

A(t) = sup{[X2

1+Y2

1]1/2(t, r) ; r∈R}

v(t) = sup{2

t+ 2 |˙

λ|+(|˜µ|+|µ0|)eµ−λ(t, r) ; r∈R}

h(t) = sup{[|˙

λ0|e−λ+ (|µ0|| ˜µ|+|λ0|| ˜µ|+|˜

µ0|)eµ−2λ](t, r) ; r∈R}

If in addition to the assumptions of Proposition 1.7 the quantities X1and Y1

satisfy (1.19) and (1.20) and agree with e−λ∂rXand e−λ∂rYrespectively for

t= 1 then

1) If t∈]T, 1],T≥0, we have the estimate

A(t)≤A0+ 2 Z1

(v(s)A(s) + h(s)K(s))ds (1.27)

2) If t≥1the analogous estimate holds with the limits tand 1exchanged in

the integral in (1.27).

Proof: Analogous to the proof of Proposition 1.7, using this time Lemma

1.9.

Note that the factor e−λin the definition of X1and Y1is very important.

Without it the above derivation would not work since the derivative λ0would

occur in b1and b4.

Lemma 1.11 Let D+and D−be defined as in lemma 1.6 and define

X2=eµX,Y2=eµY. If Xand Yhold system (1.12)-(1.13) and the field

equations (1.3)-(1.4) are satisfied, then X2and Y2satisfy

D+X2=eµ[k

t−4πt(ρ−p)]X2−e−µ

tY2(1.28)

D−Y2=−e−µ

tX2+eµ[k

t−4πt(ρ−p)]Y2(1.29)

Proof: We have

D+X2=D+(eµX) = eµD+X+ (D+eµ)X

=eµ[(−˙

λ−1

t)e−µ−µ0e−λ]X−Y

t+ ( ˙µ+µ0eµ−λ)X

= ( ˙µ−˙

λ−1

t)X−Y

=e−µ( ˙µ−˙

λ−1

t)X2−e−µ

tY2

Subtract the field equations (1.3) and (1.4) to obtain

2te−2µ(˙

λ−˙µ)+2k−2e−2µ= 8πt2(ρ−p)

i.e

˙µ−˙

λ=ke2µ+ 1

t−4πte2µ(ρ−p).

Substituting this into the above equation gives (1.28). If we replace ∂rby −∂r,

the wave equation is invariant, D+and D−,X2and Y2interchange respectively

and we can write equation (1.29).

Remark 1.12 Let

B0= 2 sup{(|X2|+|Y2|)(1, r) ; r∈R}

B(t) = sup{(|X2|2+|Y2|2)1/2(t, r) ; r∈R}

l(t) = sup{1

t+e2µ[|k|

t+ 4πt(ρ−p)](t, r) ; r∈R}

If (X2, Y2)is solution of (1.28)-(1.29), then we obtain analogously to (1.18),

the estimate

B(t)≤B0+ 2 Z1

l(s)B(s)ds (1.30)

with t∈]T, 1],T > 0or t≥1.

1.3 The reduced system

We first solve each equation of the auxiliary system introduced in the previous

section, when the other unknowns are fixed (in the form to be used later for the

iteration). In order to clarify our statements, we introduce the notations φ1,φ2

in place of ˙

φ,φ0.

Proposition 1.13 1) Let ¯

f,¯

λ,¯µ,˜µ,¯

φ1,¯

φ2be regular for (t, r)∈I×R,

I⊂]0,∞[. Substitute f,λ,µ,˙

φ,φ0respectively by ¯

f,¯

λ,¯µ,¯

φ1,¯

φ2in ρand p

to define ¯ρand ¯p. Suppose that 1∈I,◦

f∈C1(R2×[0,∞)),◦

λ,◦

µ∈C1(R)and

are periodic of period 1in r. Assume that:

e−2◦

µ(r)+k

t−k+8π

tZ1

s2¯p(s, r)ds > 0,(t, r)∈I×R(1.31)

then the system

∂tf+e¯µ−¯

λw

p1 + w2+F/t2∂rf−(˙

λw +e¯µ−¯

λ˜µp1 + w2+F/t2)∂wf= 0 (1.32)

e−2µ(2t˙

λ+ 1) + k= 8πt2¯ρ(1.33)

e−2µ(2t˙µ−1) −k= 8πt2¯p(1.34)

has a unique regular solution (f, λ, µ)on I×Rwith f(1) = ◦

f,λ(1) = ◦

λ,

µ(1) = ◦

µ. This solution is given by

f(t, r, w, F) = ◦

f((R, W)(1, t, r, w, F), F ) (1.35)

where (R, W )is the solution of the characteristic system

ds(r, w) = ( e¯µ−¯

λw

p1 + w2+F/t2,−˙

λw −e¯µ−¯

λ˜µp1 + w2+F/t2) (1.36)

satisfying (R, W)(t, t, r, w, F) = (r, w);

e−2µ(t,r)=e−2◦

µ(r)+k

t−k+8π

tZ1

s2¯p(s, r)ds (1.37)

λ(t, r) = 4πte2µ¯ρ(t, r)−1 + ke2µ(t, r)

2t(1.38)

λ(t, r) = ◦

λ(r)−Z1

λ(s, r)ds (1.39)

If I=]T, 1] (respectively I= [1, T[) with T∈[0,1[ (respectively T∈]1,∞[), then

there exists some T∗∈[T, 1] (respectively T∗∈]1, T]) such that condition (1.31)

holds on ]T∗,1]×R(respectively [1, T∗[×R). T∗is independent of ¯pif I=]T, 1],

whereas it depends on ¯pif I= [1, T [.

2) Let λ,µ,¯

λ,¯µbe regular; let C,Dbe regular as ˜µ(see definition 1.4).

Set

X=φ1e−µ−φ2e−λ, Y =φ1e−µ+φ2e−λ(1.40)

and define the operators ¯

D+,¯

D−as D+,D−in Lemma 1.6, with λ,µsubsti-

tuted respectively by ¯

λ,¯µ. Assume that ψ∈C1(R),◦

φ∈C2(R)are periodic of

period 1. Then the system ¯

D+X=C(1.41)

D−Y=D(1.42)

has a unique regular solution (φ1, φ2) such that (φ1, φ2)(1) = (ψ, ◦

φ0).

Proof: 1) the proof of this point is the same as that of Proposition 2.4 in [15].

The only thing added is the existence of T∗we now prove and that will replace

in the case of local existence, the hypothesis ◦

µ≤0 if k=−1 in [15].

If I=]T, 1]; since ¯p≥0, the left hand side of (1.31) is bounded from below by

h(t, r) = e−2◦

µ(r)+k

t−k. If k∈ {0,1}, we have, since 1

t≥1, h(t, r)≥e−2◦

µ(r)>0

and we can take T∗=T. If k=−1, since ◦

µis bounded, there exists β > 0,

such that h(1, r) = e−2◦

µ(r)> β. By the continuity of t7→ h(t, r) at t= 1, we

conclude that:

∃T∗∈]T, 1] such that e−2µ(t,r)≥h(t, r)>β

2, t ∈]T∗,1] and ¯p≥0.(1.43)

If I= [1, T[, define h(t, r) to be all the left hand side of (1.31) and proceed as

above in the case k=−1, to obtain T∗that depends this time on ¯p.

2) The system (1.41)-(1.42) in (X, Y ) is a first order linear hyperbolic sys-

tem, and the existence of a unique solution with the prescribed data (X, Y )(1) =

(e−◦

µψ−e−

◦

λ◦

φ0, e−◦

µψ+e−

◦

λ◦

φ0) is given by a theorem of Friedrichs [10]. (See also

the paper of Douglis [9] where existence of C1solutions of hyperbolic systems

in one space dimension was proved for C1initial data in the more general quasi-

linear case.) We then deduce from the relations (1.40) that define a bijection

(X, Y )7→ (φ1, φ2), the existence of a unique regular solution (φ1, φ2) of (1.41)-

(1.42) such that (φ1, φ2)(1) = (ψ, ◦

φ0). This completes the proof of Proposition

1.13.

In fact, since the equations for Xand Yare decoupled, it is not necessary

to use existence results for hyperbolic systems in the above proof; it suffices to

solve a parameter-dependent ordinary differential equation. The above proce-

dure has the advantage that it can easily be generalized to problems where the

corresponding equations are coupled as they are, for instance, in the case of a

nonlinear scalar field.

Now we show that the solution of the subsystem consisting of equations

(1.2), (1.3), (1.4) and (1.7) also satisfies equations (1.5) and (1.6), and that the

auxiliary system is equivalent to the full system.

Proposition 1.14 1) Let (f, λ, µ, φ)be a regular solution of (1.2), (1.3), (1.4)

and (1.7) on some time interval I∈]0,∞[with 1∈I, and let the initial data

satisfy (1.5) for t= 1 with ◦

φ, ψ ∈C1(R), in particular ◦

µ∈C2(R). Then (1.5)

and (1.6) hold for all t∈I, in particular µ∈C2(I×R).

2) Let (f, λ, µ, ˜µ, φ1, φ2)be a regular solution of equations (1.14),(1.3), (1.4),

(1.15), (1.16) and (1.17) with the initial data (f, λ, µ, φ1, φ2)(1) = (◦

f, ◦

λ, ◦

µ, ψ, ◦

φ0)

that are as in proposition 1.13 and satisfy (1.5) for t= 1. Then there exists a

unique regular function φsatisfying φ(1) = ◦

φ,˙

φ(1) = ψsuch that (f, λ, µ, φ)

solves the full system (1.2)-(1.11). The function φis given by:

φ(t, r) = ◦

φ(r) + Rt

1φ1(s, r)ds.

Proof: 1) Firstly, we prove that (1.5) holds. ¿From equations (1.9), (1.2), (1.7)

and integration by parts, it follows that, since supp f(t, r, ., .) is compact :

p0(s, r)s2ds =πZt

1Z∞

−∞ Z∞

p1 + w2+F/s2∂rf(s, r, w, F)dFdwds

+Zt

(−µ0˙

φ2e−2µ−λ0φ02e−2λ+˙

φ0˙

φe−2µ+φ0φ00e−2λ)s2ds

=πZt

1Z∞

−∞ Z∞

[−eλ−µw∂tf+ ( ˙

λw2eλ−µ+µ0wp1 + w2+F/t2)∂wf]dFdwds

+Zt

[−µ0˙

φ2e−2µ−λ0φ02e−2λ+˙

φ0˙

φe−2µ+φ0¨

φe−2µ+ ( ˙

λ−˙µ)φ0˙

φe−2µ

sφ0˙

φe−2µ+ (λ0−µ0)e−2λφ02]s2ds

=πZ∞

−∞ Z∞

[−eλ−µwf]s=t

s=1dFdw +πZt

1Z∞

−∞ Z∞

(˙

λ−˙µ)eλ−µwfdFdwds

+πZt

1Z∞

[( ˙

λw2eλ−µ+µ0wp1 + w2+F/t2)f]∞

−∞dFds

−πZt

1Z∞

−∞ Z∞

(2 ˙

λweλ−µ+µ0p1 + w2+F/t2+µ0w2

p1 + w2+F/s2)fdFdwds

+Zt

[−µ0˙

φ2e−2µ−µ0e−2λφ02+˙

φ0˙

φe−2µ+φ0¨

φe−2µ+ ( ˙

λ−˙µ)φ0˙

φe−2µ]ds +Zt

2sφ0˙

φe−2µds

i.e

p0(s, r)s2ds =πZ∞

−∞ Z∞

[−eλ−µwf]s=t

s=1dFdw −πZt

1Z∞

−∞ Z∞

(˙

λ+ ˙µ)eλ−µwfdFdwds

−πZt

1Z∞

−∞ Z∞

µ0(p1 + w2+F/t2+w2

p1 + w2+F/s2)fdFdwds

+Zt

[−µ0˙

φ2e−2µ−µ0e−2λφ02+˙

φ0˙

φe−2µ+φ0¨

φe−2µ+ ( ˙

λ−˙µ)φ0˙

φe−2µ]ds + [s2φ0˙

φe−2µ]s=t

s=1

−Zt

s2(−2 ˙µφ0˙

φe−2µ+φ0¨

φe−2µ+˙

φ0˙

φe−2µ)ds

=πZ∞

−∞ Z∞

[−eλ−µwf]s=t

s=1dFdw −πZt

1Z∞

−∞ Z∞

(˙

λ+ ˙µ)eλ−µwfdFdwds

−πZt

1Z∞

−∞ Z∞

µ0(p1 + w2+F/t2+w2

p1 + w2+F/s2)fdFdwds

+ [s2φ0˙

φe−2µ]s=t

s=1 +Zt

(˙

λ+ ˙µ)φ0˙

φe−2µ−Zt

µ0(˙

φ2e−2µ+φ02e−2λ)s2ds

Therefore,

p0(s, r)s2ds = [−eλ−µj(s, r)s2]s=t

s=1−Zt

(˙

λ+ ˙µ)eλ−µj(s, r)s2ds−Zt

µ0(ρ+p)(s, r)s2ds

(1.44)

Adding equations (1.3) and (1.4), we have

λ+ ˙µ= 4πte2µ(ρ+p).(1.45)

¿From equation (1.4), we obtain

e−2µ(t,r)=e−2◦

µ(r)+k

t−k+8π

tZ1

s2p(s, r)ds

and differentiating this with respect to r, yields :

tµ0e−2µ=◦

µ0e−2◦

µ+ 4πZt

s2p0(s, r)ds (1.46)

Substituting for the last integral by (1.44), using (1.45) and assuming the fact

that the constraint equation (1.5) holds for t= 1, relation (1.46) implies

−te2µ(µ0+ 4πeλ+µjt) = 4πZt

(ρ+p)s2(µ0+ 4πeλ+µjs)ds

and since the left hand side is zero at t= 1, we obtain

µ0+ 4πteλ+µj= 0

on I, i.e (1.5) holds for all t∈I. In particular this relation shows that µis C2

with respect to rwith

µ00 = (λ0+µ0)µ0−4πteλ+µj0.

Now we prove that equation (1.6) holds. From (1.10), (1.2), (1.7) and integration

by parts we obtain the identity

j0(t, r) = π

t2eλ−µZ∞

−∞ Z∞

[−p1 + w2+F/t2∂tf

+ ( ˙

λwp1 + w2+F/t2+eµ−λµ0(1 + w2+F/t2))∂wf]dFdw

+ (λ0+µ0)e−(λ+µ)˙

φφ0−e−(λ+µ)(˙

φ0φ0+˙

φφ00)

=−π

t2eλ−µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw

+π

t2eλ−µZ∞

[( ˙

λwp1 + w2+F/t2+eµ−λµ0(1 + w2+F/t2))f]∞

−∞

−π

t2eλ−µZ∞

−∞ Z∞

[˙

λp1 + w2+F/t2+˙

λw2

p1 + w2+F/t2+eµ−λµ02w]fdFdw

+ 2µ0e−(λ+µ)˙

φφ0−e−(λ+µ)˙

φ0φ0−eλ−3µ[˙

φ¨

φ+ ( ˙

λ−˙µ+2

t)˙

φ2]

i.e

j0(t, r) = −π

t2eλ−µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw −2µ0π

t2Z∞

−∞ Z∞

wfdFdw

−˙

λeλ−µπ

t2Z∞

−∞ Z∞

(p1 + w2+F/t2+w2

p1 + w2+F/t2)fdFdw

+ 2µ0e−(λ+µ)˙

φφ0−e−(λ+µ)˙

φ0φ0−eλ−3µ[˙

φ¨

φ+ ( ˙

λ−˙µ+2

t)˙

φ2]

=−π

t2eλ−µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw −2µ0π

t2Z∞

−∞ Z∞

wfdFdw

−˙

λeλ−µ(ρ+p−˙

φ2e−2µ

−φ02e−2λ)+2µ0e−(λ+µ)˙

φφ0−e−(λ+µ)˙

φ0φ0−eλ−3µ[˙

φ¨

φ+ ( ˙

λ−˙µ+2

t)˙

φ2]

=−π

t2eλ−µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw −˙

λeλ−µ(ρ+p−˙

φ2e−2µ

−φ02e−2λ)−2µ0j−e−(λ+µ)˙

φ0φ0−e−3µ+λ(˙

φ¨

φ−(˙

λ−˙µ+2

t)˙

φ2).

Since by (1.8)

˙ρ(t, r) = −2π

t3Z∞

−∞ Z∞

0p1 + w2+F/t2fdFdw

+π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw

+π

t2Z∞

−∞ Z∞

(−F

t3)f

p1 + w2+F/t2dFdw

+e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)

=−2ρ

t+1

t(e−2µ˙

φ2+e−2λφ02)−q

t+1

t(e−2µ˙

φ2−e−2λφ02)

+π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw +e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)

=−2ρ

t−q

t+2e−2µ

t˙

φ2+e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)

+π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw ;

¿From (1.3) we obtain

λ(t, r) = 4πte2µρ(t, r)−1 + ke2µ(t, r)

and differentiating with respect to tyields:

λ= 4πe2µρ+ 8πt ˙µe2µρ+1 + ke2µ

2t2+ 4πte2µ˙ρ−k˙µe2µ

= 4πe2µρ+ 2 ˙µ(˙

λ+1 + ke2µ

2t)−˙µ

tke2µ+1 + ke2µ

2t2

+ 4πte2µ{−2ρ

t−q

t+2

te−2µ˙

φ2+e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)}

+ 4π2

te2µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw

=˙µ−˙

t−4πqe2µ+ 2 ˙

λ˙µ+ 8π˙

φ2+ 4πt(−˙µ˙

φ2+˙

φ¨

φ)+4πte2µ−2λ(−˙

λφ02+˙

φ0φ0)

+ 4π2

te2µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw.

