Mild solutions for the one dimensional Nordstrom-Vlasov system

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Mild solutions for the one dimensional Nordstrom-Vlasov system

Mihai Bostan

To cite this version:

Mihai Bostan. Mild solutions for the one dimensional Nordstrom-Vlasov system. Nonlinearity, IOP

Publishing, 2007, 20 (5), pp.1257-1281. �hal-00594961�

Mild solutions for the one dimensional Nordstr¨om-Vlasov system

Mihai Bostan

(31st October 2006)

Abstract

The Nordstr¨om-Vlasov system describes the evolution of a population of self-gravitating collisionless particles. We study the existence and uniqueness of mild solution for the Cauchy problem in one dimension. This approach does not require any derivative for the initial particle density. For any initial particle density uniformly bounded with respect to the space variable by some function with finite kinetic energy and any initial smooth data for the field equation we construct a global solution, preserving the total energy. Moreover the solution propagates with finite speed. The propagation speed coincides with the light speed.

Keywords: Nordstr¨om equation, Vlasov equation, weak/mild solutions, global existence.

AMS classification: 35B35, 35A05, 35B45, 82D10.

∗Laboratoire de Math´ematiques de Besan¸con, UMR CNRS 6623, Universit´e de Franche-Comt´e, 16 route de Gray, 25030 Besan¸con Cedex France. E-mail : mbostan@math.univ-fcomte.fr, Mihai.Bostan@iecn.u-nancy.fr

1 Introduction

Consider a population of particles interacting by fields created collectively. We assume that the collisions are so rare such that we can neglect them meaning that there is no direct interaction between particles. The fields acting on the particles depend on the physical model. Typical examples of such collisionless gases occur in plasma physics and in astrophysics. In plasma physics the particles interact by electro-magnetic forces and the dynamics of the system is described by the Vlasov- Maxwell equations. If the particle velocities are small compared to the light speed then we can neglect the magnetic field and use the quasi-electrostatic approximation given by Vlasov-Poisson equations, cf. [11], [22], [3]. If the particles interact by gravitational forces the evolution of the system is given by the Einstein-Vlasov equations. The Cauchy problem of the former system is well understood, cf. [5], [12], [13], [14], [15], [17]. The Einstein-Vlasov system is much more difficult. The reader can refer to [1] for a recent review, see also [19], [20], [21]. The main application of these equations concern the stellar dynamics: the systems to be considered are galaxies and the particles are stars.

We analyze here a different relativistic model obtained by coupling the Vlasov equation to the Nordstr¨om scalar gravitation theory [18]. Let F = F (t, x, p) ≥ 0 denote the density of particles in the phase-space, where t ∈ R represents time, x ∈ R

position and p ∈ R

momentum, with N ∈ {1, 2, 3}. This density satisfies the Vlasov equation

∂

F + v(p) · ∇

F −

(∂

φ + v (p) · ∇

φ)p + (1 + |p|

)

∇

φ

· ∇

F = 0, coupled to the wave equation

∂

φ − ∆

φ = −e

Z

R^N

F (t, x, p) (1 + |p|

)

dp.

Here v(p) =

denotes the relativistic velocity of a particle with momentum

p. We normalize the physical units such that the rest mass of each particle, the

gravitational constant and the speed of the light are all equal to unity. For more details on this model see [6]. It is convenient to rewrite the system in terms of the unknowns (f, φ) where f (t, x, p) = e

F (t, x, p). The system becomes

∂

f + v(p) · ∇

f −

(Sφ)p + (1 + |p|

)

∇

φ

· ∇

f = (N + 1)f(Sφ), (1)

∂

φ − ∆

φ = −µ(t, x), (2) µ(t, x) =

Z

R^N

f (t, x, p)

(1 + |p|

)

dp, (3)

where S = ∂

+ v(p) · ∇

is the free-transport operator. We consider the initial conditions

f(0, ·, ·) = f

, φ(0, ·) = ϕ

, ∂

φ(0, ·) = ϕ

. (4) The system (1), (2), (3), (4) will be referred to as the Nordstr¨om-Vlasov system.

