Nordström - Vlasov

Mihai Bostan
email: mbostan@math.univ-fcomte.fr

Mild solutions for the one dimensional

Keywords: Nordström equation, Vlasov equation, weak/mild solutions, global existence AMS classification: 35B35, 35A05, 35B45, 82D10

The Nordström-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.

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. [START_REF] Degond | Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity[END_REF], [START_REF] Schaeffer | The classical limit of the relativistic Vlasov-Maxwell system[END_REF], [START_REF] Bostan | Asymptotic behavior of weak solutions for the relativistic Vlasov-Maxwell equations with large light speed[END_REF]. 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. [START_REF] Bouchut | Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system[END_REF], [START_REF] Diperna | Global weak solutions of the Vlasov-Maxwell system[END_REF], [START_REF] Glassey | On the 'one and one-half dimensional' relativistic Vlasov-Maxwell system[END_REF], [START_REF] Glassey | The two and one-half dimensional relativistic Vlasov-Maxwell system[END_REF], [START_REF] Glassey | Singularity formation in a collisionless plasma could only occur at high velocities[END_REF], [START_REF] Klainerman | A new approach to study the Vlasov-Maxwell system[END_REF]. The Einstein-Vlasov system is much more difficult. The reader can refer to [START_REF] Andréasson | The Einstein-Vlasov system/Kinetic theory[END_REF] for a recent review, see also [START_REF] Rein | Selfgravitating Systems in Newtonian theory -the Vlasov-Poisson system[END_REF], [START_REF] Rendall | An introduction to the Einstein-Vlasov system[END_REF], [START_REF] Rendall | The Einstein-Vlasov system (The Einstein equations and the large scale behavior of gravitational fields)[END_REF]. 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 Nordström scalar gravitation theory [START_REF] Nordström | Zur Theorie der Gravitation vom Standpunkt des Relativitätsprinzips[END_REF]. Let F = F (t, x, p) ≥ 0 denote the density of particles in the phase-space, where t ∈ R represents time,

x ∈ R N position and p ∈ R N momentum, with N ∈ {1, 2, 3}. This density satisfies the Vlasov equation

∂ t F + v(p) • ∇ x F -(∂ t φ + v(p) • ∇ x φ)p + (1 + |p| 2 ) -1 2 ∇ x φ • ∇ p F = 0,
coupled to the wave equation

∂ 2 t φ -∆ x φ = -e (N +1)φ(t,x)
R N F (t, x, p)

(1 + |p| 2 ) 1 2
dp.

Here v(p) = p (1+|p| 2 ) 1/2 denotes the relativistic velocity of a particle with momentum p. We normalize the physical units such that the rest mass of each particle, the 2 gravitational constant and the speed of the light are all equal to unity. For more details on this model see [START_REF] Calogero | Spherically symmetric steady states of galactic dynamics in scalar gravity[END_REF]. It is convenient to rewrite the system in terms of the unknowns (f, φ) where f (t, x, p) = e (N +1)φ(t,x) F (t, x, p). The system becomes

∂ t f + v(p) • ∇ x f -(Sφ)p + (1 + |p| 2 ) -1 2 ∇ x φ • ∇ p f = (N + 1)f (Sφ), (1)
∂ 2 t φ -∆ x φ = -µ(t, x), ( 2
)
µ(t, x) = R N f (t, x, p) (1 + |p| 2 ) 1 2 dp, (3)
where S = ∂ t + v(p) • ∇ x is the free-transport operator. We consider the initial conditions

f (0, •, •) = f 0 , φ(0, •) = ϕ 0 , ∂ t φ(0, •) = ϕ 1 . ( 4
)
The system (1), ( 2), ( 3), (4) will be referred to as the Nordström-Vlasov system.

Recently Calogero and Rein proved in [START_REF] Calogero | On classical solutions of the Nordström-Vlasov system[END_REF] 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 [START_REF] Glassey | Singularity formation in a collisionless plasma could only occur at high velocities[END_REF]. In one dimension they obtained the global existence and uniqueness for smooth compactly supported initial particle density f 0 ∈ C 1 c (R 2 ) and smooth initial conditions ϕ 0 ∈ C 2 b (R), ϕ 1 ∈ C 1 b (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 [START_REF] Calogero | The nonrelativistic limit of the Nordström-Vlasov system[END_REF]. The existence of global weak solution for the Nordström-Vlasov system in three dimensions has been studied as well, see [START_REF] Calogero | Global weak solutions to the Nordström-Vlasov system[END_REF]. One of the key points is the smoothing effect due to momentum averaging, see [START_REF] Diperna | Global weak solutions of the Vlasov-Maxwell system[END_REF], [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF].

The aim of this paper is to prove a more general existence and uniqueness result for the Nordström-Vlasov system in one dimension under less restrictive hypotheses.

Our method allows us to remove the support compactness of the initial density f 0 and the C 1 regularity of it. We assume that f 0 is bounded uniformly with respect to

x by some function g 0 = g 0 (p) having finite kinetic energy and we construct a unique global solution (f, φ) where f is solution by characteristics (or mild solution) of [START_REF] Andréasson | The Einstein-Vlasov system/Kinetic theory[END_REF] and φ is a classical solution of [START_REF] Bostan | Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem[END_REF]. We call such a solution (f, φ) a mild solution for the Nordström-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 Nordström-Vlasov system, see [START_REF] Bostan | Mild solutions for the relativistic Vlasov-Maxwell system for laser-plasma interaction[END_REF], [START_REF] Carrillo | Global solutions for the one-dimensional Vlasov-Maxwell system for laserplasma interaction[END_REF]. Results for initial boundary value problems can be obtained as well by this method, see [START_REF] Bostan | Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem[END_REF] 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 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2 ) and that there is some function g 0 ∈ L ∞ (R) such that 0 ≤ f 0 (x, p) ≤ g 0 (p), ∀ (x, p) ∈ R 2 , sign(•)g 0 (•) is nonincreasing on R, and R (1 + |p|)g 0 (p) dp < +∞. Then there is a unique mild solution (f ≥ 0, φ) ∈ L ∞ (]0, T [; L 1 (R 2 )) × W 2,∞ (]0, T [×R), ∀ T > 0 of the Nordström-Vlasov system (1), (2), (3), (4) with N = 1. Moreover 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 uniqueness 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 k 0 , ϕ k 0 , ϕ k 1 ) k∈{1,2}
satisfy the hypotheses of Theorem 1.1 and denote by (f k , φ k ) k∈{1,2} the corresponding global solutions of the one dimensional Nordström-Vlasov system. Moreover suppose that

R |p| 2 g k 0 (p) dp < +∞, k ∈ {1, 2}. Then for any R > 0 there is a constant C R depending on R, ϕ k 0 W 2,∞ (R) , ϕ k 1 W 1,∞ (R) , f k 0 L 1 (R 2 ) , 2 m=0 |p| m g k 0 L 1 (R) , k ∈ {1, 2} such that max t∈[0,R],|x|≤R-t (|φ 1 -φ 2 | + |∂ x φ 1 -∂ x φ 2 | + |∂ t φ 1 -∂ t φ 2 |) (t, x) ≤ C R D R 0 , where D R 0 = ϕ 1 0 -ϕ 2 0 W 1,∞ ([-R,R]) + ϕ 1 1 -ϕ 2 1 L ∞ ([-R,R]) + R -R R (1+|p|)|f 1 0 -f 2 0 |dpdx. Corollary 1.1 Assume that the initial conditions (f k 0 , ϕ k 0 , ϕ k 1 ) k∈{1,2}
satisfy the hypotheses of Theorem 1.1 and denote by (f k , φ k ) k∈{1,2} the corresponding global solutions of the one dimensional Nordström-Vlasov system. Suppose also that

f 1 0 (x, p) = f 2 0 (x, p), ϕ 1 0 (x) = ϕ 2 0 (x), ϕ 1 1 (x) = ϕ 2 1 (x), x ∈ [-R, R], p ∈ R. Then we have f 1 (t, x, p) = f 2 (t, x, p), φ 1 (t, x) = φ 2 (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 Nordström-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 candidates, than the compactly supported smooth functions. Second, we highlight the finite propagation speed of the solutions. For the numerical point of view this property 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 solution. 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 Nordström-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.

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

∂ t f +v(p)•∂ x f -(Sφ)p + (1 + p 2 ) -1 2 ∂ x φ •∂ p f = 2f (Sφ), (t, x, p) ∈]0, T [×R 2 , ( 5
)
with the initial condition

f (0, x, p) = f 0 (x, p), (x, p) ∈ R 2 . ( 6
)
For any (t, x, p) ∈ [0, T ] × R 2 we introduce the system of characteristics for ( 5)

dX ds = v(P (s)), dP ds = -P (s) (∂ t φ + v(P (s))∂ x φ) (s, X(s)) - ∂ x φ(s, X(s)) (1 + |P (s)| 2 ) 1 2
, [START_REF] Calogero | The nonrelativistic limit of the Nordström-Vlasov system[END_REF] with the conditions

X(s = t) = x, P (s = t) = p. ( 8
)
Notice that if φ is a C 1 function and ∂ t φ, ∂ x φ are Lipschitz with respect to x ∈ R uniformly on t in [0, T ], then there is a unique C 1 solution for ( 7), [START_REF] Calogero | On classical solutions of the Nordström-Vlasov system[END_REF], denoted (X(s; t, x, p), P (s; t, x, p)) or simply (X(s), P (s)). The divergence of the field appearing in the right hand side terms of ( 7) is given by

Div (x,p) v(p), -(Sφ)p -(1 + p 2 ) -1 2 ∂ x φ = -∂ t φ -v(p)∂ x φ, (9)
and by observing that d ds φ(s, X(s)) = ∂ t φ(s, X(s)) + v(P (s))∂ x φ(s, X(s)) we deduce that the jacobian matrix J(s) := ∂(X(s;t,x,p),P (s;t,x,p))

∂(x,p)

has the determinant given by det J(s) = det ∂(X(s; t, x, p), P (s; t, x, p)) ∂(x, p) = e -φ(s,X(s;t,x,p))+φ(t,x) = 0.

Multiplying ( 5) by e -2φ (t,x) and by taking into account that

S(f e -2φ ) = e -2φ Sf + f S(e -2φ ) = e -2φ Sf -2f e -2φ Sφ,
we obtain

∂ t (f e -2φ ) + v(p)∂ x (f e -2φ ) -(Sφ)p + (1 + p 2 ) -1 2 ∂ x φ ∂ p (f e -2φ ) = 0.
We deduce formally that f e -2φ is constant along any characteristic of (7) and we have the usual definition

Definition 2.1 Assume that φ ∈ C 1 ([0, T ] × R), ∂ t φ, ∂ x φ ∈ L ∞ (]0, T [; W 1,∞ (R))
and f 0 is a measurable function on R 2 . The solution by characteristics (or mild solution) of (5), ( 6) is given by f (t, x, p) = f 0 (X(0; t, x, p), P (0; t, x, p))e 2φ(t,x)-2φ(0,X(0;t,x,p)) , (t, x, p) ∈ [0, T ] × R 2 .

In the following proposition we recall some immediate properties of the mild solution

Proposition 2.1 Assume that φ ∈ C 1 ([0, T ] × R) ∩ L ∞ (]0, T [×R), ∂ t φ, ∂ x φ ∈ L ∞ (]0, T [; W 1,∞ (R)
), f 0 is measurable. Denote by f the mild solution of (5), [START_REF] Calogero | Spherically symmetric steady states of galactic dynamics in scalar gravity[END_REF].

1) If f 0 is nonnegative then f is nonnegative ;
2) If f 0 is bounded then f is bounded and we have

f L ∞ (]0,T [×R 2 ) ≤ e 4 φ L ∞ (]0,T [×R) f 0 L ∞ (R 2 ) ; 3) If f 0 ∈ L 1 (R 2 ) then f ∈ L ∞ (]0, T [; L 1 (R 2 )) and R R |f (t, x, p)| dp dx ≤ e 2 φ L ∞ (]0,T [×R) f 0 L 1 (R 2 ) .
In particular the conservation of the total mass holds

R R |f (t, x, p)|e -φ(t,x) dp dx = R R |f 0 (x, p)|e -φ(0,x) dp dx, t ∈ [0, T ] ; 4) If f 0 ∈ L 1 (R 2
) then for any continuous bounded function ψ we have

T 0 R R f (t, x, p)ψ(t, x, p) dp dx dt = R R f 0 (x, p)
T 0 ψ(t, X(t; 0, x, p), P (t; 0, x, p))

× e φ(t,X(t;0,x,p))-φ(0,x) dt dp dx ;

5) If (1 + |p|)f 0 ∈ L 1 (R 2 ) then (1 + |p|)f ∈ L ∞ (]0, T [; L 1 (R 2 )) and R R (1 + |p|)|f (t, x, p)| dp dx ≤ e 2 φ L ∞ 1 + e 2 φ L ∞ t 0 ∂ x φ(s) L ∞ ds f 0 L 1 (R 2 ) + e 4 φ L ∞ R R |p||f 0 (x, p)| dp dx ; 6) If f 0 ∈ L 1 (R 2
) then the mild solution f satisfies the following weak formulation

T 0 R R f (t, x, p)e -φ(t,x) ∂ t θ + v(p)∂ x θ -pSφ + (1 + p 2 ) -1 2 ∂ x φ ∂ p θ dp dx dt + R R f 0 (x, p)e -φ(0,x) θ(0, x, p) dp dx = 0, ( 11
)
for any θ ∈ C 1 c ([0, T [×R 2 ). In particular, if (1 + |p|)f 0 ∈ L 1 (R 2 ), the above formu- lation holds for any θ ∈ C 1 b ([0, T ] × R 2 ) such that θ(T, •, •) = 0.
Proof. The first and second statements are trivial.

3) Assume now that f 0 ∈ L 1 (R 2 ) 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 [START_REF] Carrillo | Global solutions for the one-dimensional Vlasov-Maxwell system for laserplasma interaction[END_REF] we have

R R |f (t, x, p)| dp dx = R R |f (t, X(t), P (t))|e -φ(t,X(t))+φ(0,y) dq dy = R R
|f 0 (y, q)|e φ(t,X(t))-φ(0,y) dq dy

≤ e 2 φ L ∞ f 0 L 1 (R 2 ) .
Similarly we obtain

R R |f (t, x, p)|e -φ(t,x) dp dx = R R |f (t, X(t), P (t))|e -2φ(t,X(t))+φ(0,y) dq dy = R R
|f 0 (y, q)|e -φ(0,y) dq dy.

4) Take now ψ ∈ C 0 b ([0, T ] × R 2 )
. By the previous point we already know that f (t) ∈ L 1 (R 2 ) for any t ∈ [0, T ]. By change of variables we obtain as before

T 0 R R f ψ dp dx dt = T 0 R R
f (t, X(t), P (t))ψ(t, X(t), P (t))e -φ(t,X(t))+φ(0,y) dq dy dt

= R R f 0 (y, q)
T 0 ψ(t, X(t), P (t))e φ(t,X(t))-φ(0,y) 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

R R (1 + |p|)|f | dp dx = R R (1 + |P (t)|)|f (t, X(t), P (t))|e -φ(t,X(t))+φ(0,y) dq dy = R R
(1 + |P (t)|)|f 0 (y, q)|e φ(t,X(t))-φ(0,y) dq dy. [START_REF] Diperna | Global weak solutions of the Vlasov-Maxwell system[END_REF] By [START_REF] Calogero | The nonrelativistic limit of the Nordström-Vlasov system[END_REF] we have

d ds {P (s)e φ(s,X(s)) } = - e φ(s,X(s)) (1 + |P (s)| 2 ) 1 2
∂ x φ(s, X(s)), [START_REF] Glassey | On the 'one and one-half dimensional' relativistic Vlasov-Maxwell system[END_REF] and therefore P (t)e φ(t,X(t)) = qe φ(0,y) -t 0 e φ(s,X(s))

(1 + |P (s)| 2 ) 1 2
∂ x φ(s, X(s))ds.