Combining all these relations gives :

e−2λ(µ00 +µ0(µ0−λ0)) −e−2µ(¨

λ+ ( ˙

λ+1

t)( ˙

λ−˙µ)) = e−2λ[µ0(µ0+λ0)−4πteµ+λj0+µ0(µ0−λ0)]

−e−2µ{˙µ−˙

t−4πqe2µ+ 2 ˙

λ˙µ+ 8π˙

φ2+ 4πt(−˙µ˙

φ2+˙

φ¨

φ)+4πte2µ−2λ(−˙

λφ02+˙

φ0φ0)

+ 4π2

te2µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw + ( ˙

λ+1

t)( ˙

λ−˙µ)}

= 2µ02e−2λ−e−2λ4πteλ+µ{−π

t2eλ−µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw

−eλ−µ˙

λ(ρ+p) + eλ−3µ˙

λ˙

φ2+e−λ−µ˙

λφ02−2µ0j−e−(λ+µ)˙

φ0φ0−eλ−3µ¨

φ˙

φ−eλ−3µ(˙

λ−˙µ+2

t)˙

φ2}

−e−2µ[−˙

t−4πqe2µ+ 2 ˙

λ˙µ+˙µ

t+ 8π˙

φ2+ 4πt(−˙µ˙

φ2+˙

φ¨

φ)+4πte2µ−2λ(−˙

λφ02+˙

φ0φ0)

+ 4π2

te2µZ∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw]−e−2µ(˙

λ+1

t)( ˙

λ−˙µ)

= 2µ02e−2λ+˙

λ(ρ+p)4πt + 8πtµ0eµ−λj+˙

te−2µ+ 4πq −2e−2µ˙

λ˙µ−˙µ

te−2µ−e−2µ(˙

λ+1

t)( ˙

λ−˙µ))

= 2µ02e−2λ+˙

λ(˙

λ+ ˙µ)e−2µ+ 8πtµ0eµ−λ(−1

4πte−µ−λµ0) + ˙

te−2µ+ 4πq −2e−2µ˙

λ˙µ−˙µ

te−2µ−e−2µ˙

λ2

+e−2µ˙

λ˙µ−˙

te−2µ+˙µ

te−2µ

= 4πq

2) subtract the two equations (1.16)-(1.17) to obtain

φ2−φ0

1= (˜µ−µ0)φ1(1.47)

Let us first prove that µ0= ˜µ. Consider equation (1.44) and write p=p1+p2

where p1and p2are the contributions to pmade by fand φrespectively. We

obtain from [15] that studies the case φ= 0, and in which pcorresponds to p1

here, using definitions (1.8), (1.9) and (1.10) of ρ,p,j:

s2p0

1(s, r)ds =−Zt

(˙

λ+ ˙µ)eλ−µs2(j+e−λ−µφ1φ2)ds −[eλ−µs2(j+e−λ−µφ1φ2)]t

−Zt

˜µ[(ρ+p)−(e−2µφ2

1+e−2λφ2

2)]s2ds

We have

D+(φ1e−µ−φ2e−λ) = ˙

φ1e−2µ−φ0

2e−2λ−˙µφ1e−2µ+λ0φ2e−2λ+( ˙

λφ2−µ0φ1+φ0

1−˙

φ2)e−µ−λ

¿From (1.16),

D+(φ1e−µ−φ2e−λ) = (−˙

λ−2

t)φ1e−2µ+˙

λφ2e−µ−λ−˜µe−µ−λφ1+ ˜µe−2λφ2

We deduce from these two relations that

φ0

2e−2λ=˙

φ1e−2µ−˙µφ1e−2µ+λ0φ2e−2λ−µ0φ1e−µ−λ

+ (φ0

1−˙

φ2)e−µ−λ−(−˙

λ−2

t)φ1e−2µ+ ˜µe−µ−λφ1−˜µe−2λφ2

Then,

s2p0

2(s, r)ds =Zt

s2(−µ0φ2

1e−2µ−λ0φ2

2e−2λ+φ0

1φ1e−2µ+φ0

2φ2e−2λ)ds

=Zt

s2[−µ0φ2

1e−2µ−λ0φ2

2e−2λ+φ0

1φ1e−2µ

+φ2(˙

φ1e−2µ−˙µφ1e−2µ+λ0φ2e−2λ−µ0φ1e−µ−λ

+ (φ0

1−˙

φ2)e−µ−λ−(−˙

λ−2

t)φ1e−2µ+ ˜µe−µ−λφ1−˜µe−2λφ2)]ds

=Zt

s2[−µ0φ2

1e−2µ+φ0

1φ1e−2µ+φ2˙

φ1e−2µ+ ( ˙

λ−˙µ)e−2µφ1φ2

+ (φ0

1−˙

φ2)φ2e−µ−λ+ (˜µ−µ0)φ1φ2e−µ−λ−˜µe−2λφ2

2]ds +Zt

2sφ1φ2e−2µds

Integrate the last term of this relation by parts and using (1.47), we obtain

s2p0

2(s, r)ds =Zt

s2[−µ0φ2

1e−2µ+φ0

1φ1e−2µ+φ2˙

φ1e−2µ+ ( ˙

λ−˙µ)e−2µφ1φ2

+ (φ0

1−˙

φ2)φ2e−µ−λ+ (˜µ−µ0)φ1φ2e−µ−λ−˜µe−2λφ2

2]ds + [s2φ1φ2e−2µ]t

−Zt

s2(−2 ˙µφ1φ2e−2µ+˙

φ1φ2e−2µ+φ1˙

φ2e−2µ)ds

= [s2φ1φ2e−2µ]t

1+Zt

[( ˙

λ+ ˙µ)e−2µφ1φ2+ (φ0

1φ1−φ1˙

φ2)e−2µ

+ (φ0

1−˙

φ2)φ2e−µ−λ−µ0φ2

1e−2µ−˜µφ2

2e−2λ+ (˜µ−µ0)φ1φ2e−µ−λ]s2ds

= [s2φ1φ2e−2µ]t

1+Zt

[( ˙

λ+ ˙µ)e−2µφ1φ2+ (φ0

1−˙

φ2)(φ1e−2µ+φ2e−µ−λ)

−µ0φ2

1e−2µ−˜µφ2

2e−2λ+ (˜µ−µ0)φ1φ2e−µ−λ]s2ds

= [s2φ1φ2e−2µ]t

1+Zt

[( ˙

λ+ ˙µ)e−2µφ1φ2−˜µ(φ2

1e−2µ+φ2

2e−2λ)]s2ds

Then, using (1.45),

s2p0

2(s, r)ds = [−eλ−µjs2]t

1−Zt

4πeλ+µjs(ρ+p)s2ds −Zt

˜µ(ρ+p)s2ds

(1.46) becomes :

tµ0e−2µ=◦

µ0e−2◦

µ+ 4π(e

◦

λ−◦

µ◦

j−eλ−µjt2)+4πZt

(4πseλ+µj+ ˜µ)(ρ+p)s2ds

and using the definition (1.15) of ˜µ:

tµ0e−2µ=e−2◦

µ(◦

µ0+ 4πe

◦

λ+◦

µ◦

j) + t˜µe−2µfor all t∈I⊂]0,∞[,

Hence, if (1.5) holds for t= 1, then, tµ0e−2µ=t˜µe−2µand µ0= ˜µ.

Now we prove the existence of φ. Define φby : φ(t, r) = ◦

φ(r)+Rt

1φ1(s, r)ds.

Then φ(1) = ◦

φ,˙

φ(1) = φ1(1) = ψand ˙

φ=φ1. Now (1.47) implies, since

µ0= ˜µ, that ˙

φ2=φ0

1, hence the relation φ2(1) = ◦

φ0implies φ0=φ2.

The relation µ0= ˜µalso implies that the systems (1.12)-(1.13) and (1.16)-(1.17)

are identical. Then a direct calculation, using the fact that (φ1, φ2) satisfies the

system (1.12)-(1.13) shows that φsatisfies (1.7). 

We conclude this section with a proposition dealing with the solvability of the

constraint equation (1.5) for t= 1. Let ˜

ψ=e−µψ.

Proposition 1.15 Given a function ◦

λ(r), a non-negative function ¯

f(r, w, F)

and functions ◦

φ(r)and ˜

ψ(r), all periodic in rand regular, there exists a function

◦

µ(r), periodic in rand regular, such that the constraint equation

◦

µ0=−4πe

◦

λ+◦

µ◦

holds for a non-negative function ◦

f. It can be assumed that ◦

f=¯

f+aΦ, where

Φ(r, w, F)is a fixed function, independent of the particular choice of input data,

and ais a suitable constant.

Proof: This can be proved just as in [27], with Φ chosen as in that reference.



This result shows that it is possible to produce a plentiful supply of initial data.

It cannot be applied to produce data with f= 0. A way of doing that is to

adjust ˜

ψ=¯

ψ+bΦ (bis a suitable constant) instead of adjusting f. Φ(r)≥0

and suppΦ ⊂I,Ian interval in which ∂r◦

φ(r)6= 0.

Remark 1.16 1) The two previous propositions show that the coupled system

(1.2)-(1.3)-(1.4)-(1.5)-(1.6)-(1.7) reduces to the subsystem (1.14), (1.3), (1.4),

(1.15), (1.16), (1.17) on which we concentrate in the next chapter.

2) In the following, only one norm on function spaces is used, namely the L∞-

norm, which is denoted by ||.||. For example if f∈Ck(I×[0,∞[), we define

||f(t)|| = sup{|f(t, r)|, r ∈[0,∞[}.

Chapter 2

Local existence and

continuation of solutions

In this chapter using an iteration we prove the local existence and uniqueness of

solutions of the Einstein-Vlasov-scalar field system together with continuation

criteria.

2.1 Iteration

Let us first use the solution (f, λ, µ, ˜µ, φ1, φ2) of the auxiliary system consisting

of the equations (1.14), (1.3), (1.4), (1.15), (1.16) and (1.17), to construct a

sequence of iterative solutions as follows. Define ◦

˜µ:= ◦

µ0,λ0(t, r) := ◦

λ(r),

µ0(t, r) := ◦

µ(r), ˜µ0(t, r) := ◦

˜µ(r), g0(t, r) = ψ(r), h0(t, r) = ◦

φ0for t∈]0,1],

r∈R. If λn−1,µn−1, ˜µn−1are already defined and regular on ]T∗,1] ×Rthen

let

Gn−1(t, r, w, F):= weµn−1−λn−1

p1 + w2+F/t2,−˙

λn−1w−eµn−1−λn−1˜µn−1p1 + w2+F/t2!

(2.1)

and denote by (Rn, Wn)(s, t, r, w, F) the solution of the characteristic system

ds(R, W) = Gn−1(s, R, W, F)

with initial data

(Rn, Wn)(t, t, r, w, F) = (r, w); (t, r, w, F)∈]0,1] ×R2×[0,∞[ ;

note that Fis constant along characteristics. Define

fn(t, r, w, F) := ◦

f((Rn, Wn)(1, t, r, w, F), F),(2.2)

that is, fnis the solution of

∂tfn+weµn−1−λn−1

p1 + w2+F/t2∂rfn−(˙

λn−1w+eµn−1−λn−1˜µn−1p1 + w2+F/t2)∂wfn= 0

(2.3)

with fn(1) = ◦

f. Define ρn,pn,jn,qnby the formulas (1.8), (1.9), (1.10) and

(1.11) with f,λ,µ,˙

φ,φ0respectively replaced by fn,λn−1,µn−1,gn−1,hn−1,

(n≥1). Using Proposition 1.13, 1), define µnand λnto be the solutions of

e−2µn(t,r)=e−2◦

µ(r)+k

t−k+8π

tZ1

s2pn(s, r)ds (2.4)

λn(t, r) = 4πte2µn(t,r)ρn(t, r)−1 + ke2µn

2t(2.5)

λn(t, r) = ◦

λ(r)−Z1

λn(s, r)ds (2.6)

and set

˜µn(t, r) = −4πte(µn+λn)(t,r)jn(t, r) (2.7)

Notice that, by Proposition 1.13, the right hand side of (2.4) is positive on

]T∗,1], ∀n. Now define gnand hnusing Proposition 1.13, 2) to satisfy the

conditions that the quantities

Xn=e−µngn−e−λnhn, Yn=e−µngn+e−λnhn

are solutions of the system

D+

n−1Xn=an−1Xn−1+bn−1Yn−1(2.8)

D−

n−1Yn=bn−1Xn−1+cn−1Yn−1(2.9)

where D+

n−1,D−

n−1,an−1,bn−1and cn−1are defined in the same way as D+,

D−, ˜a,b, ˜c(see Lemma 1.6), with µ,λ,˙

φ,φ0, ˜a,b, ˜csubstituted respectively

by µn−1,λn−1,gn−1,hn−1,an−1,bn−1,cn−1. Now K0and A0being defined in

Propositions 1.7 and 1.10, we introduce the following quantities that are similar

to those defined in those propositions:











Kn(t) = sup{(g2

ne−2µn+h2

ne−2λn)1

2(t, r) ; r∈R}

An(t) = sup{e−λn[(g0

n−µ0

ngn)2e−2µn+ (h0

n−λ0

nhn)2e−2λn]1/2(t, r) ; r∈R}

mn−1(t) = sup{2

t+ (|˙

λn−1|+|˜µn−1|eµn−1−λn−1)(t, r) ; r∈R}

vn−1(t) = sup{2

t+ 2 |˙

λn−1|+(|˜µn−1|+|µ0

n−1|)eµn−1−λn−1(t, r) ; r∈R}

βn−1(t) = sup{[|˙

λ0

n−1|e−λn−1+ (|µ0

n−1|| ˜µn−1|+|λ0

n−1|| ˜µn−1|

+|˜µ0

n−1|)eµn−1−2λn−1](t, r) ; r∈R}

(2.10)

Now we proceed for (2.8)-(2.9) the same way as we did for (1.16)-(1.17) to

establish the inequality (1.18) and we obtain the following analogous inequality:

Kn(t)≤K0+ 2 Z1

mn−1(s)Kn−1(s)ds (2.11)

We can use (2.8)-(2.9) to establish a system for (e−λn−1∂rXn, e−λn−1∂rYn) anal-

ogous to (1.19)-(1.20) from which we deduce the following inequality which is

analogous to (1.27)

An(t)≤A0+ 2 Z1

(vn−1(s)An−1(s) + βn−1(s)Kn−1(s))ds (2.12)

Throughout the document, we use the fact that by (2.2), kfn(t)k=k◦

fkfor

n∈Nand t∈]T∗,1]. The numerical constant Cmay change from line to line

and does not depend on nor tor the initial data. In order to prove the local

existence theorem, we prove respectively in the next two propositions :

- a uniform bound on the momenta in the support of distribution functions fn,

and a uniform bound of the first derivatives with respect to rof the functions

fn,λn,µn,gn,hn;

- the convergence of the iterates.

Proposition 2.1 We take ◦

fas in proposition 1.13 and such that

supp◦

f⊂[0, W0]×[0, F0], W0>0, F0>0.(2.13)

then there exist nonnegative constants T1, T2such that the quantities

Qn(t) = sup{| w|; (r, w, F)∈suppfn(t)}for all t ∈[T1,1].

Bn(t) = sup{k ∂rfn(s)k+A2

n−1(s); t≤s≤1}for all t ∈[T2,1].

and Kn(t)for all t∈[T1,1] are uniformly bounded in n.

Proof: Firstly we bound Qn(t) and Kn(t). On suppfn(t), we have

p1 + w2+F/t2≤p1 + Q2

n+F0/t2≤1

t(1 + F0)(1 + Qn(t)) (2.14)

and thus

kρn(t)k ≤ π

t2ZQn(t)

−Qn(t)ZF0

t(1 + F0)(1 + Qn(t))||fn(t)||dFdw + (Kn−1(t))2

≤C

t3(1 + F0)2(1 + Qn(t))2k◦

fk+(Kn−1(t))2;

kpn(t)k,kjn(t)k ≤ π

t2ZQn(t)

−Qn(t)ZF0

Qn(t)||fn(t)||dFdw + (Kn−1(t))2

≤C

t2(1 + F0)(1 + Qn(t))2k◦

fk+(Kn−1(t))2

¿From (1.43), we have, setting C0=β

e−2µn(t, r)≥C0

t(2.15)

Using (2.5)-(2.7) and (2.15) we have

|˙

λn(s, r)| ≤ 4πseµn(s,r)|ρn(s, r)|+1 + s/C0

≤C

[(1 + F0)2(1 + Qn(s))2

sk◦

fk+(Kn−1(s))2] + 1 + C0

2sC0

;

and

|˜µneµn−λn(s, r)| ≤ 4πs2

s2(1 + F0)(1 + Qn(s))2k◦

fk+Kn−1(s)2]

≤C

[(1 + F0)(1 + Qn(s))2k◦

fk+Kn−1(s)2]

Thus

|˙

Wn+1(s)| ≤ | ˙

λn(s)|+|˜µn|eµn−λn(s)p1 + w2+F/s2

≤C

[(1 + F0)2(1 + Qn(s))2

sk◦

fk+(Kn−1(s))2+1

s]|Wn+1(s)|

[(1 + F0)(1 + Qn(s))2k◦

fk+Kn−1(s)2]1 + F0

s(1 + |Wn+1(s)|)

≤C

[(1 + F0)2(1 + Qn(s))2

s(1+ k◦

fk)(1 + Kn−1(s))2](1 + |Wn+1(s)|)

≤C1

s(1 + Qn(s))2(1 + Kn−1(s))2(1 + |Wn+1(s)|)

This implies after integration over [t, 1],

Wn+1(t)≤W0+C1Z1

s(1 + Qn(s))2(1 + Kn−1(s))2(1 + |Wn+1(s)|)ds;

then

Qn+1(t)≤W0+C1Z1

s(1 + Qn(s))2(1 + Kn−1(s))2(1 + Qn+1(s))ds (2.16)

with C1=C

C0(1 + F0)2(1+ k◦

fk). Next we have, using (2.5)-(2.7) and (2.15)

|˙

λn(s, r)|+|(˜µneµn−λn)(s, r)|≤ C1

(1 + Qn(s))2

s(1 + Kn−1(s))2(2.17)

we then deduce from (2.11) and (2.17) that

Kn+1(t)≤K0+C1Z1

(1 + Qn(s))2

s(1 + Kn−1(s))2Kn(s)ds (2.18)

Add (2.16) and (2.18) to obtain

Qn+1(t)+Kn+1(t)≤W0+K0+C1Z1

s(1+Qn(s))2(1+Kn−1(s))2(1+Qn+1(s)+Kn(s))ds

(2.19)

Now define Hn(t) := sup{Qm(t) + Km(t) ; m≤n}. (Hn)n∈Nis an increasing

sequence. Then,

Qn(t) + Kn(t)≤Hn(t)≤Hn+1(t),

Qn+1(t) + Kn+1(t)≤W0+K0+C1Z1

s(1 + Hn+1(s))5ds and

Qm(t) + Km(t)≤W0+K0+C1Z1

s(1 + Hn+1(s))5ds for m≤n;

which imply :

Hn+1(t)≤W0+K0+C1Z1

s(1 + Hn+1(s))5ds

Let z1be the left maximal solution of the equation

z1(t) = W0+K0+C1Z1

s(1 + z1(s))5ds

which exists on some interval ]T1,1] with T1∈[T∗,1[. By comparing the solution

of the integral inequality with that of the corresponding integral equation it

follows that

Hn+1(t)≤z1(t), t ∈]T1,1[, n ∈N.

