Martin Czejka
email: martin.czejka@univie.ac.at

J Schueller

A Farkouh

B Gruenberger

W Scheithauer

Plasma disposition of capecitabine and its metabolites 5'DFCR and 5'DFUR in a standard and dose-intensified monotherapy regimen

Keywords: capecitabine, metabolites, pharmacokinetics, intensified schedule

Purpose: In view of a potential gain in anticancer activity in advanced colorectal cancer (ACRC), there has been considerable interest in using a higher than the approved standard dose of capecitabine (CCB) combined with oxaliplatin. This pharmacokinetic study was designed to evaluate whether CCB is metabolized at the same extent when administered as a monotherapy in two different dose regimens, comparing standard dose (CCB 1) and intensified dose (CCB 2).

Patients and methods: Seven patients suffering from ACRC received subsequently two CCB schedules: In the standard schedule 1250 mg/m 2 CCB p.o. twice daily for two weeks was administered, after a pause of one week a dose intensified CCB 2 schedule was given: 1750 mg/m 2 CCB p.o. twice daily for one week to be followed by one week rest. Due to this paired cross over design a direct comparison for each single patient was feasible.

Results: In both schedules, mean peak plasma concentrations of CCB occurred at about 50 min, those of metabolites shortly later (range, 54 -80 min). Peak plasma concentrations were about 10 % (CCB, DFCR) and 40 % (DFUR) higher in the CCB 2 regimen. According to the higher dose of CCB in the dose intensified regimen (+ 40 %), the AUC last values increased by 34 % (CCB), 20 % (DFCR) and 58 % (DFUR), respectively.

Conclusion:

The results indicate that higher doses of CCB are metabolized approximately dose-dependent compared to the standard dose. No indices for a saturation of metabolizing processes or any significant delay of elimination rate was observed. The immediate 5FU precursor DFUR was formed at a 50 % higher extent (expressed as AUC last values) than in the standard CCB 1 schedule. From the pharmacokinetic point of view this increased formation rate suggests clinical importance in regard to metabolic activation of CCB.

Introduction

The oral fluoropyrimidine carbamate capecitabine (CCB) is an equally effective and tolerable, but more convenient alternative to i.v. 5-fluorouracil (5-FU) / folinic acid. In fact, this compound has been designed to generate 5-FU preferentially within the tumour and to mimic continuous infusion of 5-FU [START_REF] Miwa | Design of a novel oral fluoropyrimidine carbamate, capecitabine, which generates 5FU selectively in tumors by enzymes concentrated in human liver and cancer tissue[END_REF][START_REF] Schüller | Preferential activation of capecitabine in tumor following oral administration to colorectal cancer patients[END_REF], while having an easier handling [START_REF] Mayer | Should capecitabine replace infusional fluorouracil and leucovorin when combined with oxalilpatin in metastatic colorectal cancer[END_REF].

CCB is rapidly and extensively absorbed through the gastrointestinal wall as an intact molecule, and rapidly metabolised to 5-FU via a three-step enzymatic cascade [START_REF] Schüller | Preferential activation of capecitabine in tumor following oral administration to colorectal cancer patients[END_REF][START_REF] Kono | Activation of 5'deoxy-5-fluorouridine by thymidine phosphorylase in human tumours[END_REF]. Firstly CCB is metabolized into 5'-deoxy-5-fluorocytidine (DFCR) by the human carboxyesterases isoenzyme 2 (hCES2), primarily in the liver. DFCR is converted to 5'-deoxy-5-fluorouridine (DFUR) by cytidine deaminase (CytDA) in tumor cells and in the liver. Finally, DFUR is metabolized into the cytotoxic 5-FU by thymidine phosphorylase (TP), which is significantly more active in the tumour tissue than in the adjacent healthy tissue [START_REF] Schüller | Preferential activation of capecitabine in tumor following oral administration to colorectal cancer patients[END_REF].

CCB is given either as a single agent or in combination with irinotecan (XELIRI) or oxaliplatin (OxPt; XELOX), which (based on different mechanisms of action) has resulted in increased therapeutic efficacy without major toxic overlap [START_REF] Twelves | Capecitabine as first-line treatment in colorectal cancer. Pooled data from two large, phase III trials[END_REF][START_REF] Cassidy | XELOX (capecitabine plus oxaliplatin): active first-line therapy for patients with metastatic colorectal cancer[END_REF].