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

|P (t)| ≤ |q|e φ(0,y)-φ(t,X(t)) + e -φ(t,X(t)) t 0 e φ(s,X(s)) |∂ x φ(s, X(s))| ds ≤ |q|e 2 φ L ∞ + e 2 φ L ∞ t 0 ∂ x φ(s) L ∞ . ( 14
)
Combining ( 12), ( 14) yields

R R (1 + |p|)|f (t, x, p)| dp dx ≤ e 2 φ L ∞ 1 + e 2 φ L ∞ t 0 ∂ x φ(s) L ∞ ds f 0 L 1 (R 2 ) + e 4 φ L ∞ R R |p||f 0 (x, p)| dp dx. 6) For any θ ∈ C 1 c ([0, T [×R 2 ) consider ψ(t, x, p) = e -φ(t,x) ∂ t θ + v(p)∂ x θ -pSφ + (1 + p 2 ) -1 2 ∂ x φ ∂ p θ .
Notice that ψ(s, X(s), P (s)) = e -φ(s,X(s)) d ds {θ(s, X(s), P (s))}. By applying 4) one gets

T 0 R R f e -φ(t,x) ∂ t θ + v(p)∂ x θ -pSφ + (1 + p 2 ) -1 2 ∂ x φ ∂ p θ dp dx dt = R R f 0 (x, p) T 0 e -φ(0,x) d ds {θ(s, X(s), P (s))} ds dp dx = - R R
f 0 (x, p)e -φ(0,x) θ(0, x, p) dp dx.

If (1+|p|)f 0 ∈ L 1 (R 2 ) we know that (1+|p|)f ∈ L ∞ (]0, T [; L 1 (R 2 )). Take χ ∈ C 1 c (R), χ(u) = 0 if |u| ≥ 2, χ(u) = 1 if |u| ≤ 1, χ ≥ 0 and χ R (•) = χ( • R ), R > 0.
In order to prove [START_REF] Degond | Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity[END_REF]

for θ ∈ C 1 b ([0, T ] × R 2 ), θ(T, •, •) = 0, apply them with θ R = θχ R (x)χ R (p) and let R → +∞.
Remark 2.1 Notice that (5) can be written

∂ t (f e -φ ) + ∂ x (v(p)f e -φ ) -∂ p (pSφ + (1 + p 2 ) -1 2 ∂ x φ)f e -φ = 0,
which justifies formally the weak formulation [START_REF] Degond | Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity[END_REF]. In particular the total mass R R f (t, x, p)e -φ(t,x) 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

u 1,∞ = u ∞ + u ∞ for any function u ∈ W 1,∞ (R). Proposition 2.2 Assume that φ k ∈ C 1 ([0, T ] × R) ∩ L ∞ (]0, T [×R), ∂ t φ k , ∂ x φ k ∈ L ∞ (]0, T [; W 1,∞ (R)), k ∈ {1, 2}. For any (x, p) ∈ R 2 consider (X k (t), P k (t)) = (X k (t; 0, x, p), P k (t; 0, x, p)) the characteristics corresponding to φ k , k ∈ {1, 2}. Then for any t ∈ [0, T ] there is a constant C depending on sup s∈[0,t],k∈{1,2} { φ k (s) L ∞ + ∂ x φ k (s) L ∞ + ∂ 2 x φ k (s) L ∞ } such that for any s ∈ [0, t] we have |X 1 (s) -X 2 (s)| + |P 1 (s)e φ 1 (s,X 1 (s)) -P 2 (s)e φ 2 (s,X 2 (s)) | ≤ C |p(φ 1 -φ 2 )(0, x)| + C t 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ.
The following result is a direct consequence of the above proposition (see the Appendix for proof details)

Corollary 2.1 Under the hypotheses of Proposition 2.2, there is a constant C depending on

sup s∈[0,T ],k∈{1,2} { φ k (s) ∞ + ∂ x φ k (s) ∞ + ∂ t φ k (s) ∞ + ∂ 2 x φ k (s) ∞ + ∂ 2 xt φ k (s) ∞ } such that we have for any t ∈ [0, T ] 2 k=1 (-1) k e φ k (t,X k (t)) (1 + |P k (t)| 2 ) 1 2 ≤ C φ 1 (t) -φ 2 (t) ∞ + t 0 φ 1 (s) -φ 2 (s) 1,∞ ds + C |p(φ 1 -φ 2 )(0, x)|, 2 k=1 (-1) k e φ k (t,X k (t)) (1 + |P k (t)| 2 ) 1 2 ∂ x φ k (t, X k (t)) ≤ C( φ 1 (t) -φ 2 (t) 1,∞ + t 0 φ 1 (s) -φ 2 (s) 1,∞ ds) + C |p(φ 1 -φ 2 )(0, x)|, 2 k=1 (-1) k e φ k (t,X k (t)) (1 + |P k (t)| 2 ) 1 2 ∂ t φ k (t, X k (t)) ≤ C( φ 1 (t) -φ 2 (t) ∞ + ∂ t φ 1 (t) -∂ t φ 2 (t) ∞ + t 0 φ 1 (s) -φ 2 (s) 1,∞ ds + |p(φ 1 -φ 2 )(0, x)|).
3 Fixed point application

Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2
), f 0 ≥ 0 and take an arbitrary T > 0. Since φ satisfies a one dimensional wave equation, a natural definition for the fixed point application is Fφ = φ for any φ in the set

D 1 = {φ ∈ C 1 ([0, T ]×R) : (φ, ∂ t φ)(0, •) = (ϕ 0 , ϕ 1 ), ∂ t φ, ∂ x φ ∈ L ∞ (]0, T [; W 1,∞ (R))},
where φ is given by d'Alembert formula

φ(t, x) = 1 2 {ϕ 0 (x + t) + ϕ 0 (x -t)} + 1 2 x+t x-t ϕ 1 (y) dy - 1 2 t 0 x+(t-s) x-(t-s) µ(s, y) dy ds, (t, x) ∈ [0, T ] × R, (15)
the function µ is given by

µ(t, x) = R f (t, x, p) (1 + p 2 ) 1 2 dp, (t, x) ∈ [0, T ] × R, (16)
and f is the mild solution of ( 5), ( 6) corresponding to φ.

Estimate of F

Since the initial density f 0 is nonnegative, we know that any mild solution f is nonnegative and by [START_REF] Glassey | Singularity formation in a collisionless plasma could only occur at high velocities[END_REF] we deduce the a priori estimate

φ(t, x) ≤ ϕ 0 ∞ + t ϕ 1 ∞ , (t, x) ∈ [0, T ] × R. ( 17
)
Actually we can restrict the application F to the set

D 2 = D 1 ∩ {φ : -ϕ 0 ∞ -T ϕ 1 ∞ - T 2 e 2 ϕ 0 ∞+T ϕ 1 ∞ f 0 L 1 ≤ φ(t, x) ≤ ϕ 0 ∞ + T ϕ 1 ∞ , ∀(t, x) ∈ [0, T ] × R},
and we check easily that for any φ ∈ D 2 we have for (t,

x) ∈ [0, T ] × R -ϕ 0 ∞ -T ϕ 1 ∞ - T 2 e 2 ϕ 0 ∞ +T ϕ 1 ∞ f 0 L 1 ≤ F φ(t, x) ≤ ϕ 0 ∞ + T ϕ 1 ∞ . ( 18
)
Indeed, for any φ ∈ D 2 , 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 x+(t-s) x-(t-s) µ(s, y) dy ≤ R R f (s, y, p) 1 + p 2 dp dy ≤ e sup φ(s,•) R R f (s, y, p)e -φ(s,y) dp dy = e sup φ(s,•) R R f 0 (y, p)e -ϕ 0 (y) dp dy ≤ e 2 ϕ 0 ∞ +T ϕ 1 ∞ f 0 L 1 .
Therefore ( 15), ( 17) imply (18).

Estimate of DFφ

The d'Alembert formula [START_REF] Glassey | Singularity formation in a collisionless plasma could only occur at high velocities[END_REF] implies

∂ t φ(t, x) = 1 2 {ϕ 0 (x + t) -ϕ 0 (x -t)} + 1 2 {ϕ 1 (x + t) + ϕ 1 (x -t)} - 1 2 t 0 {µ(s, x + t -s) + µ(s, x -(t -s))} ds, ( 19
)
and

∂ x φ(t, x) = 1 2 {ϕ 0 (x + t) + ϕ 0 (x -t)} + 1 2 {ϕ 1 (x + t) -ϕ 1 (x -t)} - 1 2 t 0 {µ(s, x + t -s) -µ(s, x -(t -s))} ds. ( 20
)
In order to estimate the L ∞ norms of ∂ t φ, ∂ x φ we need to estimate the L ∞ norm of µ. In particular we have to assume that µ 0

(•) = R f 0 (•,p) (1+p 2 ) 1 2 dp ∈ L ∞ (R). It is convenient to suppose that there is some function g 0 ∈ L ∞ (R) such that 0 ≤ f 0 (x, p) ≤ g 0 (p), (x, p) ∈ R 2 , sign(•)g 0 (•) is nonincreasing on R, (21)
and

R g 0 (p) (1 + p 2 ) 1 2 dp < +∞. ( 22
) Proposition 3.1 Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2 ) and
(21), ( 22) hold. Then for any φ ∈ D 2 we have

µ(t) L ∞ ≤ C 1 + C 2 t 0 ∂ x φ(τ ) L ∞ dτ, t ∈ [0, T ],
where

C 1 = e 6M min(1, e 4M ) R g 0 (p) (1 + p 2 ) 1 2 dp, C 2 = 2e 6M g 0 L ∞ ,
and

M = ϕ 0 ∞ + T ϕ 1 ∞ + T 2 e 2 ϕ 0 ∞ +T ϕ 1 ∞ f 0 L 1 . Proof. For (t, x, p) ∈ [0, T ] × R 2 consider (X(s), P (s)) = (X(s; t, x, p), P (s; t, x, p)), s ∈ [0, T ]
. By [START_REF] Glassey | On the 'one and one-half dimensional' relativistic Vlasov-Maxwell system[END_REF] we deduce that for any s ∈ [0, T ] we have

e φ(s,X(s)) P (s) = e φ(t,x) p - s t e φ(τ,X(τ )) (1 + |P (τ )| 2 ) 1 2 ∂ x φ(τ, X(τ )) dτ. ( 23
)
In particular we obtain e φ(0,X(0

)) P (0) = e φ(t,x) p + t 0 e φ(τ,X(τ )) (1 + |P (τ )| 2 ) 1 2 ∂ x φ(τ, X(τ )) dτ. ( 24
)
We denote by M a bound for φ ∞ , for example

M = ϕ 0 ∞ + T ϕ 1 ∞ + T 2 e 2 ϕ 0 ∞ +T ϕ 1 ∞ f 0 L 1 ,
and for any t ∈ [0, T ] we introduce the notation

R(t) = t 0 ∂ x φ(τ ) ∞ dτ.
Observe that for any p ≥ e 2M R(t) formula ( 24) implies e φ(0,X(0)) P (0) ≥ e -M p -e M R(t) ≥ 0, and thus

P (0; t, x, p) ≥ e -2M p -R(t) ≥ 0. ( 25
)
Similarly, if p ≤ -e 2M R(t), formula (24) implies -e φ(0,X(0)) P (0) ≥ e -M (-p)e M R(t) ≥ 0. In this case we obtain

-P (0; t, x, p) ≥ e -2M (-p) -R(t) ≥ 0. ( 26
)
We can estimate µ as follows

µ(t, x) = R f (t, x, p) (1 + p 2 ) 1 2 1 {|p|<e 2M R(t)} dp + R f (t, x, p) (1 + p 2 ) 1 2 1 {|p|≥e 2M R(t)} dp = µ 1 (t, x) + µ 2 (t, x), (t, x) ∈ [0, T ] × R. ( 27
)
By the definition of the mild solution we have

µ 1 (t, x) = R
f 0 (X(0; t, x, p), P (0; t, x, p))

(1 + p 2 )
1 2 e 2φ(t,x)-2φ(0,X(0;t,x,p))

1 {|p|<e 2M R(t)} dp ≤ e 4M
R g 0 (P (0; t, x, p))

(1 + p 2 )
1 2

1 {|p|<e 2M R(t)} dp ≤ 2e 6M g 0 ∞ R(t), (t, x) ∈ [0, T ] × R. (28)
In order to estimate µ 2 observe that (25), ( 26) and the monotonicity of g 0 yields f 0 (X(0; t, x, p), P (0; t, x, p)) ≤ g 0 (P (0; t, x, p))

≤ g 0 (e -2M p -R(t)), ∀ p ≥ e 2M R(t), respectively f 0 (X(0; t, x, p), P (0; t, x, p)) ≤ g 0 (P (0; t, x, p)) ≤ g 0 (e -2M p+R(t)), ∀ p ≤ -e 2M R(t).
Therefore we have as before

µ 2 (t, x) = R f (t, x, p) (1 + p 2 ) 1 2 1 {p≤-e 2M R(t)} dp + R f (t, x, p) (1 + p 2 )
1 2

1 {p≥e 2M R(t)} dp ≤ e 4M 2 k=1 R
g 0 (P (0; t, x, p))

(1 + p 2 ) 1 2 1 {(-1) k p≥e 2M R(t)} dp ≤ e 4M 2 k=1 R g 0 (e -2M p -(-1) k R(t)) (1 + p 2 ) 1 2 1 {(-1) k p≥e 2M R(t)} dp = I -(t) + I + (t). ( 29
)
By direct computations (use the changes of variable e -2M p + R(t) = q ≤ 0, respectively e -2M p -R(t) = q ≥ 0) one gets

I ± (t) ≤ ±e 6M min(1, e 4M ) ±∞ 0 g 0 (p) (1 + p 2 ) 1 2 dp. ( 30
)
Finally from ( 27), (28), (29), (30) one gets

µ(t) ∞ ≤ e 6M min(1, e 4M ) R g 0 (p) (1 + p 2 ) 1 2 dp + 2e 6M g 0 ∞ R(t), t ∈ [0, T ]. ( 31
)
Now we obtain easily estimates for the L ∞ norms of ∂ t φ, ∂ x φ. Indeed, (20) and Proposition 3.1 imply for any t ∈ [0, T ]

∂ x φ(t) ∞ ≤ ϕ 0 ∞ + ϕ 1 ∞ + t 0 µ(s) ∞ ds ≤ ϕ 0 ∞ + ϕ 1 ∞ + t 0 {C 1 + C 2 R(s)}ds ≤ ϕ 0 ∞ + ϕ 1 ∞ + C 1 T + C 2 T t 0 ∂ x φ(s) ∞ ds. ( 32
)
We restrict the application F to

D 3 = D 2 ∩{φ : max( ∂ x φ(t) ∞ , ∂ t φ(t) ∞ ) ≤ ( ϕ 0 ∞ + ϕ 1 ∞ +C 1 T )e C 2 T t , t ∈ [0, T ]}.
By using (32) we check easily that for any φ ∈ D 3 we have

∂ x Fφ(t) ∞ = ∂ x φ(t) ∞ ≤ ( ϕ 0 ∞ + ϕ 1 ∞ + C 1 T )e C 2 T t , ∀ t ∈ [0, T ]. (33)
Notice that for any φ ∈ D 3 we have by ( 19)

∂ t Fφ(t) ∞ ≤ ϕ 0 ∞ + ϕ 1 ∞ + t 0 µ(s) ∞ ds ≤ ϕ 0 ∞ + ϕ 1 ∞ + C 1 T + C 2 T t 0 ∂ x φ(s) ∞ ds ≤ ( ϕ 0 ∞ + ϕ 1 ∞ + C 1 T )e C 2 T t , ∀ t ∈ [0, T ]. (34)
By (31) we deduce also that for any φ ∈ D 3 we have

µ(t) ∞ ≤ C 1 + C 2 t 0 ( ϕ 0 ∞ + ϕ 1 ∞ + C 1 T )e C 2 T s ds = C 1 + ( ϕ 0 ∞ + ϕ 1 ∞ + C 1 T ) e C 2 T t -1 T . ( 35
)