Since Qn(t) + Kn(t)≤Hn+1(t), we obtain

Kn(t), Qn(t)≤z1(t), t ∈]T1,1[, n ∈N.

And all the quantities which were estimated against Qnand Knin the above

argument are bounded by certain powers of z1on ]T1,1]. Namely |˙

λn(t)|,|ρn(t)|,

|pn(t)|,|jn(t)|,|˜µneµn−λn(t)|,g2

n−1e−2µn−1(t), h2

n−1e−2λn−1(t) are bounded. (2.4)

shows that e−2µnis bounded and by (2.15), e2µnis bounded. (2.6) shows that

|λn(t)|is bounded. Consequently eλnand e−λnare bounded. Then gnand hn

are bounded. We conclude that, there exists a continuous function C2(t) which

depends only on z1as an increasing function, such that

(kµn(t)k,kλn(t)k,k˙

λn(t)k,kρn(t)k,kpn(t)k,

kjn(t)k,k˜µneµn−λnk,kgn(t)k,khn(t)k ≤ C2(t)(2.20)

Now we bound Bn(t). We have, using (2.20), the estimates

ρ0

n(t) = π2

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂rfndFdw

+ (−µ0

n−1g2

n−1+gn−1g0

n−1)e−2µn−1+ (−λ0

n−1h2

n−1+hn−1h0

n−1)e−2λn−1

=π2

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂rfndFdw

+ (−µ0

n−1gn−1+g0

n−1)e−µn−1gn−1e−µn−1+ (−λ0

n−1hn−1+h0

n−1)e−λn−1hn−1e−λn−1

≤π2

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂rfndFdw +g2

n−1e−2µn−1

+ (−µ0

n−1gn−1+g0

n−1)2e−2µn−1+ (−λ0

n−1hn−1+h0

n−1)2e−2λn−1+h2

n−1e−2λn−1

≤π2

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂rfndFdw +g2

n−1e−2µn−1+h2

n−1e−2λn−1

+e−2λn−1[(−µ0

n−1gn−1+g0

n−1)2e−2µn−1+ (−λ0

n−1hn−1+h0

n−1)2e−2λn−1]e2λn−1

and

kρ0

n(t)k≤ C2(t)(C3+Bn(t))

We deduce in the same way as previously :

kρ0

n(t)k,kp0

n(t)k,kj0

n(t)k,kµ0

n(t)k,k˙

λ0

n(t)k,kλ0

n(t)k≤ C2(t)(C3+Bn(t))

(2.21)

k˜µ0

neµn−λnk≤ C2(t)(C3+Bn(t)) (2.22)

Following step 2 of the proof of theorem 3.1 in [15], we have

|∂rGn(s, r, w, F)| ≤ C2(s)(C3+Bn(s));

|∂wGn(s, r, w, F)| ≤ C2(s)

and differentiate the characteristic system of the Vlasov equation with respect

to r:

ds∂r(R, W)n+1(s, t, r, w, F )|=|∂r(R, W)n+1(s, t, r, w, F).∂rGn(s, Rn+1, Wn+1, F)|

≤ |∂r(R, W)n+1(s, t, r, w, F )|C2(s)(C3+Bn(s))

therefore for (r, w, F)∈suppfn+1(t)∪suppfn(t), we obtain by Gronwall’s in-

equality

|∂r(R, W)n+1(1, t, r, w, F )| ≤ exp Z1

C2(s)(C3+Bn(s))ds.

¿From the definition of fin (2.2), we obtain

||∂rfn+1(t, r, w, F)|| =||∂r(R, W )n+1(1, t, r, w, F).∂(r, w)◦

f((R, W)n+1(1, t, r, w, F )) ||

≤ ||∂(r, w)◦

f||sup{|∂r(R, W)n+1(1, t, r, w, F )|; (r, w, F )∈suppfn+1(t)}

≤ ||∂(r, w)◦

f||exp Z1

C2(s)(C3+Bn(s))ds.

(2.23)

with C3=k◦

λ0k+k◦

µ0e−2◦

µk+1. We use (2.20), (2.21), (2.22) and estimate

(2.12) to obtain

An+1(t)≤A0+Z1

C2(s)(C3+Bn(s))(1 + An)(s)ds

Let Dn(t) := sup{Am(t)|m≤n}and En(t) := sup{Bm(t)|m≤n}.{Dn}and

{En}are increasing sequences. Therefore

1 + An+1(t)≤A0+1+Z1

C2(s)(C3+En(s))(1 + Dn+1(s))ds (2.24)

then we deduce by replacing nby any m≤nin (2.24)

1 + Dn+1(t)≤A0+1+Zt

C2(s)(C3+En(s))(1 + Dn+1(s))ds;

which gives:

Dn+1(t)≤2(A0+ 1) exp Z1

C2(s)(C3+En(s))ds. (2.25)

Now add (2.23)-(2.25) to obtain

Bn+1(t)≤(2(A0+ 1)+ k∂r,w ◦

fk) exp Z1

C2(s)(C3+En(s))ds

and deduce by replacing nby every m≤nthat:

En+1(t)≤C4exp Z1

C2(s)(C3+En+1(s))ds

where C4= 2(A0+ 1)+ k∂(r,w)◦

fk. Let z2be the left maximal solution of

z2(t) = C4exp Z1

C2(s)(C3+z2(s))ds

i.e

˙z2(t) = −C2(t)(C3+z2(t))z2(t), z2(1) = C4;

which exists on an interval ]T2,1] ⊂]T1,1]. Then we have

En+1(t)≤z2(t), t ∈]T2,1], n ∈N

and so

An(t), Bn(t)≤z2(t), t ∈]T2,1], n ∈N

and all the quantities estimated against Bnabove can be bounded in terms of

z2on ]T2,1], uniformly in n.

Remark 2.2 The sequences λn,µn,fn,˜µneµn−λn,ρn,pn,jn,gn,hn,λ0

n,µ0

n,g0

n, h0

n,˙

λn,˙µn,˙gn,˙

hn,˙

λn0,ρ0

n,p0

n,j0

n,˜µn0, are uniformly bounded in the

L∞−norm by a function of ton [T∗

1,1] with T∗

1= max(T1, T2).

In order to prove the convergence of the iterates in the following proposition,

we introduce auxiliary variables ˜gnand ˜

hndefined by ˜gn=gne−µn,

hn=hne−λn, for n∈N.

Proposition 2.3 Let [T3,1] ⊂[T2,1], be an arbitrary compact subset on which

the previous estimates hold. Then on such an interval, the iterates converge

uniformly.

Proof: Define for t∈[T3,1]:

αn(t) := sup{k (fn+1 −fn)(s)k+k(˜gn+1 −˜gn)(s)k+k(˜

hn+1 −˜

hn)(s)k

+k(λn+1 −λn)(s)k+k(µn+1 −µn)(s)k;t≤s≤1}

and let Cdenote a constant which may depend on the functions z1and z2

introduced previously. If we consider the new quantities

Xn= (˜gn+1 −˜gn)−(˜

hn+1 −˜

hn); ˜

Yn= (˜gn+1 −˜gn)+(˜

hn+1 −˜

hn),

then we obtain by subtracting the system (2.8)-(2.9) written for n+ 1 and n

(see appendix), the new system

D+

n˜

Xn=an˜

Xn−1+bn˜

Yn−1+Fn(2.26)

D−

n˜

Yn=bn˜

Xn−1+cn˜

Yn−1+Gn(2.27)

where

Fn= (an−an−1+bn−bn−1)˜gn−1+ (an−1−an+bn−bn−1)˜

hn−1

+ (e−µn−1−e−µn)(˙

˜gn−˙

hn)+(e−λn−1−e−λn)(˜g0

n−˜

and substitute in Fn,˜

nand ˜g0

nrespectively by −˜

nand −˜g0

nto obtain Gn.

Now let

θn(t) = sup{| ˜gn+1 −˜gn|+|˜

hn+1 −˜

hn|;r∈R}

Thus similarly to (2.11), we have :

θn(t)≤2Z1

(mn(s)θn−1(s) + sup{eµn(|Fn(s, r)|+|Gn(s, r)|); r∈R})ds

(2.28)

Using the mean value theorem to express the differences e−µn−e−µn−1,

e−λn−e−λn−1and remark 2.2, then (2.28) gives

|˜gn+1 −˜gn|+|˜

hn+1 −˜

hn|≤ CZ1

(αn−1+|˜µn−˜µn−1|+|˙

λn−˙

λn−1|)(s)ds

(2.29)

The expressions of ρn,pn,jnyield, using proposition 2.1, that

¿From (2.5) and (2.7), we have respectively

|˙

λn−˙

λn−1|(t) = 4πt |(e2µn−e2µn−1)ρn+ (ρn−ρn−1)e2µn−1|(t)

≤Cαn−1(t)(2.30)

|˜µn−˜µn−1|(t) = 4πt|(eλn+µn−eλn−1+µn−1)jn+ (jn−jn−1)eλn−1+µn−1|(t)

≤Cαn−1(t)

(2.31)

Using the two previous inequalities, (2.29) gives

(|˜gn+1 −˜gn|+|˜

hn+1 −˜

hn|)(t)≤CZ1

αn−1(s)ds (2.32)

By the mean value theorem, (2.4) gives :

|µn+1 −µn|(t)≤CZ1

αn(s)ds (2.33)

(2.6) gives :

|λn+1 −λn|(t) = Z1

t|˙

λn+1 −˙

λn|ds ≤CZ1

αn(s)ds (2.34)

Now from (2.30)-(2.31), (2.33)-(2.34), the mean value theorem and the fact that

tαn−1(s)ds ≤Cαn−1(t), we deduce

|Gn−Gn−1|(s, r, w, F)≤Cαn−1(s)

which implies for (r, w, F)∈suppfn−1(t)∪suppfn(t),

ds((R, W)n+1−(R, W)n)|(s, t, r, w, F) =|Gn−Gn−1|(s, r, w, F)≤Cαn−1(s)

Integrating over [t, 1] and using the fact that

|(R, W)n+1 −(R, W )n|(t, t, r, w, F) = 0 gives

|(R, W)n+1 −(R, W )n|(1, t, r, w, F)≤CZ1

αn−1(s)ds

This implies using (2.2) and the mean value theorem

|(fn+1 −fn)(t)| ≤| ∂r,w ◦

f|| (R, W)n+1 −(R, W )n|(1, t, r, w, F)

≤C|∂r,w ◦

f|Z1

αn−1(s)ds

(2.35)

Adding (2.32), (2.33), (2.34), (2.35), then

αn(t)≤CZ1

(αn(s) + αn−1(s))ds ;n≥1.

By Gronwall’s inequality αn(t)≤CR1

tαn−1(s)ds ;

and by induction

αn(t)≤Cn+1 (1 −t)n

n!≤Cn+1

n!for n ∈N, t ∈[T3,1]

Since the series PCn+1

n!converges, we deduce the convergence of Pαnwhich

implies that αn→0 for n→ ∞. Every difference term which appears in αn,

converges to zero. We deduce the uniform convergence of

fn, λn, µn,˜gn,˜

hn,˙

λn,˙µn,˜µn, ρn, pn, jn.(2.36)

And in L∞−norm, λn→λ;µn→µ; ˜µn→˜µ;fn→f; ˜gn→˜g;˜

hn→˜

h.

It remains to show that the limits ˜g, ˜

h, f, λ, µ are C1,fsolves the Vlasov

equation (1.2), λ, µ solve the field equations (1.3)-(1.4), and to show the exis-

tence of a function φthat solves the wave equation (1.7). This is the subject of

the next section.

2.2 Local existence

Theorem 2.4 (local existence) Let ◦

f∈C1(R2×[0,∞[) with

◦

f(r+ 1, w, F) = ◦

f(r, w, F)for (r, w, F)∈R2×[0,∞[,◦

f≥0, and

W0:= sup{|w||(r, w, F)∈supp◦

f}<∞

F0:= sup{F|(r, w, F)∈supp◦

f}<∞

Let ◦

λ, ψ ∈C1(R),◦

µ, ◦

φ∈C2(R)with ◦

λ(r) = ◦

λ(r+ 1),◦

µ(r) = ◦

µ(r+ 1),

◦

φ(r) = ◦

φ(r+ 1) and

◦

µ0(r) = −4πe

◦

λ+◦

µ◦

j(r), r ∈R

Then there exists a unique, left maximal, regular solution (f, λ, µ, φ)of system

(1.2)-(1.11) with (f, λ, µ, φ)(1) = (◦

f, ◦

λ, ◦

µ, ◦

φ)and ˙

φ(1) = ψon a time interval

]T, 1] with T∈[0,1[.

Proof : Consider the sequences of iterates constructed at the begining of this

chapter and the limit obtained in the above proposition. We need the uniform

convergence of the derivatives of these iterates.

We know by (2.36) that ( ˙

λn) and ( ˙µn) converge uniformly. We must now show

that, λ0

n,µ0

n,˙

fn,∂rfn,∂wfn,˙

˜gn, ˜g0

n,˙

hn,˜

nalso converge uniformly. Using

(2.3), the convergence of ˙

fnwill be a consequence of that of λn,µn,˙

λn, ˜µn

∂rfn,∂wfn. Using system (2.8)-(2.9), the convergence of ˙

˜gn,˙

hnwill be a con-

sequence of that of ˜g0

n,˜

n,µn,λn, ( ˙

λn), ˜µn. We then proceed to show the

uniform convergence of λ0

n,µ0

n,∂rfn,∂wfn, ˜g0

n,˜

In what follows, we fix T4∈[T2,1], t∈[T4,1], |w|< U,F < F0,t≤s≤1.

Step 1: Convergence of (∂rfn) and (∂wfn).

Following step 4 in the proof of theorem 3.1 in [15], and using (2.36), we can

establish with minor changes, using Proposition 2.1, that if we set:

ξn(s) = e(λn−µn)(s,r)∂Rn(s, t, r, w, F) (2.37)

ηn(s) = ∂Wn(s)+(p1 + w2+F/s2eλn−µn˙

λn)∂Rn(s) (2.38)

in which ∂stands for ∂ror ∂wand s7→ (Rn(s), Wn(s)) the indicated solution of

the characteristic system associated to equation (2.3) in fn, and then: ∀ > 0,

∃N∈Nsuch that we have, for n>N:

(|ξn+1 −ξn|+|ηn+1 −ηn|)(s)≤C+CZ1

(|ξn+1 −ξn|+|ηn+1 −ηn|)(τ)dτ

(2.39)

in which C > 0 stands, as in what follows, for a constant that may change

from line to line. (2.39) implies by Gronwall’s lemma, that (ξn) and (ηn) con-

verge uniformly. Now, since the transformation (∂Rn, ∂Wn)7→ (ξn, ηn) defined

by (2.37)-(2.38) is invertible with convergent coefficients, this implies the con-

vergence of ∂r,w(Rn, Wn) and, given (2.2), the convergence of (∂rfn) and (∂wfn).

Step 2: convergence of (λ0

n),(µ0

n),(˜µ0

n),(˜g0

n),(˜

n).

We set

γn(t) = sup{|ξn+1 −ξn|(s) + |ηn+1 −ηn|(s)+ k(µ0

n+1 −µ0

n)(s)k

+k(λ0

n+1 −λ0

n)(s)k+k(˜g0

n+1 −˜g0

n)(s)k+k(˜

n+1 −˜

n)(s)k;t≤s≤1}

(2.40)

Now since (µn), (˜µn), ( ˙

λn), (˜gn), (˜

hn), (ρn), (jn) converge uniformly, we take

the above integer Nsufficiently large so that we have for n > N:

k(µn+1 −µn)(s)k,k(jn+1 −jn)(s)k,k(˜µn−˜µn−1)(s)k,k(˙

λn+1 −˙

λn)(s)k,

k(˜gn−˜gn−1)(s)k,k(˜

hn−˜

hn−1)(s)k,k(ρn+1 −ρn)(s)k≤ 

(2.41)

A) Estimation of (λ0

n), (µ0

n), (˜µ0

n). We deduce from (2.37)-(2.38), taking ∂=∂r

that:

∂Rn(s) = e(µn−λn)(s,r)ξn(s) (2.42)

∂Wn(s) = ηn(s)−(p1 + w2+F/s2˙

λn)ξn(s) (2.43)

Let us first consider ρ0

n,j0

n,p0

nthat involve ∂rfn, (˜gn˜

hn)0,1

2(˜g2

n+˜

n). We have,

using (2.2), (2.42), (2.43)

k(∂rfn+1 −∂rfn)(s)k ≤k ∂r,w ◦

fk(|∂rRn+1 −∂rRn|+|∂rWn+1 −∂rWn|)(s)

≤Ck∂r,w ◦

fk(|eµn+1−λn+1 ξn+1 −eµn−λnξn|

+|ηn+1 −ηn|+|˙

λn+1ξn+1 −˙

λnξn|)(s).