The combination of CCB and OxPt in fact has demonstrated promising synergistic activity in ACRC. Various XELOX regimens using different dosages of CCB from 750 to 2500 mg/m 2 have been applied. The most frequently used regimens are CCB 1000 mg /m 2 p.o. twice daily for 14 days plus OxPt 130 mg/m 2 i.v. on day 1 every 3 weeks [START_REF] Clarke | Randomized phase III study of capecitabine plus oxaliplatin compared with fluorouracil/folinic acid plus oxaliplatin as first-line therapy for metastastic colorectal cancer[END_REF] or CCB 1000 mg /m 2 p.o. twice daily for 14 days plus OxPt 70 mg/m 2 i.v. once every week [START_REF] Porschen | Capecitabine plus oxaliplatin compared with fluorouracil and leucovorin plus oxaliplatin: A randomized comparison in metastatic colorectal cancer -A final report of the AIO colorectal study group[END_REF]. A modified XELOX schedule using dose intensified CCB (3500 mg/m 2 for one week; so called XELOX 2 schedule) was first described by Scheithauer et al [START_REF] Scheithauer | Intermittent weekly high-dose capecitabine in combination with oxaliplatin: a phase I/II study first line treatment of patients with advanced colorectal cancer[END_REF] and is now being investigated in several clinical trials.

For OxPt and bevacizumab, no influence on the pharmacokinetics of CCB standard dose was documented [START_REF] Brennan | Pharmacokinetic interactions between capecitabine, oxaliplatin and bevacizumab when used in combination for first line treatment of metastatic colorectal cancer (MCRC)[END_REF][START_REF] Schueller | Influence of bevacizumab on pharmacokinetics and metabolism of capecitabine in metastatic colorectal cancer. 19 th ICACT Congress on Anticancer Treatment[END_REF]. However until today only preliminary pharmacokinetic data of CCB after dose intensification have been described [START_REF] Schueller | Does dose escalation of capecitabine influence its metabolic conversion in metastatic colorectal cancer patients[END_REF] and it remains unclear whether saturation processes of enzymes affect sequential biotransformation of CCB into DFUR after higher doses of CCB. In this article we describe the pharmacokinetics and metabolism of CCB when given as a monotherapy at a standard dose of 2500 mg/m 2 p.o. daily for two weeks (CCB 1) compared to an intensified dose of 3500 mg/m 2 daily for 7 days (CCB 2).

Clinical study

Subjects

Seven patients (2 female, 5 male) who received palliative chemotherapy for ACRC entered this pharmacokinetic study. Their mean age was 67.8 years (range 57 -77), mean body mass was 78.4 + 17.2 kg (range 48 -104) and mean body surface area was 1.9 + 0.2 m 2 (range (1.41 -2.25).

Each patient provided written informed consent according the specifications of the local ethics committee. Inclusion criteria were as follows: ECOG performance status 0 -1; white blood cell count > 4000/µl, ANC > 1500; no renal impairment as judged by standard biochemical parameters (serum creatinine < 1.3 mg/dl); and adequate hepatic function i.e. bilirubin < 1.2 mg/dl, and SGOT, SGPT < 2.5 fold U/l (5 fold acceptable in case of liver mets, but not reached in this study) . In terms of liver function it has already been clarified that mild to moderate impairment caused by liver metastases do not significantly change the PK of CCB and its metabolites [START_REF] Twelves | Effect of hepatic dysfuntion due to liver metastases on the pharmacokinetics of capecitabine and its metabolites[END_REF].

Drugs

Capecitabine (Xeloda®) was supplied from Hoffmann La Roche (Vienna, Austria) as filmcoated tablets in two dose strengths: 150 mg and 500 mg, which were not to be split and had to be taken with water within 30 min after the ingestion of food. Compliance with the oral medication regimen was assessed in the hospital.

Pure chemical standard substances (CCB, DFCR, DFUR) were donated by Hoffmann La Roche (Basle, Switzerland) for calibration purposes of the analytical assay.

Chemotherapy schedule

Table 1 displays the two different chemotherapeutic drug dose schedules, which were investigated consecutively in all patients. By this paired cross over design with intra-patient comparison each subject served for its own control.

Blood samples

Blood samples of 5 ml were drawn from the cubital vein at the following times: 0, 30, 60, 90, 120, 180 and 210 min after ingestion of the oral CCB morning dose. Samples were collected in sodium-heparinized vacutubes and blood cells were separated by centrifuging at 2500 rpm for 5 minutes. Of the supernatant 2.0 ml were frozen at -80 o C until analysis.