Estimate of D 2 Fφ

We intend to estimate the second derivatives of φ = Fφ, for any φ ∈ D 3 . We have

∂ 2 x φ(t, x) = 1 2 {ϕ 0 (x + t) + ϕ 0 (x -t)} + 1 2 {ϕ 1 (x + t) -ϕ 1 (x -t)} - 1 2 ∂ x D + (t, x) + 1 2 ∂ x D -(t, x), ( 36
)
where

D ± (t, x) = t 0 µ(s, x ± (t -s)) ds. Obviously we have for any t ∈ [0, T ] ∂ 2 x φ(t) ∞ ≤ ϕ 0 ∞ + ϕ 1 ∞ + 1 2 ∂ x D + (t) ∞ + 1 2 ∂ x D -(t) ∞ , (37)
and thus it remains to estimate the L ∞ norms of ∂ x D ± . For any test function

ψ ∈ C 1 c (R) we can write R D ± (t, x)ψ (x) dx = t 0 R R f (s, x ± (t -s), p) (1 + p 2 ) 1 2 ψ (x) dp dx ds = t 0 R R f (s, x, p) (1 + p 2 ) 1 2 ψ (x ∓ (t -s)) dp dx ds. ( 38
)
By Proposition 2.1 we can write

R D ± (t, x)ψ (x) dx = R R f 0 (x, p)e -ϕ 0 (x) t 0 e φ(s,X(s)) ψ (X(s) ∓ (t -s)) (1 + |P (s)| 2 ) 1 2 ds dp dx, (39)
where (X(s), P (s)) = (X(s; 0, x, p), P (s; 0, x, p)), (s, x, p) ∈ [0, t] × R 2 . We need to estimate the integrals

I ± (t, x, p) = t 0 e φ(s,X(s)) (1 + |P (s)| 2 ) 1 2 ψ (X(s) ∓ (t -s)) ds = t 0 e φ(s,X(s)) (1 + |P (s)| 2 ) 1 2 1 v(P (s)) ± 1 d ds {ψ(X(s) ∓ (t -s))} ds = t 0 e φ(s,X(s)) (-P (s) ± (1 + |P (s)| 2 ) 1 2 ) d ds {ψ(X(s) ∓ (t -s))} ds = e φ(s,X(s)) (-P (s) ± (1 + |P (s)| 2 ) 1 2 )ψ(X(s) ∓ (t -s))| t 0 - t 0 d ds {e φ(s,X(s)) (-P (s) ± (1 + |P (s)| 2 ) 1 2 )}ψ(X(s) ∓ (t -s)) ds = I ± 1 ψ(X(t)) -I ± 2 ψ(x ∓ t) - t 0 I ± 3 (s)ψ(X(s) ∓ (t -s)) ds, ( 40
)
where

I ± 1 := e φ(t,X(t)) (-P (t) ± (1 + |P (t)| 2 ) 1 2 ), I ± 2 = e ϕ 0 (x) (-p ± (1 + p 2 ) 1
2 ), and

I ± 3 (s) = d ds {e φ(s,X(s)) (-P (s) ± (1 + |P (s)| 2 ) 1 2 )}.
The terms I ± 1 , I ± 2 can be estimated by

|I ± 1 | ≤ 2e φ ∞ (1 + |P (t)|), |I ± 2 | ≤ 2e ϕ 0 ∞ (1 + |p|). ( 41
)
By using [START_REF] Calogero | The nonrelativistic limit of the Nordström-Vlasov system[END_REF] and the bounds for the L ∞ norms of φ, ∂ x φ, ∂ t φ one gets by direct computations that

|I ± 3 (s)| ≤ C(1 + |P (s)|), ∀ 0 ≤ s ≤ t ≤ T, ( 42
)
for some constant C depending only on the initial conditions ϕ 0 , ϕ 1 , f 0 and T . Therefore combining (39), ( 40), ( 41), (42) yields

R D ± (t, x)ψ (x) dx ≤ C R R f 0 (x, p)(1 + |p|)|ψ(x ∓ t)| dp dx (43) + C R R f 0 (x, p)(1 + |P (t)|)|ψ(X(t))| dp dx + C R R f 0 (x, p) t 0 (1 + |P (s)|)|ψ(X(s) ∓ (t -s))| dp dx,
for any t ∈ [0, T ], ψ ∈ C 1 c (R) and for some constant depending only on the initial conditions and T . We will use the following lemma

Lemma 3.1 Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2 ) and that (21) holds for some function g 0 ∈ L ∞ (R) satisfying R (1 + |p|)g 0 (p) dp < +∞. ( 44
)
Then for any

φ ∈ D 3 we have R (1 + |p|)f (•, •, p) dp L ∞ (]0,T [×R) ≤ C 3 ,
where f is the mild solution of (5), ( 6) corresponding to φ and C 3 depends only on the initial conditions and T .

Proof. It is very similar to those of Proposition 3.1. For any (t,

x) ∈ [0, T ] × R let k(t, x) = R (1 + |p|)f (t, x, p) dp. We have k(t, x) = R (1 + |p|)f (t, x, p)1 {|p|<e 2M R(t)} dp + R (1 + |p|)f (t, x, p)1 {|p|≥e 2M R(t)} dp = k 1 (t, x) + k 2 (t, x), ( 45
)
where

M = ϕ 0 ∞ + T ϕ 1 ∞ + T 2 e 2 ϕ 0 ∞+T ϕ 1 ∞ f 0 L 1 and R(t) = t 0 ∂ x φ(s) ∞ ds.
Note that since φ belongs to D 3 , sup t∈[0,T ] R(t) can be bounded by some constant depending only on the initial conditions (observe that hypothesis (44) is stronger than ( 22)) and T . For the first term k 1 we can write

k 1 (t, x) = R f 0 (X(0; t, x, p), P (0; t, x, p))e 2φ(t,x)-2φ(0,X(0)) (1 + |p|)1 {|p|<e 2M R(t)} dp ≤ 2e 6M (1 + e 2M R(t))R(t) g 0 ∞ . ( 46
)
For the second term k 2 we have

k 2 (t, x) ≤ e 4M 2 k=1 R g 0 (P (0; t, x, p))(1 + |p|)1 {(-1) k p≥e 2M R(t)} dp ≤ e 4M 2 k=1 R g 0 (e -2M p -(-1) k R(t))(1 + |p|)1 {(-1) k p≥e 2M R(t)} dp ≤ e 6M max{e 2M , 1 + e 2M R(t)} R (1 + |p|)g 0 (p) dp. ( 47
)
Our conclusion follows from (45), ( 46), (47).

We prove also a more general result for later use

Lemma 3.2 Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2
) and that [START_REF] Rendall | The Einstein-Vlasov system (The Einstein equations and the large scale behavior of gravitational fields)[END_REF] holds for some function

g 0 ∈ L ∞ (R) satisfying R |p| m+1 g 0 (p) dp < +∞ with m ∈ N. Consider r : R 2 → [0, +∞[ verifying R r(x, p) dx ≤ α|p| + β, ∀ p ∈ R.
Then for any φ

∈ D 3 we have R R |p| m f (•, x, p)r(x, p) dp dx L ∞ (]0,T [) ≤ (α + β)C,
where f is the mild solution of (5), ( 6) corresponding to φ and C depends only on the initial conditions and T .

Proof. With the same notations as before we have for any

t ∈ [0, T ] R R |p| m f (t, x, p)r(x, p) dp dx = R R |p| m f (t, x, p)r(x, p)1 {|p|<e 2M R(t)} dp dx + R R |p| m f (t, x, p)r(x, p)1 {|p|≥e 2M R(t)} dp dx = I m 1 (t) + I m 2 (t). ( 48
)
For the first term we have

I m 1 (t) ≤ e 4M g 0 ∞ R |p| m 1 {|p|<e 2M R(t)} R r(x, p) dx dp ≤ (α + β)C m 1 , (49)
with

C m 1 = 2e 2M (3+m) (R(T )) m+1 max{1, e 2M R(T )} g 0 ∞ .
For the second term we can write

I m 2 (t) ≤ e 4M 2 k=1 R R |p| m g 0 (P (0; t, x, p))r(x, p)1 {(-1) k p≥e 2M R(t)} dp dx ≤ e 4M 2 k=1 R R |p| m g 0 (e -2M p -(-1) k R(t))r(x, p)1 {(-1) k p≥e 2M R(t)} dp dx ≤ (α + β)C m 2 , ( 50
) with C m 2 = e 2M (3+m) max{ R (|p| + R(T )) m g 0 (p) dp, e 2M R (|p| + R(T )) m+1 g 0 (p) dp}.
The conclusion follows by ( 48), ( 49), (50).

t ∈ [0, T ] R R (1 + |p|)f 0 (x, p)|ψ(x ∓ t)| dp dx = R |ψ(x ∓ t)| R (1 + |p|)f 0 (x, p) dp dx ≤ R |ψ(x ∓ t)| R (1 + |p|)g 0 (p) dp dx = R g 0 (p)(1 + |p|) dp ψ L 1 . ( 51
)
For the other terms in (43

) let us estimate R R f 0 (x, p)(1 + |P (s)|)|ψ(X(s) ∓ (t - s))| dp dx for any 0 ≤ s ≤ t ≤ T . By changing variables along characteristics (see Proposition 2.1) we obtain R R f 0 (x, p)(1 + |P (s)|)|ψ(X(s) ∓ (t -s))| dp dx = R R f (s, X(s), P (s))e 2ϕ 0 (x)-2φ(s,X(s)) (1 + |P (s)|)|ψ(X(s) ∓ (t -s))| dp dx = R R f (s, X(s), P (s))e ϕ 0 (x)-φ(s,X(s)) (1 + |P (s)|)|ψ(X(s) ∓ (t -s))| det J(s) dp dx ≤ e 2 φ ∞ R |ψ(X ∓ (t -s))| R f (s, X, P )(1 + |P |) dP dX ≤ e 2 φ ∞ C 3 ψ L 1 . ( 52
)
Combining ( 43), ( 51), (52

) we obtain that R D ± (t, x)ψ (x) dx ≤ C 4 ψ L 1 , ∀ ψ ∈ C 1 c (R)
, for some constant C 4 depending only on the initial conditions and T . We deduce that ∂ x D ± ∞ ≤ C 4 and therefore (37) implies

∂ 2 x φ ∞ ≤ ϕ 0 ∞ + ϕ 1 ∞ + C 4 =: C 5 . ( 53
)
Observe that from [START_REF] Rein | Selfgravitating Systems in Newtonian theory -the Vlasov-Poisson system[END_REF] we have

∂ 2 xt φ(t, x) = 1 2 {ϕ 0 (x + t) -ϕ 0 (x -t)} + 1 2 {ϕ 1 (x + t) + ϕ 1 (x -t)} - 1 2 ∂ x D + (t, x) - 1 2 ∂ x D -(t, x),
and therefore we obtain the same bound

∂ 2 xt φ ∞ ≤ ϕ 0 ∞ + ϕ 1 ∞ + C 4 = C 5 .
For estimating the second derivative ∂ 2 t φ we can write

|∂ 2 t φ(t, x)| = |∂ 2 x φ(t, x) -µ(t, x)| ≤ C 5 + µ ∞ .
By using (35) one gets

∂ 2 t φ ∞ ≤ C 6 , ( 54
)
for some constant depending only on the initial conditions and T .

Remark 3.2

The computations of the paragraphs 3.1, 3.2, 3.3 show that under the hypotheses of Lemma 3.1 D 3 is left invariant by the application F and we have

sup φ∈D 3 ∂ 2 x φ ∞ , ∂ 2 xt φ ∞ , ∂ 2 t φ ∞ , R f (•, •, p)(1 + |p|) dp ∞ < +∞, 20
where φ = Fφ and f is the mild solution of (5), [START_REF] Calogero | Spherically symmetric steady states of galactic dynamics in scalar gravity[END_REF]. It is convenient to restrict one more time the application F to the set

D 4 = D 3 ∩ {φ : max{ ∂ 2 x φ ∞ , ∂ 2 xt φ ∞ } ≤ C 5 , ∂ 2 t φ ∞ ≤ C 6 },
which is also left invariant by F.

Estimate of Fφ 1 -F φ 2

The following step is to estimate Fφ 1 -F φ 2 in terms of φ 1 -φ 2 for any φ 1 , φ 2 ∈ D 4 .

Consider φ k ∈ D 4 , k ∈ {1, 2} and denote by f k , k ∈ {1, 2} the mild solutions of ( 5), ( 6) corresponding to φ k , k ∈ {1, 2}. For any ψ ∈ L 1 (R) and t ∈ [0, T ] we have

R ( φ1 (t, y) -φ2 (t, y))ψ(y) dy = - 1 2 R t 0 R R (f 1 -f 2 )(s, x, p) (1 + p 2 ) 1 2 ψ(y) × 1 {|x-y|<t-s} dp dx ds dy (55) = - 1 2 t 0 R R (f 1 -f 2 )(s, x, p) θ(s, x) (1 + p 2 ) 1 2
dp dx ds, where θ(s, x) = x+(t-s)

x-(t-s) ψ(y) dy. Denote by (X k (s), P k (s)) the characteristics corresponding to φ k , k ∈ {1, 2}. We obtain by Proposition 2.

1 2 R ( φ1 (t, y) -φ2 (t, y))ψ(y) dy = 2 k=1 (-1) k R R f 0 (x, p)e -ϕ 0 (x) t 0 θ(s, X k (s)) (1 + |P k (s)| 2 ) 1 2
× e φ k (s,X k (s)) ds dp dx.

(56)

By using Remark 3.2 and Corollary 2.1 (with

φ 1 , φ 2 satisfying φ 1 (0) = φ 2 (0) = ϕ 0 )
we deduce that there is a constant C depending only on the initial conditions and

T such that 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 ≤ C φ 1 (s) -φ 2 (s) ∞ + s 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ . ( 57
)
We deduce that

| 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 θ(s, X k (s))| ≤ |θ(s, X 1 (s))| 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 + e φ 2 (s) ∞ 2 k=1 (-1) k θ(s, X k (s)) ≤ C ψ L 1 φ 1 (s) -φ 2 (s) ∞ + s 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ + C X 2 (s)-(t-s) X 1 (s)-(t-s) ψ(y) dy + C X 2 (s)+(t-s) X 1 (s)+(t-s) ψ(y) dy . ( 58
)
Combining ( 56), (58) we obtain 2

R ( φ1 -φ2 )(t, y)ψ(y) dy ≤ C t 0 R R f 0 ψ L 1 ( φ 1 (s) -φ 2 (s) ∞ + s 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ ) dp dx ds + C t 0 R R f 0 X 2 (s)-(t-s) X 1 (s)-(t-s)
ψ(y) dy dp dx ds

+ C t 0 R R f 0 X 2 (s)+(t-s) X 1 (s)+(t-s)
ψ(y) dy dp dx ds

= I 1 (t) + I 2 (t) + I 3 (t). ( 59
)
For the first term in the right hand side of (59) we have for some constant C (depending on the initial conditions and T )

I 1 (t) ≤ C ψ L 1 f 0 L 1 t 0 φ 1 (s) -φ 2 (s) 1,∞ ds. ( 60
)
In order to estimate the second term in the right hand side of (59) observe that for

any 0 ≤ s ≤ t ≤ T we have R R f 0 X 2 (s)-(t-s) X 1 (s)-(t-s) ψ(y) dy dp dx ≤ R R f 0 |ψ(y)|1 {|y-(X 1 (s)-(t-s))|<|X 1 (s)-X 2 (s)|} dy dp dx = R |ψ(y)| R R f 0 1 {|y-(X 1 (s)-(t-s))|<|X 1 (s)-X 2 (s)|} dp dx dy. (61) Since φ 1 (0) = φ 2 (0) = ϕ 0 we deduce by Proposition 2.2 that |X 1 (s)-X 2 (s)| ≤ CR(s)
where R(s) = s 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ . By using Lemma 3.1 we obtain after the change of variables along characteristics (cf. Proposition 2.1)