Recently Calogero and Rein proved in [8] that classical solutions for the initial value problem with regular data exist at least locally in time in the three dimensional case.

They give also a condition for global existence. The main tool consists in deriving representation formula for the time and spatial derivatives of φ, following the ideas in [15]. In one dimension they obtained the global existence and uniqueness for smooth compactly supported initial particle density f

∈ C

(R

) and smooth initial conditions ϕ

∈ C

(R), ϕ

∈ C

(R) for the potential (the subscript b indicates that the functions are bounded together with their derivatives up to the indicated order). Assuming that the characteristic particle velocity is small compared to the light speed leads to the gravitational Vlasov-Poisson system, i.e., a Vlasov equation coupled to a Poisson equation with opposite sign in front of the mass density. This comes by the fact that the forces are repulsive in the electromagnetic case and attractive in the gravitational case. This asymptotic regime was justified recently in [7]. The existence of global weak solution for the Nordstr¨om-Vlasov system in three dimensions has been studied as well, see [9]. One of the key points is the smoothing effect due to momentum averaging, see [12], [16].

The aim of this paper is to prove a more general existence and uniqueness result

for the Nordstr¨om-Vlasov system in one dimension under less restrictive hypotheses.

Our method allows us to remove the support compactness of the initial density f

and the C

regularity of it. We assume that f

is bounded uniformly with respect to x by some function g

= g

(p) having finite kinetic energy and we construct a unique global solution (f, φ) where f is solution by characteristics (or mild solution) of (1) and φ is a classical solution of (2). We call such a solution (f, φ) a mild solution for the Nordstr¨om-Vlasov equations. The same method was used recently for studying a reduced Vlasov-Maxwell system for laser-plasma interaction, which shares some common features with the Nordstr¨om-Vlasov system, see [4], [10]. Results for initial boundary value problems can be obtained as well by this method, see [2] for an existence and uniqueness result of the mild solution for the one dimensional Vlasov- Poisson system. Our existence and uniqueness result is the following

Theorem 1.1 Assume that ϕ

∈ W

(R), ϕ

∈ W

(R), f

∈ L

(R

) and that there is some function g

∈ L

(R) such that 0 ≤ f

(x, p) ≤ g

(p), ∀ (x, p) ∈ R

, sign(·)g

(·) is nonincreasing on R, and R

R

(1 + |p|)g

(p) dp < +∞. Then there is a unique mild solution (f ≥ 0, φ) ∈ L

(]0, T [; L

(R

)) × W

(]0, T [×R), ∀ T > 0 of the Nordstr¨om-Vlasov system (1), (2), (3), (4) with N = 1. Moreover R

R

(1 +

|p|)f(·, ·, p) dp belongs to L

(]0, T [×R), ∀ T > 0.

Another motivation for this approach is that the proofs of our existence and unique- ness result provide, after minor changes, the finite speed propagation of the solution.

This property inherit from the relativistic feature of the problem in hand. To our knowledge this is the first theoretical result in this direction.

Theorem 1.2 Assume that the initial conditions (f

, ϕ

, ϕ

)

satisfy the hy- potheses of Theorem 1.1 and denote by (f

, φ

)

the corresponding global so- lutions of the one dimensional Nordstr¨om-Vlasov system. Moreover suppose that R

R

|p|

g

(p) dp < +∞, k ∈ {1, 2}. Then for any R > 0 there is a constant C

depending on R, kϕ

k

, kϕ

k

, kf

k

, P

m=0

k |p|

g

k

, k ∈ {1, 2} such that

max (|φ

− φ

| + |∂

φ

− ∂

φ

| + |∂

φ

− ∂

φ

|) (t, x) ≤ C

D

,

where D

= kϕ

−ϕ

k

+kϕ

−ϕ

k

+ R

−R

R

R

(1+ |p|)|f

−f

|dpdx.