(2.44)

Estimate (2.44) gives, using (2.41) and since (λn),(µn),(ξn),(˙

λn) are bounded,

k(∂rfn+1 −∂rfn)(s)k≤ C(|ξn+1 −ξn|+|ηn+1 −ηn|)(s) + C (2.45)

Next we have, using (2.41) and remark 2.2 :

|(˜gn˜

hn)0−(˜gn−1˜

hn−1)0|=|˜g0

n˜

hn−˜g0

n−1˜

hn−1+ ˜gn˜

n−˜gn−1˜

n−1|

≤C +C(|˜g0

n−˜g0

n−1|+|˜

n−˜

n−1|)(2.46)

Now using (2.41) and the fact that (˜gn), (˜

hn), (˜g0

n), ˜

nare bounded, we obtain

2(˜g2

n+˜

n)0−1

2(˜g2

n−1+˜

n−1)0|=|˜g0

n˜gn−˜g0

n−1˜gn−1+˜

hn˜

n−˜

hn−1˜

n−1|

≤C(|˜g0

n−˜g0

n−1|+|˜

n−˜

n−1|) + C

(2.47)

We then deduce, from the expressions of ρn,pn,jnand using (2.40), (2.45),

(2.46), (2.47), (2.41)

k(ρ0

n+1 −ρ0

n)(s)k,k(p0

n+1 −p0

n)(s)k,k(j0

n+1 −j0

n)(s)k≤ C +C(γn+γn−1)(s)

(2.48)

Concerning (µ0

n), we obtain by taking the derivative of (2.4) with respect to r:

µ0

ne−2µn=◦

µ0e

◦

t−4π

tZ1

s2p0

n(s, r)ds

subtracting this relation written for n+ 1 and n, we obtain

µ0

n+1 −µ0

n=e2µn+1 [µ0

n(e−2µn−e−2µn+1 )−4π

tZ1

s2(p0

n+1 −p0

n)(s, r)ds]

This gives, using (2.41), (2.48) and the fact that µn,µ0

nare bounded :

k(µ0

n+1 −µ0

n)(s)k≤ C +CZ1

(γn−1(τ) + γn(τ))dτ (2.49)

Concerning (λ0

n), if we take the derivative of (2.5) with respect to r, we have

λ0

n= (8πtµ0

nρn+ 4πtρ0

n)e2µn−kµ0

te2µn(2.50)

We first deduce that ˙

λn0is bounded since µ0

n,ρn,ρ0

n,µnare bounded. Next,

subtracting (2.50) written for n+ 1 and n, we obtain :

λ0

n+1−˙

λ0

n=e2µn+1 [(µ0

n+1 −µ0

n)(8πtρn+1 −k

t)+8πtµ0

n(ρn+1 −ρn)

+ 4πt(ρ0

n+1 −ρ0

n)] + (e2µn+1 −e2µn)(4πtρ0

n−k

tµ0

n+ 8πtµ0

nρn)

using (2.40), (2.41), (2.48), and since (µn), (ρn), (µ0

n) are bounded

k(˙

λ0

n+1 −˙

λ0

n)(s)k≤ C +C(γn−1(s) + γn(s)) (2.51)

Now, (2.6) gives :

(λ0

n+1 −λ0

n)(t, r) = Zt

(˙

λ0

n+1 −˙

λ0

n)(s, r)ds

which gives, taking the norms, using (2.51) and integrating over [s, 1], with

λn+1(1, r)−λn(1, r) = 0 :

k(λ0

n+1 −λ0

n)(s)k≤ C +CZ1

(γn−1(τ) + γn(τ))dτ (2.52)

We will also need to bound ˜µ0

n+1 −˜µ0

n. If we take the derivative of (2.7) with

respect to r, we obtain, after subtracting the expressions written for n+ 1 and

˜µ0

n+1 −˜µ0

n= (eλn+1+µn+1 −eλn+µn)(j0

n+jn)−4πt[(j0

n+1 −j0

+ (µ0

n+1 −µ0

n+λ0

n+1 −λ0

n)jn+1 + (µ0

n+λ0

n)(jn+1 −jn)]eλn+1+µn+1

We then deduce from (2.40), (2.41), (2.48):

k(˜µ0

n+1 −˜µ0

n)(s)k≤ C +C(γn−1(s) + γn(s)) (2.53)

B) Estimation of (˜g0

n), (˜

n). Recall that

Xn= ˜gn−˜

hn;Yn= ˜gn+˜

hn;an= (−˙

λn−1

t)e−µn−˜µne−λn;

bn=−e−µn

t;cn= (−˙

λn−1

t)e−µn+ ˜µne−λn

(2.54)

Set

Cn=anXn+bnYn;˜

Cn= (λ0

n−µ0

n)e−λnX0

n+1 +µ0

nCn+C0

Dn=bnXn+cnYn;˜

Dn= (µ0

n−λ0

n)e−λnY0

n+1 +µ0

nDn+D0

(2.55)

Then system (2.8)-(2.9) becomes

(D+

nXn+1 =Cn

D−

nYn+1 =Dn

If we take the derivative of the above system with respect to r, a direct calcu-

lation shows that (see appendix) :

D+

nX0

n+1 =˜

Cn(2.56)

D−

nY0

n+1 =˜

Dn(2.57)

We are going to integrate (2.56)-(2.57) on characteristic curves of the wave op-

erator. Consider the following characteristic curves γ1

n,γ2

nof the wave operator,

starting from the point (s, r), i.e for every n,

˙γ1

n=eµn−λn,˙γ2

n=−eµn−λn, γ1

n(s) = γ2

n(s) = r(2.58)

We have D+

n=e−µnd

dt on γ1

nand D−

n=e−µnd

dt on γ2

n. We then have, integrat-

ing (2.56) over γ1

nand (2.57) over γ2

n+1(s) = X0

n+1(1) −Z1

eµn˜

Cn(τ, γ1

n(τ))dτ

n+1(s) = Y0

n+1(1) −Z1

eµn˜

Dn(τ, γ2

n(τ))dτ

Now if we subtract respectively each of this two relations written for n+ 1 and

n, we obtain :

(X0

n+1 −X0

n)(s) = Z1

sheµn−1˜

Cn−1(τ, γ1

n−1(τ)) −eµn˜

Cn(τ, γ1

n(τ))idτ (2.59)

(Y0

n+1 −Y0

n)(s) = Z1

sheµn−1˜

Dn−1(τ, γ2

n−1(τ)) −eµn˜

Dn(τ, γ2

n(τ))idτ (2.60)

Since eµnand ˜

Cnare bounded, we have

|eµn˜

Cn(τ, γ1

n(τ)) −eµn−1˜

Cn−1(τ, γ1

n−1(τ))|={eµn(τ, γ1

n(τ)) −eµn−1(τ, γ1

n(τ))}˜

Cn(τ, γ1

n(τ))

+{˜

Cn(τ, γ1

n(τ)) −˜

Cn−1(τ, γ1

n(τ))}eµn−1(τ, γ1

n(τ)) + {˜

Cn−1(τ, γ1

n(τ)) −˜

Cn−1(τ, γ1

n−1(τ))}

eµn−1(τ, γ1

n(τ)) + {eµn−1(τ, γ1

n(τ)) −eµn−1(τ, γ1

n−1(τ))}˜

Cn−1(τ, γ1

n−1(τ))

≤C{k (eµn−eµn−1)(τ)k+k(˜

Cn−˜

Cn−1)(τ)k

+|˜

Cn−1(τ, γ1

n(τ)) −˜

Cn−1(τ, γ1

n−1(τ))|+|eµn−1(τ, γ1

n(τ)) −eµn−1(τ, γ1

n−1(τ))|}

(2.61)

Now integrating the relation ˙γ1

n−˙γ1

n−1=eµn−λn−eµn−1−λn−1over [s, τ] yields

|γ1

n−γ1

n−1|(τ)≤Csup{k (λn−λn−1)(t)k+k(µn−µn−1)(t)k, T4≤t≤1}

(2.62)

we then deduce from (2.36) (the right hand side of (2.62) tends to zero as n

tends to ∞), the uniform continuity of (eµn−1), ( ˜

Cn−1) over the compact set

K= [T4,1] ×(γ1

n([T4,1]) ∪γ1

n−1([T4,1])) i.e

∀ > 0,∃η(n, )>0 such that |γ1

n−γ1

n−1|(τ)≤η⇒

|˜

Cn−1(τ, γ1

n(τ)) −˜

Cn−1(τ, γ1

n−1(τ))|+|eµn−1(τ, γ1

n(τ)) −eµn−1(τ, γ1

n−1(τ))|< 

And from (2.61), (2.59)-(2.60), we deduce that

|X0

n+1 −X0

n|(s)≤C +CZ1

sk(˜

Cn−˜

Cn−1)(τ)kdτ (2.63)

|Y0

n+1 −Y0

n|(s)≤C +CZ1

sk(˜

Dn−˜

Dn−1)(τ)kdτ (2.64)

Therefore, for nsufficiently large, (2.63)- (2.64) implies respectively :

−C −CZ1

sk(˜

Cn−˜

Cn−1)(τ)k≤ X0

n+1 −X0

n≤C +CZ1

sk(˜

Cn−˜

Cn−1)(τ)kdτ

−C −CZ1

sk(˜

Dn−˜

Dn−1)(τ)k≤ Y0

n+1 −Y0

n≤C +CZ1

sk(˜

Dn−˜

Dn−1)(τ)kdτ

We deduce from (2.54) (definition of Xnand Yn), by adding and subtracting

the previous inequalities, that

k(˜g0

n+1 −˜g0

n)(s)k,k(˜

n+1 −˜

n)(s)k≤ C +CZ1

[k(˜

Cn−˜

Cn−1)(τ)k

+k(˜

Dn−˜

Dn−1)(τ)k]dτ

(2.65)

Now from (2.55), (2.41) and the fact that the sequences ( ˙

λn), (λn), (µn), (˜µ0

are bounded together with their first derivatives, we have (see appendix)

k(˜

Cn−˜

Cn−1)(τ)k≤ C +C(γn−1(τ) + γn(τ)) (2.66)

and

k(˜

Dn−˜

Dn−1)(τ)k≤ C +C(γn−1(τ) + γn(τ)) (2.67)

Therefore, we deduce from (2.65), (2.66)-(2.67) that

k(˜g0

n+1 −˜g0

n)(s)k,k(˜

n+1 −˜

n)(s)k≤ C +CZ1

(γn−1(τ) + γn(τ))dτ (2.68)

C) Convergence of (λ0

n), (µ0

n), (˜µ0

n), (˜g0

n), (˜

n). Add inequalities (2.39),

(2.49),(2.52) and (2.68) and take the supremum over s∈[t, 1] to obtain using

(2.40)

γn(t)≤C +CZ1

(γn−1(s) + γn(s))ds (2.69)

Define Γn(t) = sup{γm, m ≤n}; then (Γn) is an increasing sequence and (2.69)

gives

Γn(t)≤C +CZ1

Γn(s)ds

And by Gronwall’s lemma,

Γn(t)≤C, t ∈[T4,1],n sufficiently large.

We then deduce that (Γn) converges uniformly to 0, and from (2.40)-(2.53),

(λ0

n), (µ0

n), (˜g0

n), (˜

n), (˜µ0

n) converge uniformly on [T4,1]. We deduce from sys-

tem (2.8)-(2.9), the uniform convergence of (˙

˜gn) and (˙

hn). The regularity of f,

λ,µ, ˜g,˜

h(and ˜µ) follows from step 1 and step 2. Therefore g=eµ˜gand h=eλ˜

are regular. Note that, using the convergence of the derivatives, we can prove

that the limit (f, λ, µ, ˜µ, g, h) is a regular solution of (1.14), (1.3), (1.4), (1.15),

(1.16), (1.17) and by Proposition 1.14, we conclude the existence of a regular

function φsuch that (f, λ, µ, φ) is a solution of the full system (1.2)-(1.11).

To end this theorem, we prove the uniqueness of the solution. Let ui=

(fi, λi, µi, φi), i= 1,2 be two regular solutions of the Cauchy problem for the

same initial data (◦

f, ◦

λ, ◦

µ, ◦

φ, ψ) at t= 1. Using the fact that uiis a solution

of the system, one proceeds similarly as to prove the convergence of iterates to

obtain

α(t)≤CZ1

α(s)ds

where

α(t) = sup{k f1(s)−f2(s)k+kλ1(s)−λ2(s)k+kµ1(s)−µ2(s)k

+k˜g1(s)−˜g2(s)k+k˜

h1(s)−˜

h2(s)k;s∈[t, 1]},

with ˜g1=˙

φ1eµ1; ˜g2=˙

φ2eµ2;˜

h1=φ0

1eλ1;˜

h2=φ0

2eλ2. We deduce that

α(t) = 0, for t∈]0,1]. This implies that f1=f2,λ1=λ2,µ1=µ2, ˜g1= ˜g2

and ˜

h1=˜

h2. By Proposition 1.14, we conclude that φ1=φ2.

2.3 Continuation criteria

Let us now prove continuation criteria for tdecreasing.

Theorem 2.5 Let (◦

f, ◦

λ, ◦

µ, ◦

φ, ψ)be initial data as in theorem 2.4. Let (f, λ, µ, φ)

be a solution of (1.2)-(1.11) on a left maximal interval of existence ]T, 1] for

which :

1. sup{|w||(t, r, w, F)∈suppf}<∞,

2. sup{(e−2µ˙

φ2+e−2λφ02)(t, r); t∈]T, 1]}<∞,

3. µis bounded.

Then T= 0.

If k≥0or ◦

µ≤0then condition 3is automatically satisfied.

Proof : Let (f, λ, µ, ˜µ, g, h) be a left maximal solution of the auxiliary sys-

tem (1.14), (1.3), (1.4), (1.15), (1.16), (1.17) with existence interval ]T, 1]. By

Proposition 1.14, there exists φsuch that (f, λ, µ, φ) solves (1.2)-(1.11). By

assumption

Q∗:= sup{|w||(r, w, F)∈suppf(t), t ∈]T, 1]}<∞.

We want to show that T= 0, so let us assume that T > 0 and take t1∈]T, 1[. We

will show that the system has a solution with initial data (f(t1), λ(t1), µ(t1), φ(t1),

φ(t1)) prescribed at t=t1which exists on an interval [t1−δ, t1] with δ > 0

independent of t1. By moving t1close enough to Tthis would extend our initial

solution beyond T, a contradiction to the initial solution being left maximal.

We have proved in Proposition 2.1 that such a solution exists at least on the

left maximal existence interval of the solutions (z1, z2) of

z1(t) = W(t1) + K(t1) + C1Zt1

s(1 + z1(s))5ds

z2(t) = C4exp Zt1

C2(s)(C3+z2(s))ds,

where

W(t1) := sup{|w||(r, w, F)∈suppf(t1)},

K(t1) := sup{(|˙

φ2|e−2µ+|φ02|e−2λ)1

2(t1, r) ; r∈R},

A(t1) := sup{e−λ[( ˙

φ0−µ0˙

φ)2e−2µ+ (φ00 −λ0φ0)2e−2λ]1/2(t1, r) ; r∈R},

C1=C

(1 + F0)2(1+ kf(t1)k) ; C0= inf{e−µ(t1,r);r∈R}

C3:=ke−2µ(t1)µ0(t1)k+kλ0(t1)k+1 ; C4:= 2 + 2A(t1)+ k∂(r,w)f(t1)k

and C2is an increasing function of z1. Now W(t1)≤Q∗,kf(t1)k=k◦

fk,F0

is unchanged since Fis constant along characteristics, and since t1<1, taking

t=t1in (2.4) shows that: C0(µ(t1)) ≥C0(◦

µ). Thus there exists a constant

M1>0 such that

TC1(f(t1), F0, µ(t1)) ≤M1for t1∈]T, 1].

The expressions of ρ, p, j, ˙

λ, ˜µshow since |w| ≤ Q∗, that

|ρ(s, r)|,|p(s, r)|,|j(s, r)|,|˙

λ(s, r)|,|˜µeµ−λ(s, r)| ≤ C+ (K(s))2

K(s) is bounded on ]T, 1]. We proceed as in step 6 of the proof of theorem 3.1

in [15] to prove that ∂(r,w)fis uniformly bounded on ]T, 1]. Let µ0= ˜µ; consider

the relations

µ0=−4πteλ+µj;µ00 =−[(λ0+µ0)j+j0]4πteλ+µ(2.70)

λ0=e2µ(8πtµ0ρ+kµ0

t+ 4πtρ0) ; λ0=◦

λ0+Zt

λ0(s, r)ds (2.71)

We bound ρ0,j0by quantities which depend on ∂rfand A(s). Since ∂rf,ρand

jare bounded, we deduce from the above relations that µ00,˙

λ0and consequently

v(s) and h(s) (see Proposition 1.10) are bounded each by A(s). Using inequality

(1.27) and the fact that K(s) is bounded, we obtain

A(t)≤A0+CZ1

(A(s) + 1)ds

And we deduce that A(t)≤A∗= (1 + A0)eC;t∈]T, 1]. Therefore

D:= sup{k ∂(r,w)f(t)k+A(t)|t∈]T, 1]}<∞.

From

µ0(t, r) = e2µ

t◦

µ0(r)e−2◦

µ+ 4πRt

1p0(s, r)s2ds,

(2.70), (2.71) and since K(t) and A(t) are bounded, p0is bounded and we obtain

a uniform bound C3(µ(t1), λ(t1)) ≤M3. Let y2be the left maximal solution of

y2(t) = Dexp Zt1

C∗

2(s)(M3+y2(s))ds,

where C∗

2depends on y1in the same way as C1depends on z1.y2exists on

an interval [t1−α, t1] with α > 0 independent of t1. If we choose t1such that

T > t1−αthen z1is bounded; z2≤y2by construction. In particular, z2exists

on I⊂[t1−α, t1].

From (1.37), we deduce for k≥0 that e−2µ≥e−2◦

µ+k

t−k≥e−2◦

µsince 1

t≥1.