R R f 0 1 {|y-(X 1 (s)-(t-s))|<|X 1 (s)-X 2 (s)|} dp dx ≤ R R f 0 1 {|y-(X 1 (s)-(t-s))|<CR(s)} dp dx = R R f 1 (s, X 1 (s), P 1 (s))e ϕ 0 (x)-φ 1 (s,X 1 (s)) 1 {|y-(X 1 (s)-(t-s))|<CR(s)} det J 1 (s) dp dx ≤ e ϕ 0 ∞ + φ 1 ∞ R R f 1 (s, X, P )1 {|y-(X-(t-s))|<CR(s)} dP dX = e ϕ 0 ∞+ φ 1 ∞ R 1 {|y-(X-(t-s))|<CR(s)} R f 1 (s, X, P ) dP dX ≤ 2Ce ϕ 0 ∞ + φ 1 ∞ R f 1 (•, •, p) dp ∞ s 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ. ( 62
)
From ( 61), (62) we obtain for any t ∈ [0, T ]

I 2 (t) ≤ C ψ L 1 t 0 φ 1 (s) -φ 2 (s) 1,∞ ds, (63)
for some constant depending on the initial conditions and T . A similar estimate holds for the term I 3 (t) and finally (59) implies

R ( φ1 -φ2 )(t, y)ψ(y) dy ≤ C ψ L 1 t 0 φ 1 (s) -φ 2 (s) 1,∞ ds, ( 64
)
for any t ∈ [0, T ] and ψ ∈ L 1 (R). We deduce that for any

φ 1 , φ 2 ∈ D 4 Fφ 1 (t) -Fφ 2 (t) ∞ ≤ C t 0 φ 1 (s) -φ 2 (s) 1,∞ ds. (65) 3.5 Estimate of ∂ x Fφ 1 -∂ x Fφ 2
We need to estimate also the difference of spatial derivatives

∂ x Fφ 1 , ∂ x Fφ 2 , in terms of φ 1 -φ 2 for any φ 1 , φ 2 ∈ D 4 .
With the notations of the previous paragraphs, by using (20), we have for any function ψ ∈ L 1 (R)

R (∂ x φ1 -∂ x φ2 )ψ dx = 1 2 t 0 R {µ 1 (s, x -(t -s)) -µ 2 (s, x -(t -s))}ψ(x) dxds - 1 2 t 0 R {µ 1 (s, x + (t -s)) -µ 2 (s, x + (t -s))}ψ(x) dxds = 1 2 t 0 R R (f 1 -f 2 )(s, x -(t -s), p) ψ(x) (1 + p 2 ) 1 2 dp dx ds - 1 2 t 0 R R (f 1 -f 2 )(s, x + (t -s), p) ψ(x) (1 + p 2 ) 1 2
dp dx ds

= 1 2 I 4 (t) - 1 2 I 5 (t). ( 66
)
We analyze only the term I 4 (t). The estimate for I 5 (t) follows in similar way. By Proposition 2.1 we have

I 4 (t) = t 0 R R (f 1 -f 2 )(s, x, p) ψ(x + (t -s)) (1 + p 2 ) 1 2
dp dx ds

= - R R f 0 (x, p)e -ϕ 0 (x) t 0 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 ψ(X k (s) + (t -s))ds dp dx = R R f 0 (x, p)e -ϕ 0 (x) t 0 2 k=1 (-1) k ((1 + |P k (s)| 2 ) 1 2 + P k (s))e φ k (s,X k (s)) × d ds X k (s)+(t-s) X k (t) ψ(u) du ds dp dx = R R f 0 (x, p)e -ϕ 0 (x) I 6 (t, x, p) dp dx. ( 67
)
After integration by parts with respect to the variable s one gets

I 6 (t, x, p) = 2 k=1 (-1) k ((1 + p 2 ) 1 2 + p)e ϕ 0 (x) X k (t) x+t ψ(u)du ( 68
) - t 0 2 k=1 (-1) k d ds ((1 + |P k (s)| 2 ) 1 2 + P k (s))e φ k (s,X k (s)) X k (s)+(t-s) X k (t)
ψ(u)du ds.

By direct computations we obtain

d ds (1 + |P k (s)| 2 ) 1 2 + P k (s) e φ k (s,X k (s)) = e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 (∂ t φ k -∂ x φ k ). ( 69
)
We can write

2 k=1 (-1) k d ds (1 + |P k (s)| 2 ) 1 2 + P k (s) e φ k (s,X k (s)) X k (s)+t-s X k (t) ψ(u)du = 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 (∂ t φ k -∂ x φ k ) X k (s)+t-s X k (t) ψ(u)du ≤ ψ L 1 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 (∂ t φ k -∂ x φ k ) + C X 2 (s)+t-s X 1 (s)+t-s ψ(u)du + C X 2 (t) X 1 (t) ψ(u)du . ( 70
)
By ( 67), ( 68), (70) one gets

|I 4 | ≤ C R R f 0 (x, p)(1 + |p|) X 2 (t) X 1 (t)
ψ(u)du dp dx

+ C ψ L 1 R R f 0 t 0 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 (∂ t φ k -∂ x φ k ) ds dp dx + C R R f 0 t 0 X 2 (s)+t-s X 1 (s)+t-s ψ(u)du ds dp dx = T 1 + T 2 + T 3 . ( 71
)
We estimate now one by one the three terms (T i ) 1≤i≤3 . It is easily seen, by using [START_REF] Glassey | On the 'one and one-half dimensional' relativistic Vlasov-Maxwell system[END_REF], that 1 + |p| ≤ C(1 + |P 1 (t)|). By performing similar computations as those in (62) and by using Proposition 2.2, Lemma 3.1, one gets

T 1 ≤ C R R f 0 (x, p)(1 + |P 1 (t)|) R |ψ(y)|1 {|y-X 1 (t)|≤|X 1 (t)-X 2 (t)|} dy dp dx ≤ C R |ψ(y)| R R f 1 (t, X 1 (t), P 1 (t))e ϕ 0 (x)-φ 1 (t,X 1 (t)) (1 + |P 1 (t)|) × 1 {|y-X 1 (t)|≤C t 0 φ 1 (s)-φ 2 (s) 1,∞ ds} det J 1 (t) dp dx dy ≤ 2e ϕ 0 ∞+ φ 1 ∞ C 2 R f 1 (t, •, p)(1 + |p|) dp ∞ ψ L 1 t 0 φ 1 (s) -φ 2 (s) 1,∞ ds ≤ C ψ L 1 t 0 φ 1 (s) -φ 2 (s) 1,∞ ds, ( 72
)
for some constant C depending only on the initial conditions and T . By using now Remark 3.2 and Corollary 2.1 we have

T 2 ≤ C ψ L 1 R R f 0 (x, p) t 0 { φ 1 (s) -φ 2 (s) 1,∞ + ∂ t φ 1 (s) -∂ t φ 2 (s) ∞ + s 0 φ 1 (τ ) -φ 2 (τ ) 1,∞ dτ }ds dp dx ≤ C ψ L 1 f 0 L 1 t 0 { φ 1 (s) -φ 2 (s) 1,∞ + ∂ t φ 1 (s) -∂ t φ 2 (s) ∞ }ds. ( 73
)
The term T 3 can be estimated as before (see ( 61), ( 62), ( 63)). We obtain

T 3 ≤ C ψ L 1 t 0 φ 1 (s) -φ 2 (s) 1,∞ ds. ( 74
)
Combining ( 71), ( 72), ( 73), (74) we deduce

|I 4 | ≤ C ψ L 1 t 0 { φ 1 (s) -φ 2 (s) 1,∞ + ∂ t φ 1 (s) -∂ t φ 2 (s) ∞ }ds,
for any t ∈ [0, T ], ψ ∈ L 1 and for some constant C depending only on the initial conditions and T . A similar inequality holds for I 5 and thus (66) implies

R (∂ x φ1 (t, x) -∂ x φ2 (t, x))ψ(x) dx ≤ C ψ L 1 t 0 { φ 1 (s) -φ 2 (s) 1,∞ + ∂ t φ 1 (s) -∂ t φ 2 (s) ∞ }ds, ( 75
)
saying that for any t ∈ [0, T ] and any φ 1 , φ 2 ∈ D 4 we have

∂ x Fφ 1 (t) -∂ x Fφ 2 (t) ∞ ≤ C t 0 { φ 1 (s) -φ 2 (s) 1,∞ + ∂ t φ 1 (s) -∂ t φ 2 (s) ∞ }ds. (76) 3.6 Estimate of ∂ t Fφ 1 -∂ t Fφ 2
By [START_REF] Rein | Selfgravitating Systems in Newtonian theory -the Vlasov-Poisson system[END_REF] we know that for any φ 1 , φ 2 ∈ D 4 we have

∂ t ( φ1 -φ2 )(t, x) = - 1 2 t 0 {(µ 1 -µ 2 )(s, x -(t -s)) + (µ 1 -µ 2 )(s, x + (t -s))}ds.
With the notations of the previous paragraph we obtain for any function

ψ ∈ L 1 (R) R (∂ t φ1 (t, x) -∂ t φ2 (t, x))ψ(x) dx = - 1 2 I 4 (t) - 1 2 I 5 (t).
By the computations of paragraph 3.5 one gets for any

ψ ∈ L 1 (R), φ 1 , φ 2 ∈ D 4 R ∂ t ( φ1 -φ2 )ψ(x) dx ≤ C ψ L 1 t 0 { φ 1 (s) -φ 2 (s) 1,∞ + ∂ t φ 1 (s) -∂ t φ 2 (s) ∞ }ds, implying that ∂ t Fφ 1 (t)-∂ t Fφ 2 (t) ∞ ≤ C t 0 { φ 1 (s)-φ 2 (s) 1,∞ + ∂ t φ 1 (s)-∂ t φ 2 (s) ∞ }ds. ( 77
)
4 Existence and uniqueness for the one dimensional Nordström-Vlasov equations

Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2
), f 0 ≥ 0 and that ( 21), ( 44) hold. Take an arbitrary T > 0. The application F satisfies F(D 4 ) ⊂ D 4 where the set D 4 contains all functions φ such that

φ ∈ C 1 ([0, T ] × R), φ(0, •) = ϕ 0 , ∂ t φ(0, •) = ϕ 1 , -ϕ 0 ∞ -T ϕ 1 ∞ - T 2 e 2 ϕ 0 ∞ +T ϕ 1 ∞ f 0 L 1 ≤ φ(t, x) ≤ ϕ 0 ∞ + T ϕ 1 ∞ , max{ ∂ x φ(t) ∞ , ∂ t φ(t) ∞ } ≤ ( ϕ 0 ∞ + ϕ 1 ∞ + C 1 T )e C 2 T t , ∀ t ∈ [0, T ], max{ ∂ 2 x φ(t) ∞ , ∂ 2 xt φ(t) ∞ } ≤ C 5 , ∂ 2 t φ ∞ ≤ C 6 , ∀ t ∈ [0, T ],
where the constants C 1 , C 2 are defined in Proposition 3.1 and the constants C 5 , C 6 are defined in (53), (54). For any function u ∈ W 2,∞ (]0, T [×R) we introduce the 65), ( 76),

notation |||u(t)||| := u(t) ∞ + ∂ x u(t) ∞ + ∂ t u(t) ∞ , ∀ t ∈ [0, T ]. From (
(77) we know that there is a constant C depending only on the initial conditions and T such that

|||Fφ 1 (t) -F φ 2 (t)||| ≤ C t 0 |||φ 1 (s) -φ 2 (s)||| ds, ∀ φ 1 , φ 2 ∈ D 4 . ( 78
)
We have the uniqueness result

Proposition 4.1 Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 1 ∈ W 1,∞ (R), f 0 ∈ L 1 (R 2 ), f 0 ≥ 0
and that (21), (44) hold. Then there is at most one mild solution (f, φ) (i.e., φ ∈

C 1 ([0, T ] × R), ∂ t φ, ∂ x φ ∈ L ∞ (]0, T [; W 1,∞ (R)) and f is solution by characteristics)
for the Nordström-Vlasov system.

Proof. Consider two mild solutions (f k , φ k ) k∈{1,2} for the Nordström-Vlasov system, which means that (φ k ) k∈{1,2} are fixed points for F. Repeating the computations of paragraphs 3.1, 3.2, 3.3 one gets that (φ k ) k∈{1,2} ⊂ D 4 and therefore (78) implies

|||φ 1 (t) -φ 2 (t)||| = |||Fφ 1 (t) -F φ 2 (t)||| ≤ C t 0 |||φ 1 (s) -φ 2 (s)|||ds, ∀ t ∈ [0, T ].
We conclude by Gronwall lemma.

We prove now the existence of mild solution for the Nordström-Vlasov system.

Proof. (of Theorem 1.1) Consider the sequence (φ n ) n≥1 ⊂ D 4 defined by

φ 0 (t, x) = 1 2 {ϕ 0 (x + t) + ϕ 0 (x -t)} + 1 2 x+t x-t ϕ 1 (y) dy, (t, x) ∈ [0, T ] × R,
and φ n+1 = Fφ n , ∀ n ≥ 0. By using (78) we deduce easily that (φ n ) n≥1 converges in

C 1 ([0, T ] × R) towards a fixed point of F. Actually φ ∈ D 4 .
Denote by f the mild solution of ( 5), ( 6) corresponding to φ and let

µ(t, x) = R f (t, x, p)(1 + p 2 ) -1 2 dp, (t, x) ∈ [0, T ] × R.
By the definition of the application F we deduce easily that (f, φ) solves the Nordström-Vlasov system.

In the following we show that the global solution constructed above preserves the total energy. We introduce the notations

ρ(t, x) := e -φ(t,x) R f (t, x, p) dp, j(t, x) := e -φ(t,x) R v(p)f (t, x, p) dp, e(t, x) := R (1 + p 2 ) 1 2 f (t, x, p) dp + 1 2 |∂ x φ(t, x)| 2 + 1 2 |∂ t φ(t, x)| 2 , π(t, x) := R pf (t, x, p) dp -∂ x φ(t, x)∂ t φ(t, x). Proposition 4.2 Assume that ϕ 0 ∈ W 2,∞ (R), ϕ 0 ∈ L 2 (R), ϕ 1 ∈ W 1,∞ (R) ∩ L 2 (R), (1 + |p|)f 0 ∈ L 1 (R 2 )
and that (21), (44) hold. Then the solution of the one dimensional Nordström-Vlasov system satisfies

∂ t ρ + ∂ x j = 0, in D ([0, +∞[×R), (79)
∂ t e + ∂ x π = 0, in D ([0, +∞[×R). ( 80
)
In particular the total mass and energy are preserved for any t > 0

R ρ(t, x) dx = R ρ(0, x) dx, R e(t, x) dx = R e(0, x) dx.
Proof. Note that by Theorem 1.1 we know that R (1+|p|)f (•, •, p) dp is bounded on [0, T ]×R, ∀ T > 0 and therefore ρ, j, e and π are well defined for (t, x) ∈ [0, +∞[×R.