Corollary 1.1 Assume that the initial conditions (f

, ϕ

, ϕ

)

satisfy the hy- potheses of Theorem 1.1 and denote by (f

, φ

)

the corresponding global solu- tions of the one dimensional Nordstr¨om-Vlasov system. Suppose also that f

(x, p) = f

(x, p), ϕ

(x) = ϕ

(x), ϕ

(x) = ϕ

(x), x ∈ [−R, R], p ∈ R. Then we have f

(t, x, p) = f

(t, x, p), φ

(t, x) = φ

(t, x), t ∈ [0, R], |x| ≤ R − t, p ∈ R.

Hence, the motivation of the question we address is two-fold. First we justify the well posedness of the one dimensional Nordstr¨om-Vlasov system in a larger class of solutions. Roughly speaking we deal with particle densities with finite mass and kinetic energy (uniformly in space) which are closer to the typical physical candi- dates, than the compactly supported smooth functions. Second, we highlight the finite propagation speed of the solutions. For the numerical point of view this prop- erty has important consequences: it allows us to localize in space when computing them.

Our paper is organized as follows. In Section 2 we analyze the Vlasov equation.

We introduce the equations of characteristics and we recall the notion of mild so- lution. We state also some important properties of the characteristics. The details of proof can be found in the Appendix. In Section 3 we construct the fixed point application and we obtain estimates for the first and second order derivatives. In Section 4 we prove the existence and uniqueness of mild solution for the Nordstr¨om- Vlasov equations in one dimension. We check also that this solution preserves the total energy. In the last section we establish the finite speed propagation property.

2 The Vlasov equation

In this section we assume that φ = φ(t, x) is a given smooth function and we recall

the notion of solution by characteristics for the Vlasov equation (1) in one dimension

∂

f +v(p)· ∂

f −

(Sφ)p + (1 + p

)

∂

φ

· ∂

f = 2f(Sφ), (t, x, p) ∈]0, T [×R

, (5) with the initial condition

f (0, x, p) = f

(x, p), (x, p) ∈ R

. (6) For any (t, x, p) ∈ [0, T ] × R

we introduce the system of characteristics for (5)

dX

ds = v(P (s)), dP

ds = −P (s) (∂

φ + v(P (s))∂

φ) (s, X(s)) − ∂

φ(s, X(s)) (1 + |P (s)|

)

, (7) with the conditions

X(s = t) = x, P (s = t) = p. (8)

Notice that if φ is a C

function and ∂

φ, ∂

φ are Lipschitz with respect to x ∈ R uniformly on t in [0, T ], then there is a unique C

solution for (7), (8), denoted (X(s; t, x, p), P (s; t, x, p)) or simply (X(s), P (s)). The divergence of the field ap- pearing in the right hand side terms of (7) is given by

Div

v(p), −(Sφ)p − (1 + p

)

∂

φ

= −∂

φ − v(p)∂

φ, (9) and by observing that

φ(s, X(s)) = ∂

φ(s, X(s)) + v (P (s))∂

φ(s, X(s)) we deduce that the jacobian matrix J (s) :=

∂(x,p)

has the determinant given by det J (s) = det

∂(X(s; t, x, p), P (s; t, x, p))

∂(x, p)

= e

6= 0. (10) Multiplying (5) by e

and by taking into account that

S(f e

) = e

Sf + f S (e

) = e

Sf − 2f e

Sφ, we obtain

∂

(f e

) + v(p)∂

(f e

) −

(Sφ)p + (1 + p

)

∂

φ

∂

(f e

) = 0.

We deduce formally that f e

is constant along any characteristic of (7) and we

have the usual definition

Definition 2.1 Assume that φ ∈ C

([0, T ] × R), ∂

φ, ∂

φ ∈ L

(]0, T [; W

(R)) and f

is a measurable function on R

. The solution by characteristics (or mild solution) of (5), (6) is given by

f (t, x, p) = f

(X(0; t, x, p), P (0; t, x, p))e

, (t, x, p) ∈ [0, T ] × R

. In the following proposition we recall some immediate properties of the mild solution Proposition 2.1 Assume that φ ∈ C

([0, T ] × R) ∩ L

(]0, T [×R), ∂

φ, ∂

φ ∈ L

(]0, T [; W

(R)), f

is measurable. Denote by f the mild solution of (5), (6).