Sampling was performed in the first week following CCB 1 standard dose on day one and five. After crossing over to CCB 2 (dose intensified schedule) sampling was repeated during the 4 th week, again on day one and five. Since no cumulative effect from day one to five was observed, the respective data from the seven patients were summarised/analysed together (N = 14).

Sample clean-up and analytics

Separation of analytes from matrix compounds was performed by a sensitive and selective solid phase extraction method using Oasis® HLB C18 cartridges (Waters Inc.).

For quantification of CCB, DFCR and DFUR in plasma samples we used reversed phase high performance liquid chromatography as previously reported [START_REF] Zufía | Simple determination of capecitabine and its metabolites by liquid chromatography with ultraviolet detection in a single injection[END_REF][START_REF] Farkouh | Pharmacolinetics and metabolism of capecitabine combined with oxaliplatin: in-vitro and in-vivo investigations[END_REF]. External calibration was performed by using pooled human plasma samples spiked with pure chemical standards.

Biometric calculations

Curve fitting of plasma concentration versus time date was performed by the pharmacokinetic software WinNonlin Professional 4.0 (Pharsight Corporation, CA 94041, USA) using the Nelder-Mead nonlinear iterative least square algorithm. For pharmacokinetic modeling of CCB and metabolites, a non-compartment model with extravascular input (model 3 of the WinNonlin library) was chosen. For calculation of AUC and AUMC the linear trapezoidal rule was applied for the ascending part of the concentration-time curve and the log-linear trapezoidal rule for the descending part of the concentration-time curve.

The following pharmacokinetic parameters had been calculated for CCB, DFCR and DFUR, (Vz and Cl tot were not calculated for CCB metabolites). Potential statistically significant differences in plasma concentrations or pharmacokinetic parameters between both CCB schedules, were evaluated by use of the independent Student's t-test (two sided, paired, homoscedastic, p-level 0.05). Insert C in the lower part of figure 1 shows the sum of plasma concentrations of CCB and metabolites: as can be seen from this concentration time curve for the CCB 2 schedule, the sum of plasma concentrations was nearly 50 % higher than in the standard CCB 1 schedule.

Results

Mean

Mean factor obtained after dividing plasma concentrations high/low was 1.32 (range from

Discussion

The aim of this pharmacokinetic study was to determine the plasma disposition and PK parameters of CCB and its metabolites using the approved standard single dose of 2500 mg/m 2 /d (CCB 1) compared to a dose intensified schedule of 3500 mg/m 2 /d (CCB 2). Such dose intensification in combination with oxaliplatin was first described in the clinical setting by Scheithauer et al [16] in a randomized phase II trial in ACRC, which was based on preclinical data in human tumor xenografts. In this study the -so called -XELOX 2 regimen was defined as bimonthly oxaliplatin 85mg/m 2 combined with CCB 1750 mg/m 2 bid for seven days repeated every two weeks. A superior outcome with higher response rates and longer progression free survival compared to standard XELOX was described. Meanwhile, this modified XELOX schedule with CCB dose intensification has been / continues to be investigated in several clinical trials: Gruenberger et al [START_REF] Gruenberger | Bevacizumab, capecitabine, and oxaliplatin as neoadjuvant therapy for patients with potentially curable metastatic colorectal cancer[END_REF] used the XELOX 2 schedule combined with bevacizumab in a phase II trial of ACRC patients with potentially resectable liver metastases as neoadjuvant treatment. In the first US trial using a higher CCB dose (3000 mg/m 2 for one week) combined with oxaliplatin and bevacizumab, this (A -ICOX) regimen was found to be fairly good tolerated, and a phase III trial comparing this schedule and standard XELOX was initiated [START_REF] Lembersky | Phase II study of bevacizumab (BEV), weekly (wkly) intermittent capecitabine (Cap), and oxaliplatin (Ox) on an every-2-week schedule for patients with untreated advanced colorectal cancer (CRC)[END_REF]. Recently Kornek et al [START_REF] Kornek | Limited treatment duration and re-induction of the same regimen upon progression versus treatment until progession with Xelox + bevacizumab in patients with advanced colorectal cancer[END_REF] presented preliminary data of a randomized phase II trial combining XELOX 2 and bevacizumab every two weeks in ACRC until progression (PD) versus its use for a limited treatment duration followed by reinduction upon PD. These examples of clinical trials underline the need for testing the PK of CCB and its metabolites at a dose which is about 40% higher than the approved single dose of CCB.