Let us prove now the continuity equation. For any θ ∈ C 1 c ([0, +∞[×R) we have by Proposition 2.1

+∞ 0 R {ρ(t, x)∂ t θ + j(t, x)∂ x θ} dx dt = R R f 0 e -ϕ 0 (x) +∞ 0 d dt θ(t, X(t)) dt dp dx = - R ρ(0, x)θ(0, x) dx,
and thus ∂ t ρ + ∂ x j = 0 holds in D ([0, +∞[×R). The conservation of the total mass has been already checked. In order to prove (80) we intend to apply the weak formulation [START_REF] Degond | Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity[END_REF] with the test function

ψ(t, x, p) = e φ(t,x) θ(t, x)(1 + p 2 ) 1 2 , for any θ ∈ C 1 c ([0, +∞[×R).
Actually ψ is not a C 1 b function but we can apply first the weak formulation with ψ R = ψ χ R (p) and then let R → +∞. We skip these standard arguments. We find that

- R R (1 + p 2 ) 1 2 θ(0, x)f 0 (x, p) dp dx = +∞ 0 R R (1 + p 2 ) 1 2 (∂ t θ + v(p)∂ x θ)f dp dx dt + +∞ 0 R R {(1 + p 2 ) 1 2 (∂ t φ + v(p)∂ x φ) -v(p)(pSφ + (1 + p 2 ) -1 2 ∂ x φ)}θf dp dx dt = +∞ 0 R R ((1 + p 2 ) 1 2 ∂ t θ + p∂ x θ)f dp dx dt + +∞ 0 R µ(t, x)θ(t, x)∂ t φ dx dt. ( 81
)
By multiplying the wave equation by θ(t, x)∂ t φ one gets

+∞ 0 R θ 1 2 ∂ t |∂ t φ| 2 -∂ 2 x φ ∂ t φ dx dt + +∞ 0 R µ(t, x)θ(t, x)∂ t φ dx dt = 0.
After integration by parts we obtain

- 1 2 +∞ 0 R {|∂ t φ| 2 + |∂ x φ| 2 }∂ t θ dx dt - 1 2 R {|ϕ 1 (x)| 2 + |ϕ 0 (x)| 2 }θ(0, x) dx + +∞ 0 R ∂ x φ∂ t φ∂ x θ dx dt + +∞ 0 R µ(t, x)θ(t, x)∂ t φ dx dt = 0. (82)
Combining ( 81), (82) yields

- +∞ 0 R {e(t, x)∂ t θ + π(t, x)∂ x θ} dx dt - R e(0, x)θ(0, x) dx = 0,
and thus (80) holds. In order to prove that the total energy is conserved, observe

that ∂ t φ, ∂ x φ ∈ L ∞ (]0, T [; L 2 (R)), ∀ T > 0. Indeed, since ϕ 0 , ϕ 1 ∈ L 2 (R)
, by using the representation formula for ∂ x φ, ∂ t φ (see [START_REF] Rendall | An introduction to the Einstein-Vlasov system[END_REF], ( 19)) it is sufficient to check that

D ± (t, x) = t 0 µ(s, x ± (t -s)) ds belong to L 2 (R) for any t ∈ [0, T ], T > 0.
This is obvious since we have for any t ∈ [0, T ]

D ± (t) ∞ ≤ t R f (•, •, p) dp L ∞ (]0,T [×R) , D ± (t) L 1≤ t R R f (•, x, p) dp dx L ∞ (]0,T [) . In particular ∂ x φ ∂ t φ ∈ L ∞ (]0, T [; L 1 (R)), T > 0. By Proposition 2.1 we know also that (1 + p 2 ) 1 2 f ∈ L ∞ (]0, T [; L 1 (R 2 )) for any T > 0. Take now η ∈ C 1 c ([0, +∞[) and apply (80) with θ(t, x) = η(t)χ R (x), R > 0. We obtain - +∞ 0 R e(t, x)η χ R dx dt - +∞ 0 R π(t, x) η(t) R χ x R dx dt - R e(0, x)η(0)χ R dx = 0.
After letting R → +∞ one gets -+∞ 0 dη dt R e(t, x) dx dt -η(0) R e(0, x) dx = 0 and thus R e(t, x) dx = R e(0, x) dx, ∀ t ≥ 0.

Finite speed propagation

Assume that (f k 0 , ϕ k 0 , ϕ k 1 ) k∈{1,2} satisfy the hypotheses of Theorem 1.2 and denote by (f k , φ k ) k∈{1,2} the global solutions of the one dimensional Nordström-Vlasov system.

We intend to estimate (φ 1 -φ 2 ) with respect to (f

1 0 -f 2 0 , ϕ 1 0 -ϕ 2 0 , ϕ 1 1 -ϕ 2 1
). In particular we deduce that the solution of the Nordström-Vlasov system propagates with finite speed, which coincides with the waves speed in [START_REF] Bostan | Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem[END_REF]. Surely, the key point here is that we are dealing with relativistic particles and therefore the characteristics

0 W 2,∞ (R) , ϕ k 1 W 1,∞ (R) , f k 0 L 1 (R 2 ) , 2 m=0 |p| m g k 0 L 1 (R) , k ∈ {1, 2}. As in formula (55) one gets R (φ 1 (t, y) -φ 2 (t, y))ψ dy ≤ ( ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + R ϕ 1 1 -ϕ 2 1 L ∞ ([-R,R]) ) ψ L 1 + 1 2 |I 7 (t)|, (83)
where

I 7 (t) = t 0 R R (f 1 -f 2 )(s, x, p) θ(s, x) (1 + p 2 ) 1 2
dp dx ds, θ(s, x) = x+(t-s)

x-(t-s) ψ(y) dy.

By Proposition 2.1 one have

t 0 R R f k (s, x, p) θ(s, x) (1 + p 2 ) 1 2 dp dxds = R R f k 0 (x, p)e -ϕ k 0 (x) t 0 θ(s, X k (s)) (1 + |P k (s)| 2 ) 1 2
× e φ k (s,X k (s)) ds dp dx, where (X k , P k ) are the characteristics corresponding to φ k , k ∈ {1, 2}. Observe that for any x > R and s ∈ [0, t] we have X k (s; 0, x, p) -(t -s) > R -t implying that θ(s, X k (s)) = 0. Similarly, for any x < -R and s ∈ [0, t] we have X k (s; 0, x, p) + (t -s) < -R + t implying that θ(s, X k (s)) = 0. Therefore we have θ(s, X k (s)) = θ(s, X k (s)) 1 {|x|≤R} and thus we obtain

|I 7 (t)| ≤ C R R -R R 2 k=1 (-1) k f k 0 (x, p)e -ϕ k 0 (x) dp dx ψ L 1 + R -R R f 1 0 (x, p)e -ϕ 1 0 (x) t 0 2 k=1 (-1) k θ(s, X k (s)) (1 + |P k (s)| 2 )
1 2

e φ k (s,X k (s)) ds dp dx

= |I 8 | + |I 9 (t)|. ( 84
) For any (x, p) ∈ [-R, R] × R, k ∈ {1, 2} we define s k (t, x, p) = sup{s ∈ [0, t] : |X k (s; 0, x, p)| ≤ R -s}.
Observing

(s)) = 0, k ∈ {1, 2}. Indeed, assuming that X k (s) > R -s implies X k (s) -(t -s) > R -t and if X k (s) < -(R -s) implies X k (s) + (t -s) < -(R -t). Therefore in both cases [X k (s) -(t -s), X k (s) + (t -s)] ∩ supp ψ = ∅ and thus θ(s, X k (s)) = 0, k ∈ {1, 2}.
By the above considerations and the properties of characteristics satisfying (102)

(see the comments in the Appendix) we obtain as in (59)

|I 9 (t)| = R -R R f 1 0 e -ϕ 1 0 s(t,x,p) 0 2 k=1 (-1) k θ(s, X k (s)) (1 + |P k (s)| 2 )
1 2

e φ k (s,X k (s)) ds dp dx

≤ C R ψ L 1 R -R R f 1 0 e -ϕ 1 0 s(t,x,p) 0 ( φ 1 (s) -φ 2 (s) L ∞ ([-(R-s),R-s]) + s 0 φ 1 (τ ) -φ 2 (τ ) W 1,∞ ([-(R-τ ),R-τ ]) dτ + |p(ϕ 1 0 -ϕ 2 0 )(x)|) ds dp dx + C R R -R R f 1 0 e -ϕ 1 0 s(t,x,p) 0 X 2 (s)-(t-s) X 1 (s)-(t-s)
ψ(y) dy ds dp dx We introduce the notations D R (s

+ C R R -R R f 1 0 e -ϕ
) = φ 1 (s)-φ 2 (s) L ∞ ([-(R-s),R-s]) , D R α (s) = ∂ α φ 1 (s)- ∂ α φ 2 (s) L ∞ ([-(R-s),R-s]) , α ∈ {x, t}.
The term I 10 (t) can be estimated as follows

I 10 (t) ≤ C R ψ L 1 ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (s) + D R x (s)} ds . ( 86
)
For the term I 11 (t) we proceed as in (61), (62)

I 11 (t)≤ C R R |ψ| R -R R f 1 0 s(t,x,p) 0 1 {|y-(X 1 (s)-(t-s))|≤C R (|p| ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + s 0 (D R +D R x )dτ )} ds dp dx dy ≤ C R R |ψ(y)| t 0 R R f 1 (
s, X, P )r(s, y, X, P ) dP dX ds dy, where r(s, y, X, P

) = 1 {|y-(X-(t-s))|≤C R ((1+|P |) ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + s 0 (D R (τ )+D R
x (τ ))dτ )} . Observe that for any (s, y, P ) ∈ [0, t] × R 2 we have R r(s, y, X, P

) dX ≤ α|P | + β with α = 2C R ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) , β = 2C R ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (τ ) + D R x (τ )} dτ ,
and therefore, by Lemma 3.2

I 11 (t) ≤ C R ψ L 1 ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (s) + D R x (s)} ds . (87)
A similar estimate holds for I 12 (t) and finally collecting (83), ( 84), ( 85), ( 86), (87

) implies D R (t) ≤ C R ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + ϕ 1 1 -ϕ 2 1 L ∞ ([-R,R]) + R -R R |f 1 0 -f 2 0 | dp dx + C R t 0 {D R (s) + D R x (s)} ds. ( 88
)
The next step is to estimate D R x (t). As in (66) we have

R (∂ x φ 1 -∂ x φ 2 )ψ dy ≤ ψ L 1 ( ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + ϕ 1 1 -ϕ 2 1 L ∞ ([-R,R]) ) + 1 2 |I 13 (t)| + 1 2 |I 14 (t)|, (89)
where

I 13 (t) = t 0 R R (f 1 -f 2 )(s, x -(t -s), p) ψ(x) (1 + p 2 ) 1 2
dp dx ds, and

I 14 (t) = t 0 R R (f 1 -f 2 )(s, x + (t -s), p) ψ(x) (1 + p 2 ) 1 2
dp dx ds.

Performing the same steps as in (67), (68), (69) one gets

I 13 (t) = 2 k=1 (-1) k R R f k 0 ((1 + p 2 ) 1 2 + p) X k (t) x+t ψ(u) du dp dx (90) - 2 k=1 (-1) k R R f k 0 e -ϕ k 0 t 0 e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 (∂ t φ k -∂ x φ k ) X k (s)+(t-s) X k (t)
ψ(u) du ds dp dx.

Notice that for any (s,

x, p) ∈ [0, t] × R 2 such that |x| > R we have X k (t) x+t ψ(u) du = X k (s)+(t-s) X k (t)
ψ(u) du = 0. Therefore in the above equality the integrations with respect to x can be restricted to [-R, R]. Observe also that for any s ∈]s k (t, x, p), t]

we have

X k (s)+(t-s) X k (t)
ψ(u) du = 0 and therefore we obtain

|I 13 (t)| ≤ C R ψ L 1 R -R R (1 + |p|)|f 1 0 -f 2 0 | + |f 1 0 e -ϕ 1 0 -f 2 0 e -ϕ 2 0 | dp dx + I 15 (t), (91)
where

I 15 (t) ≤ C R R -R R f 1 0 (x, p)(1 + |P 1 (t)|) X 2 (t) X 1 (t) ψ(u) du dp dx + C R ψ L 1 R -R R f 1 0 s(t,x,p) 0 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 (∂ t φ k -∂ x φ k ) ds dp dx + C R R -R R f 1 0 (x, p) s(t,x,p) 0 X 2 (s)+(t-s) X 1 (s)+(t-s) ψ(u) du ds dp dx = I 16 (t) + I 17 (t) + I 18 (t). ( 92
) Notice that X 2 (t) X 1 (t) ψ(u) du vanishes on ([-R, R] × R) -A(t), where A(t) = {(x, p) ∈ [-R, R] × R : ∃ λ ∈ [0, 1], |λX 1 (t) + (1 -λ)X 2 (t)| ≤ R -t} and if (x, p) ∈ A(t), then (X k , P k ) k∈{1,2}
verify the property (102) on [0, t]. Therefore for any (x, p) ∈ A(t) we can apply (106) and we obtain as in (72)

I 16 (t) ≤ C R R -R R f 1 0 (x, p)(1 + |P 1 (t)|)1 {(x,p)∈A(t)} X 2 (t) X 1 (t) ψ(u) du dp dx ≤ C R R |ψ(u)| R R
(1 + |P |)f 1 (t, X, P )r(u, X, P ) dP dX du, where r(u,

X, P ) = 1 {|X-u|≤C R ((1+|P |) ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (s)+D R
x (s)}ds)} . Applying now Lemma 3.2 yields

I 16 (t) ≤ C R ψ L 1 ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (s) + D R x (s)} ds . ( 93
)
Observe that at this step we have used the assumption R |p| 2 g k 0 (p) dp < +∞. Taking into account that (106) applies on [0, s(t, x, p)] for (X k , P k )(•; 0, x, p), k ∈ {1, 2} we deduce as in (73), (74) (use also Lemma 3.2 for estimating I 18 (t))

I 17 (t) ≤ C R ψ L 1 ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (s) + D R x (s) + D R t (s)} ds , ( 94
)
and

I 18 (t) ≤ C R ψ L 1 ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) + t 0 {D R (s) + D R x (s)} ds . ( 95
)
Collecting all the partial results (89), ( 91), ( 93), (94), (95) yields

D R x (t) ≤ C R D R 0 + t 0 {D R (s) + D R x (s) + D R t (s)} ds . ( 96
)
A similar estimate holds for D R t and therefore (88), (96) implies

D R (t) + D R x (t) + D R t (t) ≤ C R D R 0 + t 0 {D R (s) + D R x (s) + D R t (s)} ds .
The statement in Theorem 1.2 follows by Gronwall lemma.

Proof

(t, x) = φ 2 (t, x), t ∈ [0, R], |x| ≤ R-t. Observe also that for any t ∈ [0, R], x ∈ [-(R-t), R-t], p ∈ R we have |X k (s; t, x, p)| ≤ |x| + (t -s) ≤ R -s, s ∈ [0, t], k ∈ {1, 2}. By (106) one gets for any t ∈ [0, R], |x| ≤ R -t, p ∈ R (|X 1 -X 2 |+|Q 1 -Q 2 |)(0; t, x, p) ≤ C R |p(φ 1 -φ 2 )(t, x)|+ t 0 {D R (s) + D R x (s)} ds = 0,
implying that (X 1 , P 1 )(0; t, x, p) = (X 2 , P 2 )(0; t, x, p). We deduce easily that f 1 (t, x, p) = f 2 (t, x, p) for any t ∈ [0, R], |x| ≤ R -t, p ∈ R.

Appendix

We give here the details for the proofs of Proposition 2.2 and Corollary 2.1. If φ is smooth and (X(s), P (s)) is a characteristic corresponding to φ we denote by Q(s)

the quantity Q(s) = e φ(s,X(s)) P (s). Recall that we have (see [START_REF] Glassey | On the 'one and one-half dimensional' relativistic Vlasov-Maxwell system[END_REF])

dQ ds = - e φ(s,X(s)) (1 + |P (s)| 2 ) 1 2 ∂ x φ(s, X(s)), ∀s ∈ [0, T ]. ( 97
)
We start with the following easy lemma (the proof is left to the reader). ≤ e 2C(s) ( φ 1 (s) -φ 2 (s)

Lemma 6.1 Consider φ k ∈ C 1 ([0, T ]×R)∩L ∞ (]0, T [×R), ∂ x φ k , ∂ t φ k ∈ L ∞ (]0, T [; W 1,∞ (R)), k ∈ {1,
L ∞ + ∂ x φ 1 (s) L ∞ |X 1 (s) -X 2 (s)|) + e C(s) |Q 1 (s) -Q 2 (s)|, ( 100
)
|v(P 1 (s)) -v(P 2 (s))| ≤ 2e 2C(s) ( φ 1 (s) -φ 2 (s) L ∞ + ∂ x φ 1 (s) L ∞ |X 1 (s) -X 2 (s)|) + 2e C(s) |Q 1 (s) -Q 2 (s)|. (101)
At this point let us make some remarks which will be useful when studying the finite speed propagation property. Take 0 ≤ t ≤ R and assume that (X k , P k We introduce the notations

C(s) = max k∈{1,2} φ k (s) L ∞ , C(s) = max k∈{1,2} ∂ x φ k (s) L ∞ , C(s) = max k∈{1,2} ∂ 2 x φ k (s) L ∞ .
We deduce by Lemma 6.1, inequality (101)

d ds |X 1 (s) -X 2 (s)| ≤ 2e 2C(s) ( φ 1 (s) -φ 2 (s) L ∞ + ∂ x φ 1 (s) L ∞ |X 1 (s) -X 2 (s)|) + 2e C(s) |Q 1 (s) -Q 2 (s)|. ( 103
)
Using now (97) yields 1 2

d ds |Q 1 (s) -Q 2 (s)| 2 ≤ |Q 1 (s) -Q 2 (s)| 2 k=1 (-1) k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 2 ∂ x φ k (s, X k (s)) ,
and therefore we obtain (-1) k e φ k (s,X k (s)) ∂ x φ k (s, X k (s)) .