1) If f

is nonnegative then f is nonnegative ; 2) If f

is bounded then f is bounded and we have

kf k

≤ e

kf

k

; 3) If f

∈ L

(R

) then f ∈ L

(]0, T [; L

(R

)) and

Z

R

Z

R

|f (t, x, p)| dp dx ≤ e

kf

k

. In particular the conservation of the total mass holds

Z

R

Z

R

|f (t, x, p)|e

dp dx = Z

R

Z

R

|f

(x, p)|e

dp dx, t ∈ [0, T ] ; 4) If f

∈ L

(R

) then for any continuous bounded function ψ we have Z

0

Z

R

Z

R

f(t, x, p)ψ(t, x, p) dp dx dt = Z

R

Z

R

f

(x, p) Z

0

ψ(t, X(t; 0, x, p), P (t; 0, x, p))

× e

dt dp dx ; 5) If (1 + |p|)f

∈ L

(R

) then (1 + |p|)f ∈ L

(]0, T [; L

(R

)) and Z

R

Z

R

(1 + |p|)|f(t, x, p)| dp dx ≤ e

1 + e

Z

0

k∂

φ(s)k

ds

kf

k

+ e

Z

R

Z

R

|p||f

(x, p)| dp dx ;

6) If f

∈ L

(R

) then the mild solution f satisfies the following weak formulation Z

0

Z

R

Z

R

f(t, x, p)e

∂

θ + v (p)∂

θ −

pSφ + (1 + p

)

∂

φ

∂

θ

dp dx dt

+ Z

R

Z

R

f

(x, p)e

θ(0, x, p) dp dx = 0, (11)

for any θ ∈ C

([0, T [×R

). In particular, if (1 + |p|)f

∈ L

(R

), the above formu- lation holds for any θ ∈ C

([0, T ] × R

) such that θ(T, ·, ·) = 0.

Proof. The first and second statements are trivial.

3) Assume now that f

∈ L

(R

) and for any t ∈ [0, T ] consider the change of variables x = X(t; 0, y, q), p = P (t; 0, y, q). By using (10) we have

Z

R

Z

R

|f (t, x, p)| dp dx = Z

R

Z

R

|f(t, X(t), P (t))|e

dq dy

= Z

R

Z

R

|f

(y, q)|e

dq dy

≤ e

kf

k

. Similarly we obtain

Z

R

Z

R

|f (t, x, p)|e

dp dx = Z

R

Z

R

|f (t, X(t), P (t))|e

dq dy

= Z

R

Z

R

|f

(y, q)|e

dq dy.

4) Take now ψ ∈ C

([0, T ] × R

). By the previous point we already know that f (t) ∈ L

(R

) for any t ∈ [0, T ]. By change of variables we obtain as before

Z

0

Z

R

Z

R

f ψ dp dx dt = Z

0

Z

R

Z

R

f (t, X (t), P (t))ψ(t, X(t), P (t))e

dq dy dt

= Z

R

Z

R

f

(y, q) Z

0

ψ(t, X(t), P (t))e

dt dq dy.

5) By using one more time the change of variables x = X(t; 0, y, q), p = P (t; 0, y, q) and (10) one gets

Z

R

Z

R

(1 + |p|)|f | dp dx = Z

R

Z

R

(1 + |P (t)|)|f(t, X(t), P (t))|e

dq dy

= Z

R

Z

R

(1 + |P (t)|)|f

(y, q)|e

dq dy. (12) By (7) we have

d

ds {P (s)e

} = − e

(1 + |P (s)|

)

∂

φ(s, X(s)), (13)

and therefore

P (t)e

= qe

− Z

0

e

(1 + |P (s)|

)

∂

φ(s, X (s))ds.