In our study c max of CCB occurred earlier than described in the investigators brochure (this brochure reports t max values in the range of 2 hours): 0.49 hours for CCB 1 and merely identical 0.51 hours for CCB 2. The higher dose obviously does not delay the absorption rate of CCB into the blood stream. However, the differences in t max compared to the investigators data remain unclear, probably there are galenic factors leading to a modified release of the drug.

As with other cytotoxic drugs, the inter-patient variability of the pharmacokinetic parameters of CCB and metabolites in this study was high. Coefficients of variance ranged from 36 % to 142 % for plasma concentrations and are likely to be related to the variable expression of enzymes responsible for CCB biotransformation.

CytDA has high activity in hepatocytes, therefore large amounts of DFUR can be measured in plasma after the first liver passage of CCB. Due to this high enzymatic activity, the higher dose of CCB in the dose intensified CCB 2 schedule leads to the corresponding large amounts of the metabolite in the blood. CES in contrary has much lower enzymatic activity, therefore a smaller increase of the plasma concentration of DFCR has been observed in the CCB 2 regimen in our study.

In our study the apparent formation rate of DFUR did not show a high variation of CytDA activity. But recently an over-expression of 180 % of CytDA activity compared to general population values has been reported [START_REF] Mercier | Early severe toxicities after capecitabine intake: possible implication of a cytidine deaminase extensive metabolizer profile[END_REF]. Such a high enzymatic activity may lead to unexpected and severe toxicities. Therefore it could be of clinical importance not only to evaluate phenotyping of TP but also of CytDA in order to improve safety of Xeloda regimens.

The pharmacokinetics of CCB and its metabolites have been evaluated extensively in patients [START_REF] Reigner | Capecitabine; a review of its clinical pharmacokinetics[END_REF][START_REF] Igbal | Capecitabine: the new generation of fluoropyrimidines in colorectal cancer[END_REF][START_REF] Gieschke | Population pharmacokinetics and concentration-effect relationships of capecitabine metabolites in colorectal cancer patients[END_REF][START_REF] Guichard | Gene expression predicts differential capecitabine metabolism, impacting on both pharmacokinetics and antitumour activity[END_REF], but from 500 to 2500 mg/m²/d only. Over this range, the pharmacokinetics of CCB and its metabolite, 5'-DFCR were dose proportional and did not change over time. The increases in the AUCs of 5'-DFUR and 5-FU, however, were greater than proportional to the increase in dose and the AUC of 5-FU was 34% higher on day 14 than on day 1. CCB reached peak blood levels in about 1.5 hours (tmax) with peak DFUR levels occurring slightly later, at 2 hours. In this study, no difference between standard CCB 1 and the dose intensified CCB 2 schedule concerning the fate of CCB and its metabolites in the body was found.

Legends of tables and figures

  table 1

  t max time of peak plasma concentration [min] c max peak plasma concentration [µg/ml] c last last measurable drug concentration [µg/ml] λz terminal elimination-rate [min -1 ] t 1/2 λz half-life of terminal elimination [min] AUC last area under the concentration-time curve from 0 to c last [µg/ml * min] AUC inf area under the concentration-time curve from 0 to infinity [µg/ml * min] Vz volume of distribution [l] Cl tot total body clearance [l/min] MRT last mean-residence time from 0 to c last [min] To obtain the percent AUC amount of metabolite [AUC met ], the AUC last values of each metabolite (5'DFCR, 5'DFUR) were compared with the sum of AUCs as follows: AUC met * 100 AUC met [%] = AUC CCB+DFCR+DFUR An apparent formation-coefficient of the metabolites catalyzed by hCES [R hCES ] or CytDA [R CytDA ], respectively, has been calculated by dividing the metabolite AUC last by its precursors AUC last : AUC DFCR R hCES = AUC CCB AUC DFUR R CytDA = AUC DFCR Descriptive statistics of plasma concentrations and of pharmacokinetic data of all compounds was calculated by use of the scientific softwares Graph Pad Prism 5.0 and InStat 3.0 for Windows (Graph Pad Software, San Diego, CA 92130, USA). Probable statistical outliers in data sets had been verified by the Grubb`s test using the extreme studentized deviate method.