If k=−1 and ◦

µ≤0, then e−2µ≥e−2◦

µ−1

t+ 1 ≥1. Either, µis bounded above.

Then condition 3 holds. This complete the proof of the theorem.

Theorem 2.6 Let (f, λ, µ, φ)be a solution of (1.2)-(1.11) on a left maximal

interval of existence ]T, 1],T > 0, with initial data as in Theorem 2.5. If

1. Q(t) = sup{|w||(r, w, f)∈suppf(t), t ∈]T, 1]}<∞

2. µis bounded.

then T= 0.

If k≥0or ◦

µ≤0then condition 2is automatically satisfied.

Proof: We need to prove that K(t) is bounded for all t∈]T, 1]. Unless otherwise

specified in what follows constants denoted by Cwill be positive, may depend

on the initial data and may change their value from line to line.

We deduce from system (1.28)-(1.29) :

D+X2

2= 2eµk

t−4πt(ρ−p)X2

2−2e−µ

tX2Y2

D−Y2

2=−2e−µ

tX2Y2+ 2eµk

t−4πt(ρ−p)Y2

On the corresponding characteristic curves of the wave equation,

D+=D−=e−µd

dt and then

dtX2

2(t, γ1(t)) = 2e2µk

t−4πt(ρ−p)X2

2(t, γ1(t)) −2

tX2Y2(t, γ1(t))

dtY2

2(t, γ2(t)) = −2

tX2Y2(t, γ2(t)) + 2e2µk

t−4πt(ρ−p)Y2

2(t, γ2(t))

Integrate the first of the two previous equations on [t, 1] :

2(t, γ1(t)) = X2

2(1, γ1(1)) + 2 Z1

t{e2µ[−k

s+ 4πs(ρ−p)]X2

2+1

sX2Y2}(s, γ1(s))ds

≤X2

2(1, γ1(1)) + 2 Z1

t{e2µ[−k

s+ 4πs(ρ−p)]X2

2+1

2s(X2

2+Y2

2)}(s, γ1(s))ds

≤X2

2(1, γ1(1)) + 2 Z1

t{[1

2s+e2µ(−k

s+ 4πs(ρ−p))]X2

2+1

2sY2

2}(s, γ1(s))ds

In a similar way,

2(t, γ2(t)) ≤Y2

2(1, γ2(1))+2 Z1

t{1

2sX2

2+1

2s+e2µ(−k

s+ 4πs(ρ−p))Y2

2}(s, γ2(s))ds

Add the two previous inequalities and take the supremum over space :

B(t)2≤B(1)2+ 2 Z1

l(s)B(s)2ds (2.72)

where

B(t) = sup{(|X2|2+|Y2|2)1/2(t, r) ; r∈R}

l(t) = sup{1

t+e2µ[|k|

t+ 4πt(ρ−p)](t, r) ; r∈R}

Now subtract the two equations (1.8)-(1.9) to obtain :

ρ−p=π

t2Z∞

−∞ Z∞

(p1 + w2+F/t2−w2

p1 + w2+F/t2)fdFdw

=π

t2Z∞

−∞ Z∞

1 + F/t2

p1 + w2+F/t2fdFdw

≤π

t2Z∞

−∞ Z∞

1 + F/t2

p1 + F/t2fdFdw

≤π

t2Z∞

−∞ Z∞

0p1 + F/t2fdFdw

≤C

t3; since Q(t)<∞

Using (2.15) for the bound of eµ, we obtain :

l(t)≤1

t+|k|

tC0

Then (2.72) implies :

B(t)2≤B(1)2+ 2 Z1

(C(1 + 1

s) + 1

s)B(s)2ds

And by Gronwall’s lemma,

B(t)2≤B(1)2t−2eC(1−t)t−C≤Ct−C−2

Now, we have from (1.9),

p(s, r) = π

s2Z+∞

−∞ Z+∞

p1 + w2+F/s2f(s, r, w, F)dFdw +1

2(X2+Y2)(s, r)

≤π

s2Z+∞

−∞ Z+∞

|w|f(s, r, w, F)dFdw +1

2e−2µ(X2

2+Y2

2)(s, r)

≤C

s2+1

2e−2µB(s)2

≤C

s2+Cs−C−2e−2µ

Then using (1.37) (where ¯pis replaced by p), we obtain the estimate :

e−2µ(t,r)≤e−2◦

µ(r)+k

t−k+8π

tZ1

s2(C

s2+Cs−C−2e−2µ)ds

≤e−2◦

µ(r)+k

t−k+C

tZ1

(1 + s−Ce−2µ)ds

≤e−2◦

µ(r)+|k|

t+C

tZ1

(1 + s−Ce−2µ)ds

≤e−2◦

µ(r)+|k|

t+C

t(1 + Z1

s−Ce−2µds)

≤C

t(1 + Z1

s−Ce−2µds).

By Gronwall’s lemma, we deduce that e−2µ≤Ct−1exp[ C

1−C(1−t1−C)]. There-

fore,

(X2+Y2)(t, r) = e−2µ(X2

2+Y2

2)(t, r)≤Ct−3−Cexp[ C

1−C(1 −t1−C)]

i.e K(t) is bounded. And we conclude by Theorem 2.5 that T= 0.

We prove in the next theorem, the analogue of theorems 2.4 and 2.5 for

t≥1.

Theorem 2.7 Let (◦

f, ◦

λ, ◦

µ, ◦

φ, ψ)be initial data as in Theorem 2.4. Then there

exists a unique, right maximal, regular solution (f, λ, µ, φ)of (1.2)-(1.11) with

(f, λ, µ, φ, ˙

φ)(1) = (◦

f, ◦

λ, ◦

µ, ◦

φ, ψ)on a time interval [1, T[with T∈]1,∞]. If

sup{| w| |(t, r, w, F)∈suppf}<∞;

sup{e2µ(t,r)|r∈R, t ∈[1, T[}<∞

and

K(t)<∞

then T=∞.

Proof : We give only those parts of the proof which differ from the proof of

Theorem 2.4 for t≤1. The iterates are defined in the same way as before,

except that now (2.4) is used to define µnonly on the interval [1, Tn[, where

Tn:= sup (τ∈]1, Tn−1[|e−2◦

µ(r)+k

t−k−8π

tZt

s2pn(s, r)ds > 0, r ∈R, t ∈[1, τ]),

[1, Tn−1[ being the existence interval of the previous iterates and T0=∞. De-

fine:

Qn(t) := sup {|w|,(r, w, F)∈suppfn(t)}, t ∈[1, Tn[

En(t) := sup nse2µn(s,r)|r∈R,1≤s≤to

we obtain the estimates

p1 + w2+F/t2≤p1+(Qn(t))2+F0≤(1 + F0)(1 + Qn(t));

kρn(t)k,kpn(t)k,kjn(t)k≤ C∗

t(1 + Qn(t))2+ (Kn−1(t))2; (2.73)

and

|eµn−λn˜µn(t, r)|+|˙

λn(t, r)|≤ C∗(1 + Qn(t))2+ (1 + Kn−1(t))2(1 + En(t)).

where C∗=C(1 + F0)2(1+ k◦

fk). Thus, we have similarly to (2.16):

Qn+1(t)≤W0+C∗Zt

(1 + Qn(s))2(1 + En(s))(1 + Kn−1(s))2(1 + Qn+1(s))ds.

(2.74)

and similarly to (2.18):

Kn+1(t)≤K0+C∗Zt

(1 + Qn(s))2(1 + Kn−1(s))2Kn(s)(1 + En(s))ds. (2.75)

We deduce from the field equation (1.4) that

(2 ˙µne2µn)t=e2µn+ke4µn+ 8π(te2µn)2pn

Integrating over [1, s] and using integration by parts for the left hand side yields

the following estimate, since (2.73) holds,

En(t)≤k e2◦

µk+C∗Zt

(1 + Qn(s))2(1 + En(s))2(1 + Kn−1(s))2ds. (2.76)

Reasoning in the same way as in the proof of Proposition 2.1, we can say the

differential inequalities (2.74), (2.75), (2.76) allow us to estimate Qn,Knand

Enagainst the solution z1,z2and z3of the system

z1(t) = W0+C∗Zt

(1 + z1(s))3(1 + z2(s))2(1 + z3(s))ds,

z2(t) = K0+C∗Zt

(1 + z1(s))2(1 + z2(s))3](1 + z3(s))ds,

z3(t) =ke2◦

µk+C∗Zt

(1 + z1(s))2(1 + z2(s))2(1 + z3(s))2ds,

and in particular Tn≥Twhere [1, T[ is the right maximal existence interval

of (z1, z2, z3). One can now establish a bound on first order derivatives of

the iterates in the same way as in the proof of Proposition 2.1 and obtains a

local solution on a right maximal existence interval which is extendible if the

quantities Q(t) , E(t) =ke2µ(t)kand K(t) can be bounded.

Theorem 2.8 Let (f, λ, µ, φ)be a right maximal regular solution obtained in

Theorem 2.7. Assume that

sup{| w| |(t, r, w, F)∈suppf}<∞;

and

sup{e2µ(t,r)|r∈R, t ∈[1, T[}< C < ∞;

then T=∞.

Proof : We deduce from system (1.12)-(1.13):

D+X2= 2aX2+ 2bXY

D−Y2= 2bXY + 2cY 2

On the respectively characteristic curves of the wave equation, D+=D−=

e−µd

dt and then we obtain :

dtX2(t, γ1(t)) = 2eµ(aX2+bXY )(t, γ1(t)) (2.77)

dtY2(t, γ2(t)) = 2eµ(bXY +cY 2)(t, γ2(t)) (2.78)

¿From (2.77), we have:

dtX2(t, γ1(t)) = 2(−˙

λ−µ0eµ−λ−1

t)X2−2XY

= 8πte2µ(j−ρ)X2+1 + ke2µ

tX2−2

tX2−2XY

≤(−1

t+ke2µ

t)X2+X2+Y2

tsince j−ρ < 0 ;

If k= 0 or −1 then d

dtX2(t, γ1(t)) ≤1

tY2(t, γ1(t)).

If k= 1 then d

dtX2(t, γ1(t)) ≤1

t(CX2+Y2)(t, γ1(t)).

Since j+ρ > 0, we deduce as above, from (2.78), the estimates

dtY2(t, γ2(t)) ≤1

tX2(t, γ2(t)) for k= 0 or k=−1;

dtY2(t, γ2(t)) ≤1

t(X2+CY 2)(t, γ2(t)) for k= 1.

After integrating these two inequalities over [1, t] and taking the maximum over

space, we obtain :

K(t)2≤K(1)2+Zt

sK(s)2ds for k= 0 or k=−1;

K(t)2≤K(1)2+Zt

1 + C

sK(s)2ds for k= 1.

We deduce by Gronwall’s lemma that:

K(t)2≤K(1)2t , for k= 0 or k=−1 and t∈[1, T [ (2.79)

K(t)2≤K(1)2t1+C,for k= 1 and t∈[1, T[ (2.80)

And we conclude by Theorem 2.7 that T=∞.

Chapter 3

Global existence and

asymptotic behaviour of

solutions in the past

3.1 Global existence in the past

We prove that the solutions obtained in Theorem 2.4 exist on the whole interval

]0,1].

Theorem 3.1 Consider a solution of the Einstein-Vlasov system with k≥0

and initial data given for t= 1. Then this solution exists on the whole interval

]0,1]. If k < 0and ◦

µ≤0, the same result holds.

Proof : we follow the works of M. Weaver [32] and S.B. Tchapnda [29]. The

strategy of the proof is the following : suppose we have a solution on an interval

]T, 1] with T > 0. We want to show that the solution can be extended to the

past. By consideration of the maximal interval of existence this will prove the

assertion.

Firstly let us prove that under the hypotheses of the theorem, µis bounded

above.

From the field equation (1.4) we have for k≥0,

dt(te−2µ) = −k−8πt2p≤0.(3.1)

So te−2µcannot increase towards the future, i.e. it cannot decrease towards the

past. Thus on ]T, 1], te−2µmust remain bounded away from zero and hence µ

is bounded above.

For the case k=−1, since p(s, r)≥0, we get from (1.37), e−2µ≥e−2◦

µ−1

t+1 ≥1

which gives the upper bound of µfor ◦

µ≤0.

Now, let us prove that wis bounded.

Consider the following rescaled version of w, called u1, which has been inspired

by the works of [32] (p. 1090) and [30] (p. 5):

u1=eµ

2tw.

If we prove that µis bounded below then the boundedness of u1will imply the

boundedness of w. So let us show that µis bounded below under the assumption

that u1is bounded.

We have d

dt(te−2µ) = −k−8πt2p. (3.2)

Transforming the integral term defining pto u1as an integration variable instead

of wyields

p=Z∞

−∞ Z∞

8πte−3µu2

p1+4t2e−2µu2

1+F/t2fdFdu1+1

4(X2+Y2);

where Xand Yare defined in lemma 1.5. The integrand term in pcan then be

estimated by 4πe−2µ|u1|. We have

e2µ(ρ−p) = π

t2Z∞

−∞ Z∞

1 + F/t2

p1 + w2+F/t2e2µfdFdw

=π

t2Z∞

−∞ Z∞

(1 + F/t2)2teµ

p1+4t2e−2µu2

1+F/t2fdFdu1

≤2π

tZ∞

−∞ Z∞

(1 + F/t2)eµfdFdu1

≤2π

t3eµZ¯u1

−¯u1ZF0

(1 + F)fdFdu1

≤Ct−3¯u1eµ

where ¯u1is the maximum modulus of u1on the support of fat a given time. We

can then estimate from (2.72) l(s) by h(s) = Csup{1+s−1+s−2eµ¯u1(s, r); r∈

R}and B(t)2by B(1)2exp(R1

th(s)ds). Thus

X2+Y2≤e−2µB(t)2≤e−2µB(1)2exp(Z1

h(s)ds).

Therefore, using the bound for µand u1,pcan be estimated by Ce−2µand so

(3.2) implies that

dt(te−2µ)| ≤ C(1 + te−2µ),

integrating this with respect to tover [t, 1] yields

te−2µ(t, r)≤e−2µ(1,r)+Z1

C1 + se−2µ(s,r)ds,

which implies by the Gronwall inequality that te−2µis bounded on ]T, 1]; that

is µis bounded below on the given time interval.

The next step is to prove that u1is bounded. To this end, it suffices to get

a suitable integral inequality for ¯u1. Since u1=u1(t, r(t)), we can compute ˙u1:

˙u1=−eµ

2t2w+eµ

2tw( ˙µ+ ˙rµ0) + eµ

2t˙w

i.e.

˙u1=˙µ+ ˙rµ0−1

tu1+eµ

2t˙w(3.3)

We have

µ0=−4πteµ+λj, ˙r=eµ−λw

p1 + w2+F/t2

and

˙w= 4πte2µ(jp1 + w2+F/t2−ρw) + 1 + ke2µ

2tw

so that (3.3) implies the following :

˙u1=e2µ−4πt(ρ−p) + k

tu1+ 2πe3µj1 + F/t2

p1+4t2e−2µu2

1+F/t2.

i.e

˙u1|u1|=e2µ−4πt(ρ−p) + k

tu1|u1|+ 2πe3µj(1 + F/t2)|u1|

p1+4t2e−2µu2

1+F/t2(3.4)

In order to estimate the modulus of the first term on the right hand side of

equation (3.4), we need the estimate of e2µ(ρ−p)¯u2

1. For convenience let log+

be defined by log+(x) = log xwhen log xis positive and log+(x) = 0, otherwise.

Then estimating the integral defining ρ−pshows that

ρ−p≤C(1 + log+( ¯w)),

i.e.

ρ−p≤C(1 + log+(¯u1)−µ).

The expression −µis not under control ; however the expression we wish to

estimate contains a factor e2µ. The function µ7→ −µe2µhas an absolute maxi-

mum at −1/2 which is (1/2)e−1. Thus the first term on the right hand side of

equation (3.4) can be estimated by C¯u2

1(1 + log+(¯u1)).

Next the second term on the right hand side of equation (3.4) will be esti-

mated.

By definition

j=π

t2Z∞

−∞ Z∞

wf(t, r, w, F)dFdw −˙

φφ0e−µ−λ=j1+j2

The first term of jcan be estimated by C¯w2, i.e.

|j1| ≤ C¯u2

1e−2µ

and the second term |j2| ≤ 1

2e−2µB(t)2; so that it suffices to estimate the

quantity

(¯u2

1+B(t)2)(1 + F/t2)

p1+4t2e−2µu2

1+F/t2|u1|(3.5)

in order to estimate the second term on the right hand side of equation (3.4).

But since µand t−1are bounded on the interval being considered, the quantity

(3.5) can be estimated by C(¯u2

1+B(t)2). Thus adding the estimates for the

first and second terms on the right hand side of (3.4) allows us to deduce from

(3.4) that

|˙u1||u1| ≤ C¯u2

1(1 + log+(¯u1))+C(¯u2

1+B(t)2)

i.e

dt|u1|2| ≤ C(¯u1)2(1 + log+(¯u2

1)) + CB(t)2

Integrating over [t, 1] gives :

¯u2

1(t)≤¯u2

1(1) + CZ1

t¯u2

1(s)(1 + log+(¯u2

1(s))+B(s)2ds (3.6)

We deduce from the estimate of ρ−pand from inequality (2.72), that l(s) can

be estimated by C(1 + log+(¯u1)). We then obtain

B(t)2≤B(1)2+CZ1

(1 + log+(¯u2

1)(s))B(s)2ds (3.7)

Adding (3.6) and (3.7) gives estimate :

¯u2

1(t)+B(t)2≤¯u2

1(1)+B(1)2+CZ1

(1+¯u2

1+B(s)2)1 + log+(1 + ¯u2

1(s) + B(s)2)ds

(3.8)

Set v(t) = ¯u2

1(t) + B(t)2, then (3.8) can be written

v(t)≤v(1) + CZ1

(1 + v(s)) 1 + log+(1 + v(s))ds (3.9)

By the comparison principle for solutions of integral equations, it is enough to

show that the solution of the integral equation

a(t) = a(1) + CZ1

(1 + a(s)) 1 + log+(1 + a(s))ds

is bounded. The solution a(t) is a non-increasing function. Thus either a(t)≤e

everywhere, in which case the desire conclusion is immediate. Or there is some

T1in ]T, 1] such that e≤a(t) on ]T, T1]. We take T1maximal with that property.