We deduce that for any t ∈ [0, T ] we have

|P (t)| ≤ |q|e

+ e

Z

0

e

|∂

φ(s, X(s))| ds

≤ |q|e

+ e

Z

0

k∂

φ(s)k

. (14)

Combining (12), (14) yields Z

R

Z

R

(1 + |p|)|f (t, x, p)| dp dx ≤ e

1 + e

Z

0

k∂

φ(s)k

ds

kf

k

+ e

Z

R

Z

R

|p||f

(x, p)| dp dx.

6) For any θ ∈ C

([0, T [×R

) consider ψ(t, x, p) = e

∂

θ + v(p)∂

θ −

pSφ + (1 + p

)

∂

φ

∂

θ

.

Notice that ψ(s, X(s), P (s)) = e

{θ(s, X(s), P (s))}. By applying 4) one gets Z

0

Z

R

Z

R

f e

∂

θ + v(p)∂

θ −

pSφ + (1 + p

)

∂

φ

∂

θ

dp dx dt

= Z

R

Z

R

f

(x, p) Z

0

e

d

ds {θ(s, X(s), P (s))} ds dp dx

= −

Z

R

Z

R

f

(x, p)e

θ(0, x, p) dp dx.

If (1+|p|)f

∈ L

(R

) we know that (1+|p|)f ∈ L

(]0, T [; L

(R

)). Take χ ∈ C

(R), χ(u) = 0 if |u| ≥ 2, χ(u) = 1 if |u| ≤ 1, χ ≥ 0 and χ

(·) = χ(

), R > 0. In order to prove (11) for θ ∈ C

([0, T ] × R

), θ(T, ·, ·) = 0, apply them with θ

= θχ

(x)χ

(p) and let R → +∞.

Remark 2.1 Notice that (5) can be written

∂

(f e

) + ∂

(v(p)f e

) − ∂

(pSφ + (1 + p

)

∂

φ)f e

= 0,

which justifies formally the weak formulation (11). In particular the total mass R

R

R

R

f (t, x, p)e

dp dx is preserved for any t.

We end this section with a continuous dependence result of the characteristics with respect to φ. It will be used in the next section. The proof is standard and it is postponed to the Appendix. We use the notation kuk

= kuk

+ ku

k

for any function u ∈ W

(R).

Proposition 2.2 Assume that φ

∈ C

([0, T ] × R) ∩ L

(]0, T [×R), ∂

φ

, ∂

φ

∈ L

(]0, T [; W

(R)), k ∈ {1, 2}. For any (x, p) ∈ R

consider (X

(t), P

(t)) = (X

(t; 0, x, p), P

(t; 0, x, p)) the characteristics corresponding to φ

, k ∈ {1, 2}. Then for any t ∈ [0, T ] there is a constant C depending on sup

{kφ

(s)k

+ k∂

φ

(s)k

+ k∂

φ

(s)k

} such that for any s ∈ [0, t] we have

|X

(s) − X

(s)| + |P

(s)e

− P

(s)e

| ≤ C |p(φ

− φ

)(0, x)|

+ C

Z

0

kφ

(τ) − φ

(τ )k

dτ.

The following result is a direct consequence of the above proposition (see the Ap- pendix for proof details)

Corollary 2.1 Under the hypotheses of Proposition 2.2, there is a constant C de- pending on

sup

s∈[0,T],k∈{1,2}

{kφ

(s)k

+ k∂

φ

(s)k

+ k∂

φ

(s)k

+ k∂

φ

(s)k

+ k∂

φ

(s)k

} such that we have for any t ∈ [0, T ]

X

(−1)

e

(1 + |P

(t)|

)

≤ C

kφ

(t) − φ

(t)k

+ Z

0

kφ

(s) − φ

(s)k

ds

+ C |p(φ

− φ

)(0, x)|,

X

(−1)

e

(1 + |P

(t)|

)

∂

φ

(t, X

(t))

≤ C(kφ

(t) − φ

(t)k

+ Z

0

kφ

(s) − φ

(s)k

ds) + C |p(φ

− φ

)(0, x)|,

X

(−1)

e

(1 + |P

(t)|

)

∂

φ

(t, X

(t))

≤ C(kφ

(t) − φ

(t)k

+ k∂

φ

(t) − ∂

φ

(t)k

+

Z

0

kφ

(s) − φ

(s)k

ds + |p(φ

− φ

)(0, x)|).