As usual we have the estimate (-1) k e φ k (t,X k (t))

(1 + |P k (t)| 2 ) 1 2 ≤ C( φ 1 (t) -φ 2 (t) L ∞ + |X 1 (t) -X 2 (t)| + |Q 1 (t) -Q 2 (t)|) ≤ C φ 1 (t) -φ 2 (t) L ∞ + t 0 φ 1 (s) -φ 2 (s) W 1,∞ ds + C|p(φ 1 -φ 2 )(0, x)|,
where C depends only on max k∈{1,2} { φ k L ∞ (]0,T [;W 2,∞ (R)) }. In order to check the second inequality we can write

| 2 k=1 (-1) k e φ k (t,X k (t)) (1 + |P k (t)| 2 ) 1 2 ∂ x φ k (t, X k (t))| ≤ ∂ x φ 1 (t) L ∞ 2 k=1 (-1) k e φ k (t,X k (t)) (1 + |P k (t)| 2 ) 1 2 + e φ 2 (t) L ∞ ( ∂ x φ 1 (t) -∂ x φ 2 (t) L ∞ + ∂ 2 x φ 2 (t) L ∞ |X 1 (t) -X 2 (t)|) ≤ C |p(φ 1 -φ 2 )(0, x)| + φ 1 (t) -φ 2 (t) W 1,∞ + t 0 φ 1 (s) -φ 2 (s) W 1,∞ ds .

Remark 3 . 1

 31 If α = 0 the conclusion of Lemma 3.2 holds true by assuming only R |p| m g 0 (p) dp < +∞. Now, under the hypotheses of Lemma 3.1, by using (43) we can find estimates for the L ∞ norms of ∂ x D ± . Indeed, for the first term in the right hand side of (43) we have for any

  propagate with finite speed |X (s)| = |v(P (s))| < 1. Let us explain how we can adapt the computations in the paragraphs 3.4, 3.5, 3.6 for the above purposes. Take 0 ≤ t ≤ R and ψ ∈ L 1 (R) compactly supported in [-(R-t), R-t]. From now on the notation C R stands for various constants depending on R, ϕ k

2 k=1(- 1 )( 1 +

 211 2} and (X k , P k ) two characteristics corresponding to φ k , k ∈ {1, 2}. We denoteC(s) = max k∈{1,2} φ k (s) L ∞ . Then we have for any s ∈ [0, T ] k e φ k (s,X k (s)) ≤ e C(s) ( φ 1 (s)-φ 2 (s) L ∞ + ∂ x φ 1 (s) L ∞ |X 1 (s)-X 2 (s)|), (98) |P 1 (s) -P 2 (s)| ≤ e C(s) |Q 2 (s)|( φ 1 (s) -φ 2 (s) L ∞ + ∂ x φ 1 (s) L ∞ |X 1 (s) -X 2 (s)|) + e C(s) |Q 1 (s) -Q 2 (s)|, |P k (s)| 2 ) 1 2

2 k=1(- 1 ) 1

 211 ) are characteristics corresponding to φ k , k ∈ {1, 2}, which satisfy the following property∀ s ∈ [0, t], ∃ λ(s) ∈ [0, 1] : |λ(s)X 1 (s) + (1 -λ(s))X 2 (s)| ≤ R -s, (102)saying that at any time s ∈ [0, t] the segment between X 1 (s), X 2 (s) has non voidintersection with [-(R -s), R -s].Then all the statements in Lemma 6.1 holdwith φ 1 (s) -φ 2 (s) L ∞ replaced by φ 1 (s) -φ 2 (s) L ∞ ([-(R-s),R-s]) and ∂ x φ 1 (s) L ∞ replaced by max k∈{1,2} ∂ x φ k (s) L ∞ .Indeed, let us check this for formula (98), the same applying for the other. Denote byY (s) the number of [-(R -s), R -s] such that Y (s) = λ(s)X 1 (s) + (1 -λ(s))X 2 (s), λ(s) ∈ [0, 1]. We have k e φ k (s,X k (s)) ≤ e C(s) |φ 1 (s, X 1 (s)) -φ 2 (s, X 2 (s))| ≤ e C(s) {|φ 1 (s, Y (s)) -φ 2 (s, Y (s))| + 2 k=1 |φ k (s, Y (s)) -φ k (s, X k (s))|} ≤ e C(s) φ 1 (s) -φ 2 (s) L ∞ ([-(R-s),R-s]) + max k∈{1,2} ∂ x φ k (s) L ∞ |X 1 (s) -X 2 (s)| .Proof. (of Proposition 2.2) We have for any s ∈ [0, (s) -X 2 (s)| 2 = (v(P 1 (s)) -v(P 2 (s)))(X 1 (s) -X 2 (s))≤ |X 1 (s) -X 2 (s)||v(P 1 (s)) -v(P 2 (s))|.

d ds |Q 1 2 k=1(- 1 ) 2 ∂( 1 +

 12121 (s) -Q 2 (s)| ≤ k e φ k (s,X k (s)) (1 + |P k (s)| 2 ) 1 x φ k (s, X k (s)) , (104) ≤ e C(s) C(s) |P k (s)| 2 )

2 k=1(- 1 ) 2 k=1(- 1 ) 2 k=1(- 1 ) 0 {|X 1 0 { φ 1 0 φ 1 φ 1

 2121210101011 k e φ k ∂ x φ k ≤ C(s) k e φ k (s,X k (s)) + e C(s) k ∂ x φ k (s, X k (s)) ≤ e C(s) C(s){ φ 1 (s) -φ 2 (s) L ∞ + C(s)|X 1 (s) -X 2 (s)|} + e C(s) { ∂ x φ 1 (s) -∂ x φ 2 (s) L ∞ + C(s)|X 1 (s) -X 2 (s)|}. (105)Combining (100), (103), (104), (105) yields for any 0≤ s ≤ t ≤ T d ds {|X 1 (s) -X 2 (s)| + |Q 1 (s) -Q 2 (s)|} ≤ C{|X 1 (s) -X 2 (s)| + |Q 1 (s) -Q 2 (s)|} + C{ φ 1 (s) -φ 2 (s) L ∞ + ∂ x φ 1 (s) -∂ x φ 2 (s) L ∞ },for some constant depending on sup s∈[0,t] {C(s) + C(s) + C(s)}. We deduce that for any 0 ≤ s ≤ t ≤ T we have|X 1 (s) -X 2 (s)| + |Q 1 (s) -Q 2 (s)| ≤ C s (τ ) -X 2 (τ )| + |Q 1 (τ ) -Q 2 (τ )|}dτ + C|p(φ 1 -φ 2 )(0, x)| + C t (τ ) -φ 2 (τ ) L ∞ + ∂ x φ 1 (τ ) -∂ x φ 2 (τ ) L ∞ }dτ.By Gronwall lemma we obtain for any 0≤ s ≤ t ≤ T |X 1 -X 2 |(s) + |Q 1 -Q 2 |(s) ≤ Ce Cs |p(φ 1 -φ 2 )(0, x)| + t (τ ) -φ 2 (τ ) W 1,∞ dτ .Notice that if (X k , P k ) k∈{1,2} satisfy the property (102), then in formula (105) wecan replace φ 1 (s) -φ 2 (s) L ∞ by φ 1 (s) -φ 2 (s) L ∞ ([-(R-s),R-s]) and ∂ x φ 1 (s) -∂ x φ 2 (s) L ∞ by ∂ x φ 1 (s) -∂ x φ 2 (s) L ∞ ([-(R-s),R-s]). Finally one gets for such characteristics|X 1 (s) -X 2 (s)| + |Q 1 (s) -Q 2 (s)| ≤ Ce Cs |p(φ 1 -φ 2 )(0, x)| (τ ) -φ 2 (τ ) W 1,∞ ([-(R-τ ),R-τ ]) dτ.Proof. (of Corollary 2.1) Using (98), (100) and Proposition 2.2 yields for any t ∈ [0, T ] 2 k=1

  (•; 0, x, p), P k (•; 0, x, p)), k ∈ {1, 2} satisfy the property (102) on [0, s(t, x, p)],where s(t, x, p) = max k∈{1,2} s k (t, x, p). If s ∈]s(t, x, p), t] we claim that θ(s, X k

that the function s → |X k (s; 0, x, p)| -(R -s) is strictly increasing we deduce that |X k (s; 0, x, p)| ≤ R -s, s ∈ [0, s k (t, x, p)] and |X k (s; 0, x, p)| > R -s, s ∈]s k (t, x, p), t]. Notice that for any (x, p) ∈ [-R, R] × R the characteristics (X k

  . (of Corollary 1.1) Notice that under the hypotheses of Corollary 1.1 we can apply Theorem 1.2. Indeed, the only step in the proof of Theorem 1.2 where we need the additional hypothesis R |p| 2 g k 0 (p) dp < +∞ is in (93). But in our case ϕ 1 0 -ϕ 2 0 L ∞ ([-R,R]) = 0 and thus, by Remark 3.1, the inequality (93) still holds true by assuming only R |p|g k 0 (p) dp < +∞. We deduce that φ 1

The third inequality follows in similar manner. As before it is easily seen that, for characteristics satisfying (102), all the statements of Corollary 2.1 hold with φ 1 (s)-


Tuija Tapaninen

Pertti J Neuvonen

MD Mikko Niemi
email: mikko.niemi@helsinki.fi

Effect of ABCB1 haplotypes on the pharmacokinetics and renin-inhibiting effect of aliskiren

Keywords: Pharmacokinetics, Pharmacogenetics, ABCB1 haplotypes, Aliskiren

come

Effect of haplotypes on the pharmacokinetics and renin-inhibiting effect of aliskiren

INTRODUCTION

The direct renin inhibitor aliskiren is poorly absorbed and has a low oral bioavailability of 2-3% [1]. Aliskiren is mainly eliminated by biliary excretion [1]. In addition, about 0.4% of the oral dose is excreted unchanged into urine and about 1.4% undergoes oxidative biotransformation via CYP3A4 [1]. Aliskiren is a substrate of P-glycoprotein (also known as ABCB1) [2], an efflux transporter expressed in tissues with an excretory or barrier function [3,4]. The P-glycoprotein and CYP3A4 inhibitors, itraconazole and cyclosporine [5][6][7][8][9], have raised the area under the plasma concentration-time curve (AUC) of aliskiren by about 6.5-fold and 5-fold, respectively [10,11], supporting a major role of P-glycoprotein in limiting the intestinal absorption of aliskiren. Moreover, rifampicin, an inducer of P-glycoprotein and CYP3A4 [12,13], has reduced the AUC of aliskiren by 56% [14].

A common synonymous single nucleotide polymorphism (SNP) in the ABCB1 gene encoding P-glycoprotein, c.3435C>T, has been variably associated with reduced expression and function of P-glycoprotein in vitro and increased plasma concentrations of P-glycoprotein substrate drugs in vivo [15][16][17]. The mechanism underlying these associations is not completely understood, but it has been suggested that c.3435C>T affects ABCB1 mRNA stability [START_REF] Rao | Direct demonstration of high affinity interactions of immunosuppressant drugs with the drug binding site of the human P-glycoprotein[END_REF]. Alternatively, the associations may be explained by linkage disequilibrium between c.3435C>T and a functional sequence variation, or that c.3435C>T and sequence variations in linkage disequilibrium with it form a haplotype conferring altered P-glycoprotein activity [16,19]. The ABCB1 c.3435C>T SNP is in strong linkage disequilibrium with the c.1236C>T (synonymous) and c.2677G>T/A (p.Ala893Ser/Thr) SNPs within the same gene [17,19,20].

The variant alleles c.1236T, c.2677T, and c.3435T are common in Europeans and European Americans, with reported frequencies of occurrence ranging from 39 to 46%, 39 to 47%, and 54 to 60%, respectively [17,[19][20][21][22]. In the Finnish population, the two most common haplotypes based on these SNPs are c.1236T-c.2677T-c.3435T (TTT) and c.1236C-c.2677G-c.3435C (CGC), with frequencies of 42.7% and 34.4%, respectively [17]. The ABCB1 TTT haplotype has been associated, for example, with increased plasma concentrations of and enhanced lipid-lowering response to simvastatin (acid) and atorvastatin, compared with the ABCB1 CGC haplotype [17,23,24].

Considerable interindividual variability exists in the pharmacokinetics of aliskiren [25], but the background of this variability is unknown. Our hypothesis was that genetic variability in ABCB1 affects the pharmacokinetics of aliskiren. Therefore, we investigated the effects of the ABCB1 CGC and TTT haplotypes on the pharmacokinetics and pharmacodynamics of aliskiren.

METHODS

Subjects

Twenty-two healthy White Finnish volunteers participated in this study after giving written informed consent. They were recruited from a group of more than 700 subjects genotyped for the ABCB1 c.1236C>T (rs1128503), c.2677G>T/A (rs2032582), and c.3435C>T (rs1045642) SNPs [17]. To minimize variability in aliskiren pharmacokinetics due to other genetic variants, all subjects were genotyped for the functionally significant ABCB1 c.1199G>A (p.Ser400Asn; rs2229109) [17] and CYP3A5*3 (g.6986A>G; rs776746) [26] alleles and only noncarriers of the ABCB1 c.1199A and CYP3A5 g.6986A (CYP3A5 expressor) alleles were recruited.

Four women and seven men homozygous for the ABCB1 CGC haplotype, mean ± SD age 26 ± 4 years (range, 22-37 years), height 178 ± 9 cm (158-189 cm), and weight 74 ± 11 kg (63-95 kg), and five women and six men homozygous for the TTT haplotype, age 24 ± 2 years (21-27 years), height 175 ± 10 cm (158-186 cm), and weight 69 ± 12 kg (51-94 kg), participated in the study. Each participant's health was ascertained by medical history, clinical examination, and laboratory tests. None was on any continuous medication and none was a tobacco smoker.

Study design

The study protocol was approved by the Coordinating Ethics Committee of the Helsinki and Uusimaa Hospital District, and the National Agency for Medicines. Following an overnight fast, the subjects ingested a single 150-mg dose of aliskiren (Rasilez; Novartis, Horsham, UK) with 150 ml water. A standardized warm meal was served 4 h after aliskiren, and a standardized light meal after 7 h and 10 h. As a safety assessment, systolic and diastolic blood pressures were measured from the forearm with an automatic oscillometric blood pressure monitor (Omron M5-I, Omron Healthcare Europe BV, Hoofddorp, The Netherlands), with the participant in a sitting position, prior to and 2, 4, 7, 9, 12, and 24 h after aliskiren ingestion. The participants were under direct medical supervision for 12 h after the administration of aliskiren. No significant differences in blood pressures were observed between the haplotype groups (data not shown). Use of other drugs was prohibited for one week and use of grapefruit products for three days before aliskiren administration and 72 h thereafter. Timed blood samples (5 or 10 ml each) for aliskiren concentration measurements were drawn into tubes containing ethylenediaminetetraacetic acid (EDTA) prior to and 0.5, 1, 2, 3, 4, 5, 7, 9, 12, 24, 34, 48, and 72 h after aliskiren administration. Blood samples for the determination of plasma renin activity (5 ml each) were drawn before the administration of aliskiren and 4 h and 24 h thereafter into chilled EDTA tubes, which were placed on ice immediately after sampling. Plasma was separated within 30 min.