Then it follows on ]T, T1] that inequality

a(t)≤C 1 + ZT1

a(s)(1 + log a(s))ds!

holds for a constant C. The boundedness of a(t) follows from that of the solution

of the differential equation ˙x=Cx(1 + log x) which is exp(exp(Ct)−1). In

either case a(t) is bounded. Thus ¯u2

1and B(t)2are bounded i.e wand K(t) are

bounded. The proof of the theorem is complete using theorem 2.6.

3.2 On past asymptotic behaviour

In this section we examine the behaviour of solutions as t→0.

Firstly we follow the work of Ringstrom [24](P. S310-S311) to bound the

quantity |φ0|eµ−λby C|tlog t|−1, where C is a positive constant.

Lemma 3.2 Let A1=1

8(−˙

φ+φ

tlog t+φ0eµ−λ)2and

A2=1

8(−˙

φ+φ

tlog t−φ0eµ−λ)2with t∈]0,1[. If φsatisfies the wave equation,

then

(∂t+e−µ−λ∂r)A1=−1

4t(1 + 1

log t)[(−˙

φ+φ

tlog t)2+φ02e2µ−2λ]

2t(1 + 1

log t)φ02e2µ−2λ+1

4(˙

λ−˙µ+1

t)( ˙

φ−φ0eµ−λ)(−˙

φ+φ

tlog t+φ0eµ−λ)

(3.10)

(∂t+e−µ−λ∂r)A2=−1

4t(1 + 1

log t)[(−˙

φ+φ

tlog t)2+φ02e2µ−2λ]

2t(1 + 1

log t)φ02e2µ−2λ+1

4(˙

λ−˙µ+1

t)( ˙

φ+φ0eµ−λ)(−˙

φ+φ

tlog t−φ0eµ−λ)

(3.11)

Proof : We have,

8(∂t+e−µ−λ∂r)A1= 2[−¨

φ+˙

tlog t−φ(1 + log t)

(tlog t)2+ ( ˙µ−˙

λ)φ0eµ−λ+˙

φ0eµ−λ

+eµ−λ(−˙

φ0+φ0

tlog t+ (µ0−λ0)φ0eµ−λ+φ00eµ−λ)](−˙

φ+φ

tlog t+φ0eµ−λ)

= 2[−¨

φ+φ00e2µ−2λ+˙

tlog t−φ(1 + log t)

(tlog t)2+ ( ˙µ−˙

λ)φ0eµ−λ

+φ0eµ−λ

tlog t+ (µ0−λ0)φ0e2µ−2λ](−˙

φ+φ

tlog t+φ0eµ−λ)

i.e

8(∂t+e−µ−λ∂r)A1= 2[( ˙

λ−˙µ+1

t)˙

φ+ (µ0−λ0)φ0e2µ−2λ+˙

tlog t−φ(1 + log t)

(tlog t)2

+ ( ˙µ−˙

λ)φ0eµ−λ+φ0eµ−λ

tlog t+ (µ0−λ0)φ0e2µ−2λ](−˙

φ+φ

tlog t+φ0eµ−λ)

= 2[( ˙

λ−˙µ+1

t)˙

φ+˙

tlog t−φ

(tlog t)2−φ

t2log t

+ ( ˙µ−˙

λ)φ0eµ−λ+φ0eµ−λ

tlog t](−˙

φ+φ

tlog t+φ0eµ−λ)

= 2[( ˙

λ−˙µ+1

t)( ˙

φ−φ0eµ−λ) + 1

tφ0eµ−λ+1

tlog tφ0eµ−λ

tlog t(˙

φ−φ

tlog t) + 1

t(˙

φ−φ

tlog t)](−˙

φ+φ

tlog t+φ0eµ−λ)

= 2(−˙

φ+φ

tlog t+φ0eµ−λ)(−1

tlog t−1

t)(−˙

φ+φ

tlog t−φ0eµ−λ)

+ 2( ˙

λ−˙µ+1

t)( ˙

φ−φ0eµ−λ)(−˙

φ+φ

tlog t+φ0eµ−λ)

=−2

t(1 + 1

log t)(−˙

φ+φ

tlog t)2−φ02e2µ−2λ)

+ 2( ˙

λ−˙µ+1

t)( ˙

φ−φ0eµ−λ)(−˙

φ+φ

tlog t+φ0eµ−λ)

=−2

t(1 + 1

log t)(−˙

φ+φ

tlog t)2+φ02e2µ−2λ+4

t(1 + 1

log t)φ02e2µ−2λ

+ 2( ˙

λ−˙µ+1

t)( ˙

φ−φ0eµ−λ)(−˙

φ+φ

tlog t+φ0eµ−λ);

and (3.10) follows. If we replace rby −r, the wave equation is invariant, ∂t+

e−µ−λ∂rand ∂t−e−µ−λ∂r,A1and A2interchange; and we can write (3.11).

Lemma 3.3 Let (f, λ, µ, φ)be a left maximal solution of the Einstein-Vlasov-

scalar field system on the interval ]T, 1],0≤T < e−1. Assume that

Q(t) = sup{| w| |(r, w, F)∈suppf(t)} ≤ Ctα

for some positive constants C,αand for some t∈]T, e−1]. Then

(−˙

φ+φ

tlog t)2+φ02e2µ−2λ≤C(tlog t)−2(3.12)

Proof : Consider the two characteristic curves (t, γ1(t)) and (t, γ2(t)) of the

wave operator. Since t∈]0, e−1], the term (1 + 1

log t)φ02e2µ−2λis nonnegative

and (−˙

φ+φ

tlog t)2+φ02e2µ−2λ= 4(A1+A2), then from (3.10) :

(∂t+e−µ−λ∂r)A1(t, γ1(t)) ≥ −1

t(1 + 1

log t)(A1+A2)(t, γ1(t))

−1

4(˙

λ−˙µ+1

t)(−˙

φ+φ0eµ−λ)(−˙

φ+φ

tlog t+φ0eµ−λ)(t, γ1(t))

(3.13)

Similarly, we deduce from (3.11) that :

(∂t−e−µ−λ∂r)A2(t, γ2(t)) ≥ −1

t(1 + 1

log t)(A1+A2)(t, γ2(t))

−1

4(˙

λ−˙µ+1

t)(−˙

φ−φ0eµ−λ)(−˙

φ+φ

tlog t−φ0eµ−λ)(t, γ2(t))

(3.14)

Since Q(t)≤Ctα, we can bound ρ−pby Ct−3+α(see the proof of Theorem

2.6). e2µ≤Ct; then

(˙

λ−˙µ)(t) + 1

t=−ke2µ

t+ 4πte2µ(ρ−p)≤C(1 + t−1+α); and from (2.72), l(s)

can be bounded by s−1+C+Cs−1+α. We deduce from (2.72) (consider the

integral term in the interval [t, e−1]) that

B(t)2≤B(e−1)2exp[2 Re−1

t(s−1+C+Cs−1+α)ds] i.e B(t)2≤Ct−2. There-

fore |˙

φ(t)|and |φ0|eµ−λ(t) are bounded each by Ct−1. We can then have a lower

bound of the second term of the right hand side of each inequality (3.13) and

(3.14) which is −C(t−2+t−3+α). Then

(∂t+e−µ−λ∂r)A1(t, γ1(t)) ≥ −1

t(1 + 1

log t)(A1+A2)(t, γ1(t)) −C(t−2+t−3+α)

and

(∂t+e−µ−λ∂r)A2(t, γ2(t)) ≥ −1

t(1 + 1

log t)(A1+A2)(t, γ2(t)) −C(t−2+t−3+α)

On the corresponding characteristic, we have ∂t+e−µ−λ∂r=∂t−e−µ−λ∂r=

dt . Take the supremum in the space of each of the above two inequalities and

add them. Then

dt(A1+A2)(t, r)≥ −2

t(1 + 1

log t)(A1+A2)(t, r)−C(t−2+t−3+α)

Set u(t) = (A1+A2)(t) and v(t) = (tlog t)2(A1+A2)(t). If v(t) is bounded,

then we conclude that u(t) is bounded by C(tlog t)−2. Let us prove that v(t) is

bounded. We have :

dt = 2t(log t)u+ 2t(log t)2u+ (tlog t)2du

= (tlog t)2(du

dt +2

t(1 + 1

log t)u)

≥(tlog t)2(−Ct−2−Ct−3+α) = −C(log t)2(1 + t−1+α)

Then, v(t)≤v(e−1) + CRe−1

t(1 + s−1+α)(log s)2ds ≤v(e−1) + C. We obtain

the desired conclusion of the lemma.

Remark 3.4 In the case f= 0, we obtain from previous theorem and theorem

2.5, the global existence of solutions and the above estimates hold on the whole

interval ]0, e−1].

Remark 3.5 We have from the field equation (1.4),

e−2µ=e−2◦

µ+k

t−k+8π

tZ1

s2p(s, r)ds ≥e−2◦

µ+k

t−k

and then, for k= 0,e−2µ≥e−2◦

t≥C0

for k= 1,e−2µ≥e−2◦

µ+1

t−1≥e−2◦

t≥C0

for k=−1,e−2µ≥e−2◦

µ−1

t+ 1 ≥C0

t+ 1 ;

where

C0=(inf e−2◦

µfor k= 0 or 1

inf e−2◦

µ−1for k=−1

Thus

e2µ≤t

C0for k= 0 or k= 1 ;

and e2µ≤t

C0+tfor k=−1.

Remark 3.6 Let us prove that inf e−2◦

µ(r)=1

||e2◦

µ||.

For all r∈R,e2◦

µ(r)≤ ||e2◦

µ||, hence e−2◦

µ(r)≥1

||e2◦

µ|| ; and

inf e−2◦

µ(r)≥1

||e2◦

µ||.

For all r∈R,e−2◦

µ(r)≥inf e−2◦

µ(r), i.e e2◦

µ(r)≤1

inf e−2◦

µ(r)and

||e2◦

µ|| ≤ 1

inf e−2◦

µ(r)i.e inf e−2◦

µ(r)≤1

||e2◦

µ||. This completes the proof.

First we analyze the curvature invariant RαβγδRαβγδ called the Kretschman

scalar in order to prove that there is a spacetime singularity. Thus the spacetime

cannot be extended further.

Theorem 3.7 Let (f, λ, µ, φ)be a regular solution of the surface-symmetric

Einstein-Vlasov-scalar field system on the interval ]0,1] with data given for

t= 1. Then

(RαβγδRαβγδ)(t, r)≥4

t6inf e−2◦

µ+k2

with r∈R.

Proof We can compute the Kretschman scalar (see [15]) and obtain

RαβγδRαβγδ = 4[e−2λ(µ00 +µ0(µ0−λ0)) −e−2µ(¨

λ+˙

λ(˙

λ−˙µ))]2

t2[e−4µ˙

λ2+e−4µ˙µ2−2e−2(λ+µ)(µ0)2]

t4(e−2µ+k)2

=: K1+K2+K3

The first term K1is nonnegative and can be dropped.

Inserting the expressions

e−2µ˙

λ= 4πtρ −k+e−2µ

2t;e−2µ˙µ= 4πtp +k+e−2µ

2t;e−λ−µµ0=−4πtj

into the formula for K2yields

K2=8

t2(4πtρ −k+e−2µ

2t)2+ (4πtp +k+e−2µ

2t)2−2(−4πtj)2

t216π2t2(ρ2+p2−2j2)−4πt(ρ−p)k+e−2µ

t+(k+e−2µ)2

2t2.

Now

|j(t, r)| ≤ π

t2Z∞

−∞ Z∞

0|w|fdFdw +|˙

φ|e−µ|φ0|e−λ

≤π

t2Z∞

−∞ Z∞

(1 + w2+F/t2)1/4f1/2|w|

(1 + w2+F/t2)1/4f1/2dFdw

2(˙

φ2e−2µ+φ02e−2λ)

≤π

t2[Z∞

−∞ Z∞

0p1 + w2+F/t2fdFdw]1

2[Z∞

−∞ Z∞

p1 + w2+F/t2fdFdw]1

2(˙

φ2e−2µ+φ02e−2λ) by the Cauchy-Schwarz inequality.

≤π

2t2[Z∞

−∞ Z∞

0p1 + w2+F/t2fdFdw +Z∞

−∞ Z∞

p1 + w2+F/t2fdFdw]

2(˙

φ2e−2µ+φ02e−2λ)

≤1

2(ρ+p)(t, r).

In fact the above inequality holds in general for all choice of matter satisfying

the dominant energy condition. Therefore

ρ2+p2−2j2≥ρ2+p2−1

2(ρ+p)2=1

2ρ2+1

2p2−ρp =1

2(ρ−p)2

and

K2≥8

t28πt2(ρ−p)2−4πt(ρ−p)k+e−2µ

t+(k+e−2µ)2

2t2

≥4

t24πt(ρ−p)−k+e−2µ

t2

≥0.

Recalling the expression for e−2µwe get

e−2µ+k=e−2◦

µ(r)+k

t+8π

tZ1

s2p(s, r)ds

≥e−2◦

µ+k

t≥inf e−2◦

µ+k

thus

K3=4

t4(e−2µ+k)2≥4

t6inf e−2◦

µ+k2

and so we deduce from (3.15) that

(RαβγδRαβγδ)(t, r)≥4

t6inf e−2◦

µ(r)+k2

, t ∈]0,1], r ∈R,

and the proof is complete. 

Remark 3.8 We deduce from the above inequality that

lim

t→0(RαβγδRαβγδ)(t, r) = ∞,

uniformly in r∈R.

Next we prove that the singularity at t= 0 is a crushing singularity i.e.

the mean curvature of the surfaces of constant tblows up. In the case where

there is only a scalar field and no Vlasov contribution this singularity is velocity

dominated i.e the generalized Kasner exponents have limits as t→0.

Theorem 3.9 Let (f, λ, µ, φ)be a regular solution of the surface-symmetric

Einstein-Vlasov-scalar field system on the interval ]0,1] with initial data given

on t= 1. Let

K(t, r) := −e−µ˙

λ(t, r) + 2

t

denotes the mean curvature of the hypersurfaces of constant t. Then

K(t, r)≤ −Ct−3/2,

where Cis a positive constant.

Proof For a metric of the form ds2=−α(t, r)2dt2+gijdxidxj,

the second fundamental form of the hypersurfaces of constant tis given by

Kij =−(2α)−1∂tgij and its trace K(t, r) = gij Kij is the mean curvature of

that hypersurfaces. For gdefined by (1.1), we have

Kij =−e−µ

2∂tgij. Thus

K11 =−e−µ

2∂tg11 =−˙

λe2λ−µ;K22 =−e−µ

2∂tg22 =−te−µ;

K33 =−te−µsin2

kθ.

And K(t, r) = −(˙

λ+2

t)e−µ. We have

λ=e2µ4πtρ −k+e−2µ

2t≥ −e2µk+e−2µ

2t.(3.15)

and

K(t, r)≤k−3e−2µ

2teµ.

For k= 0 or k=−1,

K(t, r)≤ − 3

2te−µ.

and the estimate

e−2µ≥e−2◦

µ+k

implies

K(t, r)≤ − 3

2t(inf e−2◦

µ+k

t)1/2≤ −Ct−3/2where C=3

2(inf e−2◦

µ+k)1/2.

For k= 1 we have

e−2µ≥e−2◦

t>1 = k(since t < 1)

thus

K(t, r)≤eµ1−3e−2µ

2t≤(e2µ−3

2)e−µ

≤ −e−µ

t≤ −inf e−◦

t3/2

≤ −Ct−3/2where C= inf e−◦

µ.



Remark 3.10 We deduce from above that

lim

t→0K(t, r) = −∞,

uniformly in r∈R.

Theorem 3.11 Let (λ, µ, φ)be a regular solution of the spherical, plane and hy-

perbolic symmetric Einstein-scalar field system on the interval ]0,1] with initial

data given at t= 1. Then

lim

t→0

1(t, r)

K(t, r)=a(r) ; lim

t→0

2(t, r)

K(t, r)= lim

t→0

3(t, r)

K(t, r)=1

2(1 −a(r)),

uniformly in r∈R,

where K1

1(t, r)

K(t, r),K2

2(t, r)

K(t, r),K3

3(t, r)

K(t, r)

are the generalized Kasner exponents and a(r)a function of r.

Proof We have

1(t, r)

K(t, r)=t˙

λ(t, r)

t˙

λ(t, r)+2;K2

2(t, r)

K(t, r)=K3

3(t, r)

K(t, r)=1

t˙

λ(t, r)+2.

with

t˙

λ= 4πt2e2µρ−k

2e2µ−1

As we have seen previously

e2µ(t,r)≤Ct

which implies that

e2µ(t,r)→0 as t→0

Let t0∈]0, e−1] and t∈]0, t0]. From (3.11),

∂t(φ

log t) = 1

log t(˙

φ−φ

tlog t) = O(t−1(log t)−2)

so that φ(t, r)

log t=φ(t0, r)

log t0−Zt0

s−1(log s)−2ds.

The integral term of the above relation converges as t→0. Set

A(r) = lim

t→0

φ(t, r)

log t=φ(t0, r)

log t0−Zt0

s−1(log s)−2ds

Since from (3.11), ( ˙

φ−φ

tlog t) = O((t|log t|)−1), we have

t˙

φ=φ

log t+O((|log t|)−1)

so that t˙

φ→A(r) as t→0. Inequality (3.11) shows also that

φ02e2µ−2λ=O((t|log t|)−2). Using these limits, we have

t˙

λ(t, r) = 2π(t2˙

φ2+t2φ02e2µ−2λ)−k

2e2µ−1

2→2πA(r)2−1

2as t→0, uniformly in r.