3 Fixed point application

Assume that ϕ

∈ W

(R), ϕ

∈ W

(R), f

∈ L

(R

), f

≥ 0 and take an arbitrary T > 0. Since φ satisfies a one dimensional wave equation, a natural definition for the fixed point application is F φ = ˜ φ for any φ in the set

D

= {φ ∈ C

([0, T ]× R) : (φ, ∂

φ)(0, ·) = (ϕ

, ϕ

), ∂

φ, ∂

φ ∈ L

(]0, T [; W

(R))}, where ˜ φ is given by d’Alembert formula

φ(t, x) = ˜ 1

2 {ϕ

(x + t) + ϕ

(x − t)} + 1 2

Z

x−t

ϕ

(y) dy

− 1 2

Z

0

Z

x−(t−s)

µ(s, y) dy ds, (t, x) ∈ [0, T ] × R, (15) the function µ is given by

µ(t, x) = Z

R

f(t, x, p)

(1 + p

)

dp, (t, x) ∈ [0, T ] × R, (16) and f is the mild solution of (5), (6) corresponding to φ.

3.1 Estimate of F

Since the initial density f

is nonnegative, we know that any mild solution f is nonnegative and by (15) we deduce the a priori estimate

φ(t, x) ˜ ≤ kϕ

k

+ tkϕ

k

, (t, x) ∈ [0, T ] × R. (17) Actually we can restrict the application F to the set

D

= D

∩ {φ : −kϕ

k

− T kϕ

k

− T

2 e

kf

k

≤ φ(t, x)

≤ kϕ

k

+ T kϕ

k

, ∀(t, x) ∈ [0, T ] × R},

and we check easily that for any φ ∈ D

we have for (t, x) ∈ [0, T ] × R

−kϕ

k

− T kϕ

k

− T

2 e

kf

k

≤ F φ(t, x) ≤ kϕ

k

+ T kϕ

k

. (18)

Indeed, for any φ ∈ D

, by using the definition of the mild solution f and the conservation of the total mass, cf. Proposition 2.1, we obtain for any 0 ≤ s ≤ t ≤ T

Z

x−(t−s)

µ(s, y) dy ≤ Z

R

Z

R

f (s, y, p) p 1 + p

dp dy

≤ e

Z

R

Z

R

f (s, y, p)e

dp dy

= e

Z

R

Z

R

f

(y, p)e

dp dy

≤ e

kf

k

. Therefore (15), (17) imply (18).

3.2 Estimate of DF φ

The d’Alembert formula (15) implies

∂

φ(t, x) = ˜ 1

2 {ϕ

(x + t) − ϕ

(x − t)} + 1

2 {ϕ

(x + t) + ϕ

(x − t)}

− 1 2

Z

0

{µ(s, x + t − s) + µ(s, x − (t − s))} ds, (19) and

∂

φ(t, x) = ˜ 1

2 {ϕ

(x + t) + ϕ

(x − t)} + 1

2 {ϕ

(x + t) − ϕ

(x − t)}

− 1 2

Z

0

{µ(s, x + t − s) − µ(s, x − (t − s))} ds. (20) In order to estimate the L

norms of ∂

φ, ∂ ˜

φ ˜ we need to estimate the L

norm of µ. In particular we have to assume that µ

(·) = R

R f0(·,p)

(1+p²)¹²

dp ∈ L

(R). It is convenient to suppose that there is some function g

∈ L

(R) such that

0 ≤ f

(x, p) ≤ g

(p), (x, p) ∈ R

, sign(·)g

(·) is nonincreasing on R, (21)

and Z

R

g

(p)