Urine was collected up to 12 h after aliskiren ingestion. Urine aliquots and plasma were stored at -70ºC until analysis.

Determination of aliskiren concentrations and renin activity

Plasma concentrations of aliskiren were quantified by use of solid phase extraction and liquid chromatography-tandem mass spectrometry (LC/MS/MS) [10]. An aliquot of 700 µl of each plasma sample was mixed with 100 µl of 10% phosphoric acid and 50 µl internal standard solution (1000 ng/ml acebutolol in 1:1 methanol:water). Then the mixture was loaded to the extraction cartridge (Oasis MCX 1cc/30 mg, Waters, Milford, MA, USA), which had been pre-conditioned with 1 ml of methanol and 1 ml of 1% acetic acid. The cartridge was subsequently washed with 1 ml of methanol and 1 ml of 1% acetic acid. The analytes were eluted out with two times 0.6 ml of 2% ammonium hydroxide in methanol. The eluent was dried at 50°C under a nitrogen stream. The residue was dissolved in 100 µl of the mobile phase, and 15 µl of the sample was injected to the LC/MS/MS system. The chromatographic separation was carried out on an AtlantisT3 column (2.1×100 mm, 3 µm) protected by an AtlantisT3 (2.1×10 mm, 3 µm) guard cartridge (Waters). The mobile phase consisted of (A) 2 mmol/l ammonium acetate (pH 4.90) and (B) methanol. The total running time was 15 minutes with the mobile phase gradient of 0.5 minutes at 70% A, 0.5 minutes to 5% A, 3 minutes at 5% A, 0.1 minutes to 70% A, and 10.9 minutes at 70% A. The flow rate of the mobile phase was 250 µl/min. The column temperature was 40°C. Mass spectra were obtained using an Applied Biosystems SCIEX API 2000 Q Trap mass spectrometer operated in positive ion mode (Sciex Division of MDS Inc, Toronto, ON, Canada). The ion transitions were mass-to-charge ratio (m/z) 552.4 to m/z 436.3 for aliskiren and m/z 337.3 to m/z 116.3 for acebutolol.

For the determination of aliskiren concentrations in urine, an aliquot of 300 µl of each urine sample was mixed with 300 µl of fresh frozen plasma, 100 µl of 10% phosphoric acid, and 50 µl internal standard solution (1000 ng/ml acebutolol in 1:1 methanol:water). The extraction, chromatographic separation, and mass spectrometric detection were performed as described for plasma aliskiren. The lower limit of A revised manuscript for European Journal of Clinical Pharmacology, April 2010 8 quantification was 0.25 ng/ml for plasma and 9 ng/ml for urine aliskiren. The calibration curve for plasma aliskiren was linear over the range 0.25-500 ng/ml (r>0.999, weighting 1/x) and for urine aliskiren over the range 9-1800 ng/ml (r>0.999, weighting 1/x). The extraction recoveries were > 80%. The intra-day coefficients of variation were ≤ 1.2% at relevant concentrations in plasma and ≤ 4.8% in urine (n=6).

Plasma renin activity was determined by radioimmunoassay of generated angiotensin I with a commercially available method (RENCTK; DiaSorin, Saluggia, Italy) at HUSLAB laboratory (Helsinki, Finland).

Pharmacokinetics and pharmacodynamics

Aliskiren pharmacokinetics were characterized by peak plasma concentration (C max ), time to C max (t max ), elimination half-life (t ½ ), AUC 0-72 , AUC 0-∞ , the amount of aliskiren excreted into urine from 0 to 12 h (Ae), and the renal clearance (Cl renal ).

Pharmacokinetic parameters were calculated with non-compartmental methods using MK-Model, version 5.0 (Biosoft, Cambridge, UK). The terminal log-linear part of each concentration-time curve was identified visually. The elimination rate constant (k e ) was determined by linear regression analysis of the log-linear part of the plasma drug concentration-time curve. The t ½ was calculated by the equation t ½ =ln2/k e . The AUC values were calculated by a combination of the linear and log-linear trapezoidal rules with extrapolation to infinity, when appropriate, by division of the last measured concentration by k e . The Cl renal of aliskiren was calculated by the equation Cl renal =Ae/AUC 0-12 . Aliskiren pharmacodynamics were characterized by plasma renin activity at 4 h and 24 h after aliskiren intake.

Statistical analysis

The data were analyzed using SPSS 17.0 (SPSS, Chicago, IL, USA). The C max , AUC, Ae, and Cl renal data were logarithmically transformed before statistical analysis.

Statistical comparisons were made between the ABCB1 haplotypes using analysis of variance (ANOVA). Body weight was set as a covariate for C max , AUC, and Ae [27].

The t max data were compared with the Mann-Whitney test. Differences were considered statistically significant when P<0.05. The geometric mean ratio between the TTT and CGC homozygotes with 95% confidence interval (CI) was calculated for the C max , AUC, Ae, and Cl renal values, and the mean difference with 95% CI for the t ½ and renin activity variables. The number of subjects was estimated to be sufficient to detect a 50% difference in the AUC 0-∞ of aliskiren between the ABCB1 CGC and TTT homozygotes, with a power of 80% (α-level 5%).

RESULTS

ABCB1 haplotypes had no significant effect on the pharmacokinetics or renin-inhibiting effect of aliskiren (Table 1, Fig. 1). The geometric mean ratio (95% CI) of aliskiren C max and AUC 0-∞ in participants homozygous for the TTT haplotype to those in participants homozygous for the CGC haplotype were 1.14 (0.66, 1.96; P=0.631) and 1.01 (0.58, 1.76; P=0.960), respectively. As a post hoc analysis, fractional AUC values (AUC 0-1 , AUC 0-2 , AUC 0-3 , AUC 0-4 , and AUC 0-5 ) were also calculated for aliskiren, but no tendency for a difference between the haplotype groups was observed in these values

DISCUSSION

In the present study, the ABCB1 haplotypes had no significant effect on the pharmacokinetics or renin-inhibiting effect of the P-glycoprotein substrate aliskiren.

However, these genetic results do not indicate that the P-glycoprotein transporter could not be otherwise important for the pharmacokinetics of aliskiren. In vitro, aliskiren is a high affinity substrate of P-glycoprotein (K m 2.1 µmol/l), as evidenced by studies in Pglycoprotein-expressing Sf9 cell membrane vesicles [2]. In vivo, inhibitors and inducers of P-glycoprotein markedly affect aliskiren pharmacokinetics [10,11,14].

It is noteworthy that ABCB1 genotypes have explained only a fraction of the variability in P-glycoprotein expression, and the effects of ABCB1 genotypes and haplotypes on drug pharmacokinetics have been modest [15,17,28,29]. For example, the AUCs of simvastatin acid and atorvastatin have been 60% and 55% larger, respectively, in subjects homozygous for the ABCB1 TTT haplotype compared with subjects homozygous for the CGC haplotype [17]. Studies investigating the effects of the ABCB1 c.3435C>T SNP on the pharmacokinetics of the P-glycoprotein prototype substrate digoxin have revealed conflicting results, with a meta-analysis suggesting that the ABCB1 c.3435C>T SNP does not affect digoxin pharmacokinetics [12,15,[28][29][30][31].

It has been suggested that the ABCB1 c.3435C>T SNP, or its combination with the c.1236C>T or c.2677G>T/A SNP, may alter the substrate specificity of P-glycoprotein [16]. Therefore, the effect of the ABCB1 genotype on the pharmacokinetics of different P-glycoprotein substrates may vary. The present study had sufficient power to detect a 50% difference in the AUC 0-∞ of aliskiren between ABCB1 haplotypes, and thus a small difference cannot be ruled out. However, it is unlikely that a smaller than 50% difference in aliskiren AUC 0-∞ would be of clinical importance. For example, a 76% increase in aliskiren AUC by ketoconazole was not considered clinically relevant [2]. Nevertheless, the mean AUC 0-∞ values of aliskiren were almost identical in the two haplotype groups.

Similarly to previous reports [25], a high interindividual variability was observed in the C max and AUC values of aliskiren. The present results suggest that common ABCB1 haplotypes do not explain this variability. The large differences in aliskiren pharmacokinetics might be explained by variability in gastrointestinal physiology, and other genetic and non-genetic factors affecting the activity of P-glycoprotein or CYP3A, or other transporters or enzymes involved in aliskiren pharmacokinetics [2].

INTRODUCTION

The direct renin inhibitor aliskiren is poorly absorbed and has a low oral bioavailability of 2-3% [1]. Aliskiren is mainly eliminated by biliary excretion [1]. In addition, about 0.4% of the oral dose is excreted unchanged into urine and about 1.4% undergoes oxidative biotransformation via CYP3A4 [1]. Aliskiren is a substrate of P-glycoprotein (also known as ABCB1) [2], an efflux transporter expressed in tissues with an excretory or barrier function [3,4]. The P-glycoprotein and CYP3A4 inhibitors, itraconazole and cyclosporine [5][6][7][8][9], have raised the area under the plasma concentration-time curve (AUC) of aliskiren by about 6.5-fold and 5-fold, respectively [10,11], supporting a major role of P-glycoprotein in limiting the intestinal absorption of aliskiren. Moreover, rifampicin, an inducer of P-glycoprotein and CYP3A4 [12, 13], has reduced the AUC of aliskiren by 56% [14].

A common synonymous single nucleotide polymorphism (SNP) in the ABCB1 gene encoding P-glycoprotein, c.3435C>T, has been variably associated with reduced expression and function of P-glycoprotein in vitro and increased plasma concentrations of P-glycoprotein substrate drugs in vivo [15][16][17]. The mechanism underlying these associations is not completely understood, but it has been suggested that c.3435C>T affects ABCB1 mRNA stability [START_REF] Rao | Direct demonstration of high affinity interactions of immunosuppressant drugs with the drug binding site of the human P-glycoprotein[END_REF]. Alternatively, the associations may be explained by linkage disequilibrium between c.3435C>T and a functional sequence variation, or that c.3435C>T and sequence variations in linkage disequilibrium with it form a haplotype conferring altered P-glycoprotein activity [16,19]. The ABCB1 c.3435C>T SNP is in strong linkage disequilibrium with the c.1236C>T (synonymous) and c.2677G>T/A (p.Ala893Ser/Thr) SNPs within the same gene [17,19,20].

The variant alleles c.1236T, c.2677T, and c.3435T are common in Europeans and European Americans, with reported frequencies of occurrence ranging from 39 to 46%, 39 to 47%, and 54 to 60%, respectively [17,[19][20][21][22]. In the Finnish population, the two most common haplotypes based on these SNPs are c.1236T-c.2677T-c.3435T (TTT) and c.1236C-c.2677G-c.3435C (CGC), with frequencies of 42.7% and 34.4%, respectively [17]. The ABCB1 TTT haplotype has been associated, for example, with increased plasma concentrations of and enhanced lipid-lowering response to simvastatin (acid) and atorvastatin, compared with the ABCB1 CGC haplotype [17,23,24].

Considerable interindividual variability exists in the pharmacokinetics of aliskiren [25], but the background of this variability is unknown. Our hypothesis was that genetic variability in ABCB1 affects the pharmacokinetics of aliskiren. Therefore, we investigated the effects of the ABCB1 CGC and TTT haplotypes on the pharmacokinetics and pharmacodynamics of aliskiren. Deleted: The antihypertensive drug aliskiren is a substrate of ABCB1 [13]. It is poorly absorbed and has a low oral bioavailability of 2-3% [14]. Aliskiren is mainly eliminated by biliary excretion [14]. In addition, about 0.4% of the oral dose is excreted unchanged into urine and about 1.4% undergoes oxidative biotransformation via CYP3A4 [14]. The ABCB1 and CYP3A4 inhibitors, itraconazole and cyclosporine [15][16][17][START_REF] Rao | Direct demonstration of high affinity interactions of immunosuppressant drugs with the drug binding site of the human P-glycoprotein[END_REF][19], have raised the AUC of aliskiren by about 6.5-fold and 5-fold, respectively [20,21], supporting a major role of ABCB1 in limiting the intestinal absorption of aliskiren. Moreover, rifampicin, an inducer of ABCB1 and CYP3A4 [22,23], has reduced the AUC of aliskiren by 56% [24] Novartis, Horsham, UK) with 150 ml water. A standardized warm meal was served 4 h after aliskiren, and a standardized light meal after 7 h and 10 h. As a safety assessment, systolic and diastolic blood pressures were measured from the forearm with an automatic oscillometric blood pressure monitor (Omron M5-I, Omron Healthcare

Europe BV, Hoofddorp, The Netherlands), with the participant in a sitting position, prior to and 2, 4, 7, 9, 12, and 24 h after aliskiren ingestion. The participants were under direct medical supervision for 12 h after the administration of aliskiren. No significant differences in blood pressures were observed between the haplotype groups (data not shown). Use of other drugs was prohibited for one week and use of grapefruit products for three days before aliskiren administration and 72 h thereafter.

Timed blood samples (5 or 10 ml each) for aliskiren concentration measurements were drawn into tubes containing ethylenediaminetetraacetic acid (EDTA) prior to and 0.5, 1, 2, 3, 4, 5, 7, 9, 12, 24, 34, 48, and 72 h after aliskiren administration. Blood samples for the determination of plasma renin activity (5 ml each) were drawn before the administration of aliskiren and 4 h and 24 h thereafter into chilled EDTA tubes, which were placed on ice immediately after sampling. Plasma was separated within 30 min.

Urine was collected up to 12 h after aliskiren ingestion. Urine aliquots and plasma were stored at -70ºC until analysis.

Determination of aliskiren concentrations and renin activity

Plasma concentrations of aliskiren were quantified by use of solid phase extraction and liquid chromatography-tandem mass spectrometry (LC/MS/MS) [10]. An aliquot of 700 µl of each plasma sample was mixed with 100 µl of 10% phosphoric acid and 50 µl Deleted: March Deleted: Timed ethylenediaminetetraacetic acid blood samples (5-15 ml each) were drawn prior to and up to 72 h after aliskiren ingestion. Plasma was separated within 30 min. Urine was collected up to 12 h after aliskiren ingestion. The samples were stored at -70ºC until analysis. ¶ internal standard solution (1000 ng/ml acebutolol in 1:1 methanol:water). Then the mixture was loaded to the extraction cartridge (Oasis MCX 1cc/30 mg, Waters, Milford, MA, USA), which had been pre-conditioned with 1 ml of methanol and 1 ml of 1% acetic acid. The cartridge was subsequently washed with 1 ml of methanol and 1 ml of 1% acetic acid. The analytes were eluted out with two times 0.6 ml of 2% ammonium hydroxide in methanol. The eluent was dried at 50°C under a nitrogen stream. The residue was dissolved in 100 µl of the mobile phase, and 15 µl of the sample was injected to the LC/MS/MS system. The chromatographic separation was carried out on an AtlantisT3 column (2.1×100 mm, 3 µm) protected by an AtlantisT3 (2.1×10 mm, 3 µm) guard cartridge (Waters). The mobile phase consisted of (A) 2 mmol/l ammonium acetate (pH 4.90) and (B) methanol. The total running time was 15 minutes with the mobile phase gradient of 0.5 minutes at 70% A, 0.5 minutes to 5% A, 3 minutes at 5% A, 0.1 minutes to 70% A, and 10.9 minutes at 70% A. The flow rate of the mobile phase was 250 µl/min. The column temperature was 40°C. Mass spectra were obtained using For the determination of aliskiren concentrations in urine, an aliquot of 300 µl of each urine sample was mixed with 300 µl of fresh frozen plasma, 100 µl of 10% phosphoric acid, and 50 µl internal standard solution (1000 ng/ml acebutolol in 1:1 methanol:water). The extraction, chromatographic separation, and mass spectrometric detection were performed as described for plasma aliskiren. The lower limit of Deleted: March Deleted: Aliskiren plasma and urine concentrations were quantified using an Applied Biosystems SCIEX Q Trap LC/MS/MS system (Sciex Division of MDS, Toronto, ON, Canada) [20]. Acebutolol served as an internal standard. quantification was 0.25 ng/ml for plasma and 9 ng/ml for urine aliskiren. The calibration curve for plasma aliskiren was linear over the range 0.25-500 ng/ml (r>0.999, weighting 1/x) and for urine aliskiren over the range 9-1800 ng/ml (r>0.999, weighting 1/x). The extraction recoveries were > 80%. The intra-day coefficients of variation were ≤ 1.2% at relevant concentrations in plasma and ≤ 4.8% in urine (n=6).