We take a(r) = 4πA(r)2−1

4πA(r)2+3 to complete the proof . 

Chapter 4

Global existence and

asymptotic behaviour of

solutions in the future

4.1 Global existence in the future

We prove the global existence in the future in the cases of plane and hyperbolic

symmetries.

Theorem 4.1 Assume that (f, λ, µ, φ)is a right maximal regular solution of

the Einstein-Vlasov system with scalar field obtained in Theorem 2.7. Then for

k= 0 or k=−1,

sup{e2µ(t,r)|r∈R, t ∈[1, T[}<∞.

Proof : We now establish a series of estimates which will result in an upper

bound on µ. Similar estimates were used in [4] and [3]. Unless otherwise spec-

ified in what follows constants denoted by Cwill be positive, may depend on

the initial data and may change their value from line to line.

Firstly, integration of (1.4) with respect to tand the fact that pis non-negative

imply that

e2µ(t,r)="e−2◦

µ(r)+k

t−k−8π

tZt

s2p(s, r)ds#−1

≥t

C−kt, t ∈[1, T[ (4.1)

Next let us show that

eµ+λρ(t, r)dr ≤Ct, t ∈[1, T[ (4.2)

We have d

dt Z1

eµ+λρ(t, r)dr =Z1

[( ˙

λ+ ˙µ)eµ+λρ+eµ+λ˙ρ]dr

with

˙ρ=−2π

t3Z∞

−∞ Z∞

0p1 + w2+F/t2fdFdw +π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw

+π

t2Z∞

−∞ Z∞

(−F

t3)f

p1 + w2+F/t2dFdw +e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)

=−2

tρ−1

tq+1

t(e−2µ˙

φ2+e−2λφ02) + 1

t(e−2µ˙

φ2−e−2λφ02)

+π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw +e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)

=−2

tρ−1

tq+2

te−2µ˙

φ2+e−2µ(−˙µ˙

φ2+˙

φ¨

φ)

+e−2λ(−˙

λφ02+˙

φ0φ0) + π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw.

The Vlasov equation and integration by parts imply,

t2Z∞

−∞ Z∞

0p1 + w2+F/t2∂tfdFdw

=π

t2Z∞

−∞ Z∞

0p1 + w2+F/t2"(˙

λw +eµ−λµ0p1 + w2+F/t2)∂wf−eµ−λw

p1 + w2+F/t2∂rf#dFdw

=π

t2Z∞

−∞ Z∞

0h˙

λwp1 + w2+F/t2+eµ−λµ0(1 + w2+F/t2)i∂wfdFdw

−π

t2Z∞

−∞ Z∞

eµ−λw∂rfdFdw

=−eµ−λhj0+e−µ(−µ0˙

φ+˙

φ0)φ0e−λ+e−λ(−λ0φ0+φ00)˙

φe−µi

−˙

λπ

t2Z∞

−∞ Z∞

0 p1 + w2+F/t2+w2

p1 + w2+F/t2!fdFdw

−µ0eµ−λπ

t2Z∞

−∞ Z∞

2wfdFdw

And so

˙ρ=−2

tρ−1

tq−eµ−λj0−2µ0jeµ−λ−˙

λ(ρ+p) + 2

te−2µ˙

φ2

+e−2µ(−˙µ˙

φ2+˙

φ¨

φ) + e−2λ(−˙

λφ02+˙

φ0φ0)+(λ0+µ0)e−2λ˙

φφ0

−e−2λ(˙

φ0φ0+˙

φφ00)−2µ0˙

φφ0e−2λ+˙

λ(e−2µ˙

φ2+e−2λφ02)

=−2

tρ−1

tq−eµ−λ(j0+ 2µ0j)−˙

λ(ρ+p)+(˙

λ−˙µ+2

t)˙

φ2e−2µ

+ (λ0−µ0)˙

φφ0e−2λ+˙

φ(e−2µ¨

φ−e−2λφ00)

=−2

tρ−1

tq−eµ−λ(j0+ 2µ0j)−(4πtρe2µ−1 + ke2µ

2t)(ρ+p);

where we use the wave equation to substitute the term e−2µ¨

φ−e−2λφ00 and also

the expression of ˙

λ. Therefore,

dt Z1

eµ+λρ(t, r)dr =−1

tZ1

eµ+λ2ρ+q+eµ−λ(j0+ 2µ0j)−ρ+p

2(1 + ke2µ)dr

=−1

tZ1

eµ+λ2ρ+q−ρ+p

2(1 + ke2µ)−Z1

(e2µj)0dr

(4.3)

and since µand jare periodic with respect to r, we deduce that:

dt Z1

eµ+λρ(t, r)dr =−1

tZ1

eµ+λ2ρ+q−ρ+p

2(1 + ke2µ)dr

For k= 0, since q≥ −2ρ, we have q+3ρ−p

2≥q+ρ≥ −ρ; we deduce that:

dt Z1

eµ+λρ(t, r)dr ≤1

tZ1

eµ+λρ(t, r)dr

Integrating this inequality with respect to tyields (4.2) for k= 0. For k=−1,

we have, using (4.1):

dt Z1

eµ+λρ(t, r)dr =−1

tZ1

eµ+λ2ρ+q−ρ+p

2(1 −e2µ)dr

≤ −1

tZ1

eµ+λ(2ρ+q−ρ+p

2)dr −1

C+tZ1

eµ+λρ+p

2dr

≤ −1

tZ1

eµ+λ(2ρ+q)dr + (1

t−1

C+t)Z1

eµ+λρ+p

2dr

≤1

tZ1

eµ+λρdr + (1

t−1

C+t)Z1

eµ+λρdr

≤(2

t−1

C+t)Z1

eµ+λρdr

where we use the fact that 2ρ+q≥ −ρand ρ+p

2≤ρ. Integrating the above

inequality with respect to tyields (4.2) for k=−1. Using equation µ0=

−4πteµ+λj, the fact that |j| ≤ ρand (4.2) we find

|µ(t, r)−Z1

µ(t, σ)dσ |=|Z1

0Zr

µ0(t, τ)dτdσ |≤ Z1

0Z1

0|µ0(t, τ)|dτdσ

≤4πt Z1

eµ+λ|j(t, τ)|dτ ≤4πt Z1

eµ+λρ(t, τ)dτ

that is

|µ(t, r)−Z1

µ(t, σ)dσ |≤ Ct2, t ∈[1, T[, r ∈[0,1] (4.4)

Next we show that

eµ(t,r)−λ(t,r)≤Ct, t ∈[1, T [, r ∈[0,1].(4.5)

To see this observe that : relation ˙µ−˙

λ= 4πte2µ(p−ρ) + 1+ke2µ

t,p−ρ≤0

and (4.1) imply that

∂

∂teµ−λ= ( ˙µ−˙

λ)eµ−λ=eµ−λ4πte2µ(p−ρ) + 1 + ke2µ

t

≤1 + ke2µ

teµ−λ≤(1

t+k

C−kt)eµ−λ;

and integrating this inequality with respect to tyields

eµ−λ≤Ct

C−kt ≤Ct,

i.e. (4.5).

We now estimate the average of µ(t) over the interval [0,1] which in combi-

nation with (4.4) will yield the desired upper bound on µ. We use (1.4), (4.2),

(4.5) and the fact that p≤ρ,ke2µ≤0 :

µ(t, r)dr =Z1

◦

µ(r)dr +Zt

1Z1

˙µ(s, r)drds

≤C+Zt

2sZ1

[e2µ(8πs2p+k) + 1]drds

=C+1

2ln t+ 4πZt

1Z1

se2µpdrds +Zt

1Z1

ke2µ

2sdrds

≤C+1

2ln t+ 4πZt

1Z1

seµ−λeµ+λρdrds +Zt

1Z1

ke2µ

2sdrds

≤C+1

2ln t+CZt

s3ds

=C+1

2ln t+Ct4

with (4.4) this implies

µ(t, r)≤C(1 + ln t+t4+t2)≤Ct4, t ∈[1, T[, r ∈[0,1].(4.6)

Remark 4.2 we have proven that for initial data as in Theorem 2.4(local ex-

istence), the right maximal regular solution of the Einstein-Vlasov-scalar field

satisfies estimates (4.2)-(4.5)-(4.6).

In the next theorem, we prove that this solution exists on the full interval [1,∞[.

Theorem 4.3 Assume that (f, λ, µ, φ)is a solution of the full system on a right

maximal interval of existence [1, T[, then T=∞.

Proof: Assume that T < ∞. We show that under the bound of µ, we obtain the

bound of sup{|w||(t, r, w, F)∈suppf}which is in contradiction to Theorem

2.8. The proof of the bound on wis similar to the proof of [see [15], Theorem

6.2]. Let

W0:= sup{|w||(r, w, F)∈supp◦

f}<∞,

F0:= sup{F|(r, w, F)∈supp◦

f}<∞.

Except in the vacuum case we have W0>0 and F0>0. For t≥1 define

P+(t) := max{0,max{w|(r, w, F)∈suppf(t)}},

P−(t) := min{0,min{w|(r, w, F)∈suppf(t)}}.

Constants denoted by C will be positive, may depend on the initial data and may

change their value from line to line. Let (r(s), w(s), F ) be a characteristic in the

support of f. Assume that P+(t)>0 for some t∈[1, T[, and let w(t) = w > 0.

We have

˙w=−˙

λw −eµ−λµ0p1 + w2+F/s2

= (−4πse2µρ+1 + ke2µ

2s)w+ 4πse2µjp1 + w2+F/s2

= 4πse2µ(jp1 + w2+F/s2−ρw) + 1 + ke2µ

2sw

=4π2

se2µZ∞

−∞ Z∞

0˜wp1 + w2+F/s2−wq1 + ˜w2+˜

F/s2fd ˜

Fd ˜w

+1 + ke2µ

2sw−4πse2µ1

2w(˙

φ2e−2µ+φ02e−2λ) + p1 + w2+F/s2˙

φφ0e−µ−λ

(4.7)

Using the fact that K(s)2≤K(1)2sfor s∈[1, T[ (see (2.78)), we have :

2w(˙

φ2e−2µ+φ02e−2λ) + p1 + w2+F/s2˙

φφ0e−µ−λ

≥(w+p1 + w2+F/s2)˙

φφ0e−µ−λ

≥ −p1 + w2+F/s2K(s)2

≥ −C(1 + P+(s))s(4.8)

Set γ= ˜wp1 + w2+F/s2−wq1 + ˜w2+˜

F/s2. As long as w(s)>0, we have

the following estimates: if ˜w≤0 then γ≤0. If ˜w > 0 then

4π2

se2µZ∞

−∞ Z∞

γfd ˜

Fd ˜w+1 + ke2µ

2sw≤1 + ke2µ

2sw

+4π2

se2µZP+(s)

0ZF0

˜w2(1 + w2+F/s2)−w2(1 + ˜w2+˜

F/s2)

˜wp1 + w2+F/s2+wq1 + ˜w2+˜

F/s2

fd ˜

Fd ˜w

≤4π2

se2µZP+(s)

0ZF0

˜w2(1 + F/s2)

2 ˜ww fd ˜

Fd ˜w+1 + ke2µ

2sw

≤4π2

se2µZP+(s)

0ZF0

˜w(1 + F)

wfd ˜

Fd ˜w+1 + ke2µ

2sw

≤4π2F0(1 + F0)k◦

fke2µ

2s(P+(s))21

w+1 + ke2µ

2sw

≤C

s(P+(s))2

w+w(4.9)

Therefore, using (4.8) and (4.9), (4.7) yields:

˙w(s)≤C

s(P+(s))2

w(s)+w(s)+C(1 + P+(s))s2

i.e.

˙w(s)w(s)≤C

s(P+(s))2+CP+(s)(1 + P+(s))s2

dsw(s)2≤C(s−1+s2)(P+(s))2+CP+(s)s2

as long as w(s)>0. Let t1∈[1, t[ be defined minimal such that w(s)>0 for

s∈[t1, t[, then

w(t)2≤w(t1)2+CZt

[(s−1+s2)(P+(s))2+s2P+(s)]ds

If t1= 1 then w(t1)≤w0and

w(t)2≤w2

0+CZt

[(s−1+s2)(P+(s))2+s2P+(s)]ds.

If t1>1 then w(t1) = 0 (t1is the minimal) and

w(t)2≤CZt

[(s−1+s2)(P+(s))2+s2P+(s)]ds

≤w2

0+CZt

[(s−1+s2)(P+(s))2+s2P+(s)]ds.

Thus

(P+(t))2≤w2

0+CZt

[(s−1+s2)(P+(s))2+s2P+(s)]ds; for all t∈[1, T[

Now we use the fact that P+≤1

2(1 + P2

+) to obtain

(P+(t))2≤w2

0+CZt

[(s−1+s2)(P+(s))2+s2]ds

≤(w2

0+Ct3) + CZt

(s−1+s2)(P+(s))2ds; for all t∈[1, T[

If t < ∞, applying Gronwall’s inequality to this estimate implies that P+is

bounded on [1, T[.

Estimating ˙w(s) from below in the case w(s)<0 along the same lines shows

that P−is bounded as well and the proof is complete.

Remark 4.4 In the case of spherical symmetry (k= 1), there is no global

existence in the future, regardless of the size of initial data. For any solution

(f, λ, µ, φ), the estimate

e−2µ(t,r)=e−2◦

µ(r)+ 1

t−1−8π

tZt

s2p(s, r)ds ≤e−2◦

µ(r)+ 1

t−1

has to hold on the interval [1, T[. Since the right hand side of this inequality

tends to −1for t→ ∞, it follows that T < ∞. And we deduce from the previous

Theorem (|w|<∞) that ||e2µ|| → ∞ for t→T.

4.2 The future asymptotic behaviour

In this section we prove that the spacetime obtained in Theorem 4.3 (with f= 0)

in the plane symmetric case is timelike and null geodesically complete in the

expanding direction. Later on, we prove that this result holds for homogeneous

solutions of the Einstein-Vlasov-scalar field system in plane and hyperbolic sym-

metry.

Let us determine first the explicit solutions φ, µ, λ in the case f= 0 and k= 0.

4.2.1 Integration of equations

In the case k= 0 the wave equation can be reduced to a simple linear equation

as observed in [19]. This reduction goes as follows. In that case the field

equations imply that λ−µ+ log tis constant in time. It may, however, be

dependent on r. Suppose that ris replaced by a new coordinate son the

initial hypersurface. Choosing sappropriately makes the transformed quantity

λ−µ+ log tconstant on the initial hypersurface and hence everywhere. Let

us call this constant η. Then λ−µ+ log t=ηand the wave equation (1.7)

simplifies to

∂ttφ+t−1∂tφ=t2e−2η∂ssφ. (4.10)

Now with the change of coordinate rby s,√grrdr =eλdr =e˜

λds =√gssds with

λ−µ+ log t=ηi.e ds =eλ−˜

λdr =te−ηeλ−µdr and 1 = e−ηR1

0te(λ−µ)(t,r)dr

for all t≥1. In particular for t= 1, 1 = Ae−ηis the period of variables rand

sfor A=R1

0e(λ−µ)(1,r)dr.

Now if we set T=βt2(βa positive constant), then (4.10) reads

∂T T φ+T−1∂Tφ=1

4βe−2η∂ssφ. Setting Ae−η= 2πand 4βe2η= 1, we can use

the results of Jurke (cf. [11]) obtained in the case of polarized Gowdy T3-models

(here Wis replaced by φ,tby T, the period of the space coordinate is 2π). The

equations are similar with the same boundary conditions. Following this, for all

(t, r)∈[1,∞[×Rand T=t2,φmust be in this form :

φ(t, r) = (a1+ 2blog t(homogeneous case)

a1+ 2blog t+t−1α(t, r) + β(t, r) (non-homogeneous case)

(4.11)

where a1and bare constants fixed by initial values of φand ˙

φ,αand βreal-

valued C2-functions with |α|,|α0| ≤ C,|˙α| ≤ Ct,|β|,|β0| ≤ Ct−3,|˙

β| ≤

Ct−2,Ca positive constant and 1

4(t−2∂tt −t−3∂t)α(t, r) = ∂rrα(t, r). We

deduce that |˙

φφ0| ≤ Ct−1.

¿From the field equation (1.4), we have :

2t˙µ= 1 + 8πt2e2µp

i.e

˙µ=1

2t+ 2πt(˙

φ2+e2µ−2λφ02) = 1

2t+ 2πt(˙

φ2+ 4t2φ02)

¿From (1.3)

2t˙

λ=−1+8πt2e2µρ

i.e

λ=−1

2t+ 2πt(˙

φ2+e2µ−2λφ02) = −1

2t+ 2πt(˙

φ2+ 4t2φ02)

We deduce from the expression of φ, that ˙µand ˙

λare bounded each by a

positive constant C. From Theorem 15 of [11] (the hypothesis of this theorem

are satisfied : atis replaced here by ˙µ), µcan be cast for all (t, r)∈[1,∞[×R

into the form

µ(t, r) = (1

2(16πb2+ 1) log t+γif µ0= 0

νt2+δ(t, r) (non-homogeneous case) (4.12)

where νis a positive constant, γa constant fixed by initial value of µand the

function δsatisfies the inequalities, |δ(t, r)| ≤ C(1+t), |˙

δ| ≤ Ct with a positive

constant C. Note that if φ0= 0 then from equation (1.5), µ0= 0 and λ0= 0;

the solutions are independent of rand we are in the homogeneous case. Since

λ−˙µ=−t−1, we deduce that λ(t, r) = µ(t, r)−log t+◦

λ(r)−◦

µ(r).

4.2.2 Geodesic completeness

Let ]τ−, τ+[3τ7→ (xα(τ), pβ(τ)) be a geodesic whose existence interval is max-

imally extended and such that x0(τ0) = t(τ0) = 1 for some τ0∈]τ−, τ+[. We

want to show that for future-directed timelike and null geodesics, τ+= +∞.