(1 + p

)

dp < +∞. (22)

Proposition 3.1 Assume that ϕ

∈ W

(R), ϕ

∈ W

(R), f

∈ L

(R

) and (21), (22) hold. Then for any φ ∈ D

we have

kµ(t)k

≤ C

+ C

Z

0

k∂

φ(τ )k

dτ, t ∈ [0, T ], where

C

= e

p min(1, e

)

Z

R

g

(p)

(1 + p

)

dp, C

= 2e

kg

k

, and

M = kϕ

k

+ T kϕ

k

+ T

2 e

kf

k

.

Proof. For (t, x, p) ∈ [0, T ] × R

consider (X(s), P (s)) = (X(s; t, x, p), P (s; t, x, p)), s ∈ [0, T ]. By (13) we deduce that for any s ∈ [0, T ] we have

e

P (s) = e

p − Z

t

e

(1 + |P (τ)|

)

∂

φ(τ, X (τ )) dτ. (23) In particular we obtain

e

P (0) = e

p + Z

0

e

(1 + |P (τ)|

)

∂

φ(τ, X(τ )) dτ. (24) We denote by M a bound for kφk

, for example

M = kϕ

k

+ T kϕ

k

+ T

2 e

kf

k

, and for any t ∈ [0, T ] we introduce the notation R(t) = R

0

k∂

φ(τ )k

dτ. Observe that for any p ≥ e

R(t) formula (24) implies e

P (0) ≥ e

p − e

R(t) ≥ 0, and thus

P (0; t, x, p) ≥ e

p − R(t) ≥ 0. (25) Similarly, if p ≤ −e

R(t), formula (24) implies −e

P (0) ≥ e

(−p) − e

R(t) ≥ 0. In this case we obtain

−P (0; t, x, p) ≥ e

(−p) − R(t) ≥ 0. (26) We can estimate µ as follows

µ(t, x) = Z

R

f(t, x, p)

(1 + p

)

1

dp + Z

R

f (t, x, p)

(1 + p

)

1

dp

= µ

(t, x) + µ

(t, x), (t, x) ∈ [0, T ] × R. (27)

By the definition of the mild solution we have µ

(t, x) =

Z

R

f

(X(0; t, x, p), P (0; t, x, p))

(1 + p

)

e

1

dp

≤ e

Z

R

g

(P (0; t, x, p))

(1 + p

)

1

dp

≤ 2e

kg

k

R(t), (t, x) ∈ [0, T ] × R. (28) In order to estimate µ

observe that (25), (26) and the monotonicity of g

yields

f

(X(0; t, x, p), P (0; t, x, p)) ≤ g

(P (0; t, x, p)) ≤ g

(e

p − R(t)), ∀ p ≥ e

R(t), respectively

f

(X(0; t, x, p), P (0; t, x, p)) ≤ g

(P (0; t, x, p)) ≤ g

(e

p+R(t)), ∀ p ≤ −e

R(t).

Therefore we have as before µ

(t, x) =

Z

R

f (t, x, p)

(1 + p

)

1

dp + Z

R

f (t, x, p)

(1 + p

)

1

dp

≤ e

X

Z

R

g

(P (0; t, x, p))

(1 + p

)

1

dp

≤ e

X

Z

R

g

(e

p − (−1)

R(t))

(1 + p

)

1

dp

= I

(t) + I

(t). (29)

By direct computations (use the changes of variable e

p + R(t) = q ≤ 0, respec- tively e

p − R(t) = q ≥ 0) one gets

I

(t) ≤ ±e

p min(1, e

)

Z

0

g

(p)

(1 + p

)

dp. (30) Finally from (27), (28), (29), (30) one gets

kµ(t)k

≤ e

p min(1, e

)

Z

R

g

(p)

(1 + p

)

dp + 2e

kg

k

R(t), t ∈ [0, T ]. (31)