Plasma renin activity was determined by radioimmunoassay of generated angiotensin I with a commercially available method (RENCTK; DiaSorin, Saluggia, Italy) at HUSLAB laboratory (Helsinki, Finland).

Pharmacokinetics and pharmacodynamics

Aliskiren pharmacokinetics were characterized by peak plasma concentration (C max ), time to C max (t max ), elimination half-life (t ½ ), AUC 0-72 , AUC 0-∞ , the amount of aliskiren excreted into urine from 0 to 12 h (Ae), and the renal clearance (Cl renal ).

DISCUSSION

In the present study, the ABCB1 haplotypes had no significant effect on the pharmacokinetics or renin-inhibiting effect of the P-glycoprotein substrate aliskiren.

However, these genetic results do not indicate that the P-glycoprotein transporter could not be otherwise important for the pharmacokinetics of aliskiren. In vitro, aliskiren is a high affinity substrate of P-glycoprotein (K m 2.1 µmol/l), as evidenced by studies in Pglycoprotein-expressing Sf9 cell membrane vesicles [2]. In vivo, inhibitors and inducers of P-glycoprotein markedly affect aliskiren pharmacokinetics [10,11,14].

It is noteworthy that ABCB1 genotypes have explained only a fraction of the variability in P-glycoprotein expression, and the effects of ABCB1 genotypes and haplotypes on drug pharmacokinetics have been modest [15,17,28,29]. For example, the AUCs of simvastatin acid and atorvastatin have been 60% and 55% larger, respectively, in subjects homozygous for the ABCB1 TTT haplotype compared with subjects homozygous for the CGC haplotype [17]. Studies investigating the effects of the ABCB1 c.3435C>T SNP on the pharmacokinetics of the P-glycoprotein prototype substrate digoxin have revealed conflicting results, with a meta-analysis suggesting that the ABCB1 c.3435C>T SNP does not affect digoxin pharmacokinetics [12,15,[28][29][30][31].

It has been suggested that the ABCB1 c.3435C>T SNP, or its combination with the c.1236C>T or c.2677G>T/A SNP, may alter the substrate specificity of P-glycoprotein [16]. Therefore, the effect of the ABCB1 genotype on the pharmacokinetics of different P-glycoprotein substrates may vary. The present study had sufficient power to detect a 50% difference in the AUC 0-∞ of aliskiren between ABCB1 haplotypes, and thus a small difference cannot be ruled out.

However, it is unlikely that a smaller than 50% difference in aliskiren AUC 0-∞ would be of clinical importance. For example, a 76% increase in aliskiren AUC by ketoconazole was not considered clinically relevant [2]. Nevertheless, the mean AUC 0-∞ values of aliskiren were almost identical in the two haplotype groups.

Similarly to previous reports [25], a high interindividual variability was observed in the , AUC, and Ae data are estimated marginal mean (95% CI) adjusting for weight; t ½ , Cl renal , and plasma renin activity are estimated marginal mean (95% CI); t max are median (range). CI, Confidence interval; C max , peak plasma concentration; t max , time to C max ; t ½ , elimination half-life; AUC 0-72 , area under the plasma concentration-time curve from 0 to 72 h; AUC 0-∞ , area under the plasma concentration-time curve from time 0 to infinity; Ae, amount excreted into urine within 12 h; Cl renal renal clearance; Baseline, before the administration of aliskiren. * Mean difference (95% CI).

  for European Journal of Clinical Pharmacology, April 2010 6

  for European Journal of Clinical Pharmacology, April 2010 7

(

  P>0.3, data not shown). Aliskiren C max , AUC 0-∞ , Ae, and Cl renal varied 14.1-, 16.2-, 10.4-, and 3.1-fold, respectively, between individual participants.

  In conclusion, the common ABCB1 haplotypes c.1236C-c.2677G-c.3435C and c.1236Tc.2677T-c.3435T have no effects on the pharmacokinetics or pharmacodynamics of aliskiren. P e e r R e v i e w A revised manuscript for European Journal of Clinical Pharmacology, April 2010 20

Fig. 1

 1 FIGURE LEGEND

  Deleted: MarchDeleted: ABCB1 (also known as Pglycoprotein) is an efflux transporter expressed in tissues with an excretory or barrier function[1, 2]. for European Journal of Clinical Pharmacology, April 2010 4

A

  revised manuscript for European Journal of Clinical Pharmacology, April 2010 7

an

  Applied Biosystems SCIEX API 2000 Q Trap mass spectrometer operated in positive ion mode (Sciex Division of MDS Inc, Toronto, ON, Canada). The ion transitions were mass-to-charge ratio (m/z) 552.4 to m/z 436.3 for aliskiren and m/z 337.3 to m/z 116.3 for acebutolol.

A

  revised manuscript for European Journal of Clinical Pharmacology, April 2010 8

  Pharmacokinetic parameters were calculated with non-compartmental methods using MK-Model, version 5.0 (Biosoft, Cambridge, UK). The terminal log-linear part of each concentration-time curve was identified visually. The elimination rate constant (k e ) was determined by linear regression analysis of the log-linear part of the plasma drug concentration-time curve. The t ½ was calculated by the equation t ½ =ln2/k e . The AUC values were calculated by a combination of the linear and log-linear trapezoidal rules with extrapolation to infinity, when appropriate, by division of the last measured concentration by k e . The Cl renal of aliskiren was calculated by the equation Cl renal =Ae/AUC 0-12 . Aliskiren pharmacodynamics were characterized by plasma renin activity at 4 h and 24 h after aliskiren intake. Deleted: March Deleted: measured Deleted: using Deleted: radioimmunoassay Deleted: areas under the plasma concentration-time curve from 0 to 72 h ( Deleted: ) and 0 h to infinity (

CA

  max and AUC values of aliskiren. The present results suggest that common ABCB1 haplotypes do not explain this variability. The large differences in aliskiren pharmacokinetics might be explained by variability in gastrointestinal physiology, and other genetic and non-genetic factors affecting the activity of P-glycoprotein or CYP3A, or other transporters or enzymes involved in aliskiren pharmacokinetics [2]. In conclusion, the common ABCB1 haplotypes c.1236C-c.2677G-c.3435C and c.1236Tc.2677T-c.3435T have no effects on the pharmacokinetics or pharmacodynamics of aliskiren. revised manuscript for European Journal of Clinical Pharmacology, April 2010 20

21 FIGURE LEGEND Fig. 1 (

 21LEGEND1 FIGURE LEGEND

Table 1

 1 Pharmacokinetic variables of and plasma renin activity response to a single 150-mg oral dose of aliskiren in healthy subjects homozygous for the ABCB1 c.1236C-c.2677G-c.3435C (CGC/CGC genotype; n=11) or c.1236T-c.2677T-c.3435T (TTT/TTT genotype; AUC, and Ae data are estimated marginal mean (95% CI) adjusting for weight; t ½ , Cl renal , and plasma renin activity are estimated marginal mean (95% CI); t max are median (range). CI, Confidence interval; C max , peak plasma concentration; t max , time to C max ; t ½ , elimination half-life; AUC 0-72 , area under the plasma concentration-time curve from 0 to 72 h; AUC 0-∞ , area under the plasma concentration-time curve from time 0 to infinity; Ae, amount excreted into urine within 12 h; Cl renal renal clearance; Baseline, before the administration of aliskiren.

	n=11) haplotype
	Variable	ABCB1 genotype	Geometric mean ratio	P value
		CGC/CGC	TTT/TTT	(95% CI)
	Aliskiren
	C max (ng/ml)	98 (67, 143)	111 (76, 163)	1.14 (0.66, 1.96)	0.631
	t max (h)	1 (0.5-5)	0.5 (0.5-5)		0.495
	t ½ (h)	23.9 (19.0, 28.8)	22.5 (17.6, 27.4)	-1.4 (-8.3, 5.5)*	0.684
	AUC 0-72 (ng•h/ml)	419 (289, 608)	429 (296, 622)	1.02 (0.60, 1.74)	0.927
	AUC 0-∞ (ng•h/ml)	443 (301, 652)	449 (306, 661)	1.01 (0.58, 1.76)	0.960
	Ae (mg)	0.499 (0.335, 0.742) 0.516 (0.347, 0.767)	1.03 (0.58, 1.83)	0.905
	Cl renal (l/h)	1.75 (1.48, 2.06)	1.71 (1.45, 2.01)	0.98 (0.77, 1.23)	0.842
	Plasma renin activity (ng/ml/h)
	At baseline	1.73 (0.96, 2.50)	2.38 (1.61, 3.15)	0.66 (-0.43, 1.75)*	0.222
	At 4 h	0.01 (-0.09, 0.11)	0.09 (-0.01, 0.19)	0.08 (-0.06, 0.22)*	0.251
	At 24 h	1.08 (0.51, 1.65)	1.36 (0.79, 1.92)	0.28 (-0.52, 1.08)*	0.479
	C max ,

* Mean difference (95% CI).

Table 1

 1 Pharmacokinetic variables of and plasma renin activity response to a single 150-mg oral dose of aliskiren in healthy subjects homozygous for the ABCB1 c.1236C-c.2677G-c.3435C (CGC/CGC genotype; n=11) or c.1236T-c.2677T-c.3435T (TTT/TTT genotype;

	n=11) haplotype
	Variable	ABCB1 genotype	Geometric mean ratio	P value
		CGC/CGC	TTT/TTT	(95% CI)
	Aliskiren
	C max (ng/ml)	98 (67, 143)	111 (76, 163)	1.14 (0.66, 1.96)	0.631
	t max (h)	1 (0.5-5)	0.5 (0.5-5)		0.495
	t ½ (h)	23.9 (19.0, 28.8)	22.5 (17.6, 27.4)	-1.4 (-8.3, 5.5)*	0.684
	AUC 0-72 (ng•h/ml)	419 (289, 608)	429 (296, 622)	1.02 (0.60, 1.74)	0.927
	AUC 0-∞ (ng•h/ml)	443 (301, 652)	449 (306, 661)	1.01 (0.58, 1.76)	0.960
	Ae (mg)	0.499 (0.335, 0.742) 0.516 (0.347, 0.767)	1.03 (0.58, 1.83)	0.905
	Cl renal (l/h)	1.75 (1.48, 2.06)	1.71 (1.45, 2.01)	0.98 (0.77, 1.23)	0.842
	Plasma renin activity (ng/ml/h)
	At baseline	1.73 (0.96, 2.50)	2.38 (1.61, 3.15)	0.66 (-0.43, 1.75)*	0.222
	At 4 h	0.01 (-0.09, 0.11)	0.09 (-0.01, 0.19)	0.08 (-0.06, 0.22)*	0.251
	At 24 h	1.08 (0.51, 1.65)	1.36 (0.79, 1.92)	0.28 (-0.52, 1.08)*	0.479
	C max

European Journal of Clinical Pharmacology

ACKNOWLEDGEMENTS

We thank Ms Kaisa Kurkinen, MSc, for aliskiren concentration measurements, and Ms Eija Mäkinen-Pulli, Ms Lisbet Partanen, and Mr Jouko Laitila for skilled assistance.

This study was supported by grants from the Sigrid Jusélius Foundation (Helsinki, Finland).

ACKNOWLEDGEMENTS

We thank Ms Kaisa Kurkinen, MSc, for aliskiren concentration measurements, and Ms Eija Mäkinen-Pulli, Ms Lisbet Partanen, and Mr Jouko Laitila for skilled assistance.

This study was supported by grants from the Sigrid Jusélius Foundation (Helsinki, Finland).

COMPETING INTEREST

None to declare.

ABSTRACT

Purpose

This study aimed to investigate possible effects of ABCB1 haplotypes on the pharmacokinetics and renin-inhibiting effect of aliskiren.

Methods

Eleven healthy volunteers homozygous for the ABCB1 c.1236C-c.2677G-c.3435C (CGC) haplotype and eleven homozygous for the c.1236T-c.2677T-c.3435T (TTT) haplotype ingested a single 150-mg dose of aliskiren. Plasma aliskiren concentrations were measured up to 72 h, its excretion into urine up to 12 h, and plasma renin activity up to 24 h.

Results

The ABCB1 haplotypes had no significant effect on the pharmacokinetics or renininhibiting effect of aliskiren. The geometric mean ratio (95% confidence interval) of aliskiren peak plasma concentration and area under the plasma concentration-time curve from 0 h to infinity in participants homozygous for the TTT haplotype to those in participants homozygous for the CGC haplotype were 1.14 (0.66, 1.96; P=0.631) and 1.01 (0.58, 1.76; P=0.960), respectively.

Conclusions

These data suggest that the ABCB1 c.1236C-c.2677G-c.3435C and c.1236T-c.2677Tc.3435T haplotypes have no clinically meaningful effect on the pharmacokinetics of aliskiren.

KEYWORDS

METHODS

Subjects

Twenty-two healthy White Finnish volunteers participated in this study after giving written informed consent. They were recruited from a group of more than 700 subjects genotyped for the ABCB1 c.1236C>T (rs1128503), c.2677G>T/A (rs2032582), and c.3435C>T (rs1045642) SNPs [17]. To minimize variability in aliskiren pharmacokinetics due to other genetic variants, all subjects were genotyped for the functionally significant ABCB1 c.1199G>A (p.Ser400Asn; rs2229109) [17] and CYP3A5*3 (g.6986A>G; rs776746) [26] alleles and only noncarriers of the ABCB1 c.1199A and CYP3A5 g.6986A (CYP3A5 expressor) alleles were recruited.

Study design

The study protocol was approved by the Coordinating Ethics Committee of the Helsinki and Uusimaa Hospital District, and the National Agency for Medicines. Following an overnight fast, the subjects ingested a single 150-mg dose of aliskiren (Rasilez;

Statistical analysis

The data were analyzed using SPSS 17.0 (SPSS, Chicago, IL, USA). The C max , AUC, Ae, and Cl renal data were logarithmically transformed before statistical analysis.

Statistical comparisons were made between the ABCB1 haplotypes using analysis of variance (ANOVA). Body weight was set as a covariate for C max , AUC, and Ae [27].

The t max data were compared with the Mann-Whitney test. Differences were considered statistically significant when P<0.05. The geometric mean ratio between the TTT and CGC homozygotes with 95% confidence interval (CI) was calculated for the C max , AUC, Ae, and Cl renal values, and the mean difference with 95% CI for the t ½ and renin activity variables. The number of subjects was estimated to be sufficient to detect a 50% difference in the AUC 0-∞ of aliskiren between the ABCB1 CGC and TTT homozygotes, with a power of 80% (α-level 5%).

RESULTS

ABCB1 haplotypes had no significant effect on the pharmacokinetics or renin-inhibiting effect of aliskiren (Table 1, Fig. 1). The geometric mean ratio (95% CI) of aliskiren C max and AUC 0-∞ in participants homozygous for the TTT haplotype to those in participants homozygous for the CGC haplotype were 1.14 (0.66, 1.96; P=0.631) and 1.01 (0.58, 1.76; P=0.960), respectively. As a post hoc analysis, fractional AUC values (AUC 0-1 , AUC 0-2 , AUC 0-3 , AUC 0-4 , and AUC 0-5 ) were also calculated for aliskiren, but no tendency for a difference between the haplotype groups was observed in these values