Consider first the case of a timelike geodesic, i.e.,

gαβpαpβ=−m2;p0>0

with m > 0. Since dt/dτ =p0>0, the geodesic can be parameterized by the

coordinate time t. Recall that along the geodesics the variables

t,r,p0,w:= eλp1,F:= t4(p2)2+ sin2

kθ(p3)2satisfy the following system of

differential equations :

dτ =e−λw, dw

dτ =−˙

λp0w−e2µ−λµ0(p0)2,dF

dτ = 0 (4.13)

dτ =p0,dp0

dτ =−˙µ(p0)2−2e−λµ0p0w−e−2µ˙

λw2−e−2µt−3F. (4.14)

With respect to coordinate time the geodesic exists on the interval [1,∞[ since

on bounded t-intervals the Christoffel symbols are bounded and the right hand

sides of the geodesic equations written in coordinate time are linearly bounded

in p1,p2,p3. Along the geodesic we define wand Fas above. Then the relation

between coordinate time and proper time along the geodesic is given by

dτ =p0=e−µpm2+w2+F/t2,

and to control this we need to control was a function of coordinate time.

The plane symmetric case without Vlasov

Assume that w(t)>0 for some t≥1. By (4.13) and the fact that |˙

φ||φ0|(t)≤

Ct−1,e(µ−λ)(t)=e−ηt, we have as long as w(s)>0

dsw(s) = −˙

λw −e2µ−λµ0p0= 4πse2µ(jpm2+w2+F/s2−ρw) + 1

2sw

≤1

2sw−4πse2µρw +|4πse2µj(w+pm2+F/s2)|

≤1

2sw+ 4πse2µ|j|pm2+F/s2since |j| ≤ ρ

≤1

2sw+ 4πseµ−λ|˙

φ||φ0|pm2+F/s2

≤1

2sw+Cs

(4.15)

Let t0∈[1, t[ be defined minimal such that w(s)>0 for s∈[t0, t[. Then

Gronwall’s inequality shows that

w(t)≤w(t0) + CZt

sexp(Zt0

dτ

2τ)dsexp(Zt

dτ

2τ).

Now either t0= 1 and w(t0) = w(1) or t0>1 and w(t0) = 0 (t0is the minimal).

Thus

w(t)≤|w(1)|+CZt

sexp(Z1

dτ

2τ)dsexp(Zt

dτ

2τ)≤Ct2.

Estimating ˙w(s) from below in the case w(s)<0 along the same lines yields

the upper bound of −w(t).

Since |δ| ≤ C(1 + t), νt2+δ > νt2−C(1 + t) = νt2(1 −C(1+t)

νt2). For tlarge, we

choose C(1+t)

νt2<1

2i.e νt2+δ > 1

2νt2. Then along the geodesic we have :

dτ

dt =eµ

pm2+w2+F/t2=eνt2+δ

pm2+Ct4+F/t2

≥e1

2νt2

t2√m2+C+F≥νt2

2t2√m2+C+F= constant.

Since the left hand side is constant, the integral over [1,∞[ diverges.

In the homogeneous case (ν= 0, j= 0, ρ≥0), (4.15) becomes

dsw(s)≤1

2sw

and we can prove similarly as above that w(t)≤Ct1/2for t≥1. Therefore

dτ

dt =e1

2(16πb2+1) log t+γ

pm2+Ct +F/t2≥t1

2(16πb2+1)eγ

t1/2√m2+C+F≥Ct8πb2

The integral on the right hand side of the above inequality over [1,∞[ diverges.

In either case, we conclude that τ+= +∞as desired.

In the case of a future-directed null geodesic, i.e. m= 0 and p0(τ0)>0, p0is

everywhere positive and the quantity Fis again conserved. The argument can

now be carried out exactly as in the timelike case, implying that τ+= +∞. We

have proven :

Theorem 4.5 Consider initial data with plane symmetry for the Einstein-scalar

field system written in areal coordinates. Then the corresponding spacetime is

timelike and null geodesically complete in the expanding direction.

Remark 4.6 In the homogeneous plane symmetric case with only a scalar field,

ρ(t) = p(t) = 1

2e−2µ˙

φ2,(˙

λ−˙µ)(t) = −t−1and solving the wave equation, we

obtain |˙

φ(t)|=|ψ|t−1. From the field equations, ˙

λ(t) = (2πψ2−1

2)t−1and

λ(t) = ◦

λ+ (2πψ2−1

2) log t;µ(t) = ◦

µ+ (2πψ2+1

2) log t;

ρ(t) = p(t) = 1

2ψ2e−2◦

µt−3−4πψ2;j(t) = 0 ;q(t) = 2p(t).

We can compute the limiting values of the generalized Kasner exponents :

t˙

λ= 2πψ2−1

lim

t→∞

1(t, r)

K(t, r)= lim

t→∞

t˙

λ(t, r)

t˙

λ+ 2 =4πψ2−1

4πψ2+ 3 ;

lim

t→∞

2(t, r)

K(t, r)= lim

t→∞

3(t, r)

K(t, r)= lim

t→∞

t˙

λ+ 2 =2

4πψ2+ 3,

The spatially homogeneous case

Consider now the Einstein-Vlasov-scalar field system for k≤0. We study the

future behaviour for solutions which are independent of the space coordinate r.

In this case µ0= 0, φ0= 0 and equation (1.5) shows that j= 0.

We prove that w(t) is bounded for finite time tand the spacetime obtained

is geodesically complete. Since ρ≥0, j= 0 and k≤0, we have from (4.13) as

long as w(s)>0,

dsw(s) = −˙

λw −e2µ−λµ0p0= 4πse2µ(jpm2+w2+F/s2−ρw) + 1 + ke2µ

2sw

=−4πse2µρw +1 + ke2µ

2sw

≤1

2sw

and we can prove as above that w(t)≤Ct1/2for t≥1. Using (4.1), e2µ≥t

C≥

Cfor k= 0; and e2µ≥t

C+t=1

1+C/t ≥1

1+Cfor k=−1. In each case, eµ≥C.

Then along the geodesic,

dτ

dt =eµ

pm2+w2+F/t2≥C

pm2+Ct +F/t2

≥C

t1/2√m2+C+F≥Ct−1/2.

The integral on the right hand side over [1,∞[ diverges. Therefore τ+= +∞.

In the case of a future-directed null geodesic, the argument is the same as what

we done in the previous subsection.

In the homogeneous case, a number of further estimates can also be obtained.

We follow the proof of Theorem 4.1 and [17] to obtain a better bound of µwhich

depends explicitly on the data. Recall (4.3)

dt(eµ+λρ)(t) = −1

teµ+λ2ρ+q−ρ+p

2(1 + ke2µ).

Since q≥0, we obtain :

for k= 0, d

dt(eµ+λρ)(t)≤ −1

teµ+λρ(t);

thus

eµ+λρ(t)≤e

◦

µ+

◦

λ◦

ρt−1; (4.16)

for k=−1

dt(eµ+λρ)(t)≤ −1

teµ+λ2ρ−ρ+p

2−1

C0+t

ρ+p

2

≤eµ+λρ−1

t−1

C0+t

≤ − 2

C1+teµ+λρ(t)

where C0=k+e−2◦

µ(see (4.1)) and C1= max(0, C0). Thus

eµ+λρ(t)≤(C1+ 1)2e

◦

µ+

◦

λ◦

ρt−2.(4.17)

Next we have similarly to (4.5),

eµ−λ≤e

◦

µ−

◦

λtfor k = 0 (4.18)

and

eµ−λ≤e

◦

µ−

◦

λe−2◦

µt

e−2◦

µ−1 + tfor k = −1

If ◦

µ≤0 then t

e−2◦

µ−1+t≤1. If ◦

µ≥0 then e−2◦

µt

e−2◦

µ−1+t≤1. Thus in either case,

eµ−λ≤e|◦

µ|−

◦

λfor k = −1 (4.19)

Now

µ(t) = ◦

µ+Zt

˙µ(s)ds =◦

µ+Zt

2s[e2µ(8πs2p+k) + 1]ds.

For k= 0,

µ(t)≤◦

µ+1

2log t+ 4πZt

se2µρds =◦

µ+1

2log t+ 4πZt

seµ−λeµ+λρds

≤◦

µ+1

2log t+ 4πe

◦

µ+

◦

λ◦

ρe

◦

µ−

◦

λZt

sds

≤◦

µ+1

2log t+ 2πe2◦

µ◦

ρt2

i.e

µ(t)≤◦

µ+ 2πe2◦

µ◦

ρt2.(4.20)

For k=−1,

µ(t)≤◦

µ+ 4πZt

se2µρds +Zt

2s(1 −e2µ)ds

≤◦

µ+ 4πZt

seµ−λeµ+λρds +Zt

2s(1 −s

C0+s)ds

≤◦

µ+ 4π(C1+ 1)2e|◦

µ|−

◦

λ◦

ρe

◦

µ+

◦

λZt

s−1ds +Zt

2(1

s−1

C0+s)ds

≤◦

µ+1

2log(C0+1)+4π(C1+ 1)2e|◦

µ|+◦

µ◦

ρZt

s−1ds

i.e

µ(t)≤◦

µ+1

2log(C0+1)+4πe|◦

µ|+◦

µ◦

ρ(C1+ 1)2log t. (4.21)

We have proven :

Theorem 4.7 Consider spatially homogeneous solutions of plane and hyper-

bolic symmetric Einstein-Vlasov-scalar field system written in areal coordinates

and initial data given for t= 1. Then these solutions exist on the whole in-

terval [1,∞[. The corresponding spacetimes are timelike and null geodesically

complete in the expanding direction and estimates (4.16)-(4.17)-(4.18)-(4.19)-

(4.20)-(4.21) hold.

Conclusion

We have investigated in spherical, plane and hyperbolic symmetry, cosmological

solutions of the Einstein-Vlasov-scalar field system. But their are some remain-

ing problems listed as follows :

- past global existence for k=−1 without any restriction on the data;

- asymptotics in the past direction with Vlasov matter. Maybe the idea of [24]

could be used here?

- future geodesic completeness in general;

- future asymptotics. Maybe one can use the results of [26] which study the

future asymptotic behavior of massless scalar fields in a class of cosmological

vacuum spacetimes;

- in the case of a non-linear scalar field, using our approach can we obtain sim-

ilar results as in the case of non-vanishing cosmological constant?

Thus it appears that the work of this thesis can serve as a useful starting point

for future investigations.

Appendices

Appendix A

Here we establish relations (2.26) and (2.27). Relation (2.8) gives :

e−µn−1(˙

˜gn−˙

hn) + e−λn−1(˜g0

n−˜

n) = an−1(˜gn−1−˜

hn−1) + bn−1(˜gn−1+˜

hn−1)

i.e

e−µn−1˙

˜gn−e−λn−1˜

n=e−µn−1˙

hn−e−λn−1˜g0

n+(an−1+bn−1)˜gn−1+(bn−1−an−1)˜

hn−1

Therefore,

D+

n˜

Xn=D+

n(˜gn+1 −˜

hn+1)−D+

n(˜gn−˜

hn)

=an(˜gn−˜

hn) + bn(˜gn+˜

hn)−e−µn(˙

˜gn−˙

hn)−e−λn(g0

n−h0n)

= (an+bn)˜gn+ (bn−an)˜

hn−˙

˜gn(e−µn−e−µn−1) + ˜

n(e−λn−e−λn−1)

+˙

hne−µn−˜g0

ne−λn−˙

˜gne−µn−1+˜

ne−λn−1

= (an+bn)˜gn+ (bn−an)˜

hn−˙

˜gn(e−µn−e−µn−1) + ˜

n(e−λn−e−λn−1)

+˙

hne−µn−˜g0

ne−λn−e−µn−1˙

hn+e−λn−1˜g0

n−(an−1+bn−1)˜gn−1−(bn−1−an−1)˜

hn−1

= (an+bn)(˜gn−˜gn−1)+(bn−an)(˜

hn−˜

hn−1)+(an+bn−an−1−bn−1)˜gn−1

+ (bn−an+an−1−bn−1)˜

hn−1+ (e−µn−e−µn−1)(˙

hn−˙

˜gn)+(e−λn−e−λn−1)(˜

n−˜g0

=an[(˜gn−˜gn−1)−(˜

hn−˜

hn−1)] + bn[(˜gn−˜gn−1)+(˜

hn−˜

hn−1)]

+ (an+bn−an−1−bn−1)˜gn−1+ (bn−an+an−1−bn−1)˜

hn−1

+ (e−µn−e−µn−1)(˙

hn−˙

˜gn)+(e−λn−e−λn−1)(˜

n−˜g0

=an˜

Xn−1+bn˜

Yn−1+Fn

If we replace ˜

nand ˜g0

nrespectively by −˜

nand −˜g0

n, (2.8) and (2.9), D+

nand

D−

n,˜

Xnand ˜

Yninterchange respectively and we can write (2.27).

Appendix B

Let us prove relations (2.56) and (2.57). We have

∂r(D+

nXn+1) = (∂rD+

n)Xn+1 +D+

nX0

n+1

i.e

D+

nX0

n+1 =∂r(D+

nXn+1)−(∂rD+

n)Xn+1

=C0

n+µ0

ne−µn∂tXn+1 +λ0

ne−λnX0

n+1

We deduce from relation (e−µn∂t+e−λn∂r)Xn+1 =Cn, that

e−µn∂tXn+1 =Cn−e−λnX0

n+1

Therefore

D+

nX0

n+1 =C0

n+µ0

nCn+λ0

ne−λnX0

n+1 −µ0

ne−λnX0

n+1

=C0

n+ (λ0

n−µ0

n)e−λnX0

n+1 +µ0

nCn=˜

If we replace rby −r(i.e ∂rby −∂r), D+

nand D−

n,Xn+1 and Yn+1 interchange

respectively and we can write (2.57).

Appendix C

Let us prove inequalities (2.66)-(2.67). we have from (2.55)

Cn+˜

Cn−1= (λ0

n−µ0

n)e−λnX0

n+1 −(λ0

n−1−µ0

n−1)e−λn−1X0

+µ0

nCn−µ0

n−1Cn−1+˜

n−˜

n−1

Now we obtain the following relations :

i) (λ0

n−µ0

n)e−λnX0

n+1 −(λ0

n−1−µ0

n−1)e−λn−1X0

n= (λ0

n−µ0

n)e−λn(X0

n+1 −X0

+ [(λ0

n−µ0

n)e−λn−(λ0

n−1−µ0

n−1)e−λn−1]X0

= (λ0

n−µ0

n)e−λn(˜g0

n+1 −˜g0

n+˜

n−˜

n+1) + [(λ0

n−λ0

n−1)e−λn

−(µ0

n−µ0

n−1)e−λn+ (λ0

n−1−µ0

n−1)(e−λn−e−λn−1]X0

ii)µ0

nCn−µ0

n−1Cn−1= (µ0

n−µ0

n−1)Cn+µ0

n−1(Cn−Cn−1)

iii)˜

n+˜

n−1=a0

nXn+anX0

n+b0

nYn+bnY0

n−a0

n−1Xn−1−an−1X0

n−1−b0

n−1Yn−1−bn−1Y0

n−1

= (a0

n−a0

n−1)Xn+a0

n−1(Xn−Xn−1)+(an−an−1)X0

n+an−1(X0

n−X0

n−1)

+ (b0

n−b0

n−1)Yn+b0

n−1(Yn−Yn−1)+(bn−bn−1)Y0

n+bn−1(Y0

n−Y0

n−1)

iv)a0

n−a0

n−1=−µ0

n(−˙

λn−1

t−˙

λ0

n)e−µn+ (−˜µ0

n+ ˜µ0

nλ0

n)e−λn

+µ0

n−1(−˙

λn−1−1

t−˙

λ0

n−1)e−µn−1+ (−˜µ0

n−1+ ˜µ0

n−1λ0

n−1)e−λn−1

= (µ0

n−1−µ0

n)(−˙

λn−1

t)e−µn−µ0

n−1[(−˙

λn−1

t)e−µn+ (−˙

λn−1−1

t)e−µn−1]

+ ( ˙

λ0

n−1−˙

λ0

n)e−µn−˙

λ0

n−1(e−µn−e−µn−1) + (˜µ0

n−1−˜µ0

n)e−λn

−˜µ0

n−1(e−λn−e−λn−1)+(λ0

n−λ0n−1)˜µne−λn+λ0n−1(˜µne−λn−˜µn−1e−λn−1)

v)b0

n−b0

n−1=µ0

te−µn−µ0

n−1

te−µn−1

t(µ0

n−µ0

n−1)e−µn+µ0

n−1

t(e−µn−e−µn−1)

Using remark 2.2, (2.36) and (2.40), we obtain respectively from i), ii), iv) and

v) the following estimations :

|(λ0

n−µ0

n)e−λnX0

n+1 −(λ0

n−1−µ0

n−1)e−λn−1X0

n| ≤ C+C(|˜g0

n+1 −˜g0

n|+|˜

n+1 −˜

+|µ0

n−µ0

n−1|+|λ0

n−λ0

n−1|)

≤C +C(γn(τ) + γn−1(τ))

|µ0

nCn−µ0

n−1Cn−1| ≤ |µ0

n−µ0

n−1||Cn|+|µ0

n−1||Cn−Cn−1|

≤C +Cγn−1(τ)

|a0

n−a0

n−1| ≤ C+ (|µ0

n−1−µ0

n|+|˙

λ0

n−1−˙

λ0

n|+|˜µ0

n−1−˜µ0

n|+|λ0

n−λ0

n−1|)

≤C +Cγn−1(τ)

|b0

n−b0

n−1| ≤ C +C|µ0

n−µ0

n−1| ≤ C +Cγn−1(τ)

|X0

n−X0

n−1|,|Y0

n−Y0

n−1| ≤ C(|˜g0

n−1−˜g0

n|+|˜

n−1−˜

n|)≤Cγn−1(τ)

Using this, we deduce from iii) that

|˜

n+˜

n−1| ≤ C +C(γn(τ) + γn−1(τ))

Combining all the necessary estimates, (2.66) follows. Similarly, we prove in-

equality (2.67).

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