1

 1 figure 1Peak concentrations of CCB and DFCR occurred within 60 min after ingestion of tablets showing that DFCR was formed rapidly from CCB without any delay. DFUR concentrations in plasma were about two times higher than those of CCB while DFCR concentrations were very similar to those of CCB. In both regimens peak concentrations of DFUR occurred about half an hour later; this can be explained by the cascade metabolism of CCB with DFCR as the intermediate metabolite which itself represents the precursor for DFUR formation. After having reached its maximum, plasma concentrations of all compounds declined slowly.

1 .

 1 16 to 1.67) and this factor correlated closely with the dose factor of 1.40 (CCB 2 versus CCB 1). Mean logarithmic plasma concentration decay of the sum of CCB plus metabolites was strongly linear: see insert D in lower part of figure 1. For both schedules slope (k) and coefficient of regression line (R) were very similar: k = -0.095 for CCB 1 (R = 0.961, p = 0.0023) and k = -0.134 for CCB 2 (R = 0.969, p = 0.0015).

Figure 1 :Figure 2 :

 12 Figure 1: Mean plasma concentration time curves in the CCB 1 (A) and CCB 2 (B) schedule, the sum of CCB, DFCR and DFUR plasma concentrations in CCB 1 and CCB 2 (C) and log plasma concentrations of the sum of CCB, DFCR and DFUR (D)

Table 2

 2 AUC last and AUC inf values differed distinctly between the two CCB schedules: + 34 % for CCB, + 20 % for DFCR and + 58 % for DFUR; the latter difference was statistically significant (p < 0.05). These changes in AUC last values correlated very close to the percent changes in c max values with a coefficient of correlation R = 0.966. Percent distribution of

	presents an overview for the non-compartment pharmacokinetic data of CCB, DFCR
	and DFUR obtained in both schedules.
	table 2
	Peak plasma concentrations of DFUR were significantly higher (+ 51%, p < 0.05) in the dose
	intensified CCB 2 schedule, while CCB and DFCR maximum concentrations only differed by
	15 % (CCB) and 10 % (DFCR), respectively. In both schedules, c last values of all three
	compounds were very similar, indicating that distribution, metabolic and elimination
	processes have completed already after 3 hours. The intensified CCB 2 dose obviously does
	not cause an inhibition of these ADME processes.
	The PK parameters t 1/2 λz and MRT of CCB were in the same order of magnitude in both
	regimens. Contrary to CCB, half-lives of DFCR and DFUR showed a great inter-patient
	variability represented by high standard deviations of λz, t 1/2 λz and MRT.

Table 1 :

 1 Applied capecitabine treatment schedules

Table 2 :

 2 Mean pharmacokinetic parameters (+ SD) of CCB, DFCR and DFUR

Table 2 :

 2 Mean pharmacokinetic parameters (+ SD) of CCB, DFCR and DFUR

	parameter	CCB CCB 1	CCB 2	DFCR CCB 1	CCB 2	DFUR CCB 1
	c max	7,57 (4,36)	8,68 (4,71)	6,50 (2,83)	7,05 (3,98)	9,34 (4,07)*	14,
	t max	49 (22)	51 (30)	54 (17)*	64 (33)	69 (37)
	c last	0,43 (0,52)	0,56 (0,76)	0,96 (0,64)	1,01 (0,96)	2,70 (1,83)	2,1
	λz	0,032 (0,014)	0,029 (0,011)	0,015 (0,006)	0,014 (0,007)	0,013 (0,018)	0,018
	t 1/2 λz	26 (11)	28 (12)	42 (24)	61 (39)	63 (60)
	AUC last	507 (325)	681 (511)	532 (191)	638 (273)	793 (318)*	125

Assuming the AUC´s of CCB + DFCR + DFUR as 100%, the percentage of the respective AUC "CCB 2" versus "CCB 1" was nearly identical for CCB 26% versus 27%, for DFCR 26% versus 29%, for DFUR 48% versus 44%. The same effect could be demonstrated for each of 7 concentration time points. Therefore, from the pharmacokinetic point of view, dose escalation of CCB seems feasible as no significant change of its metabolic conversion could be documented. Safety and efficacy of XELOX 2 using a dose intensification of CCB is currently investigated in several clinical trials.


Mihai Bostan
email: mbostan@univ-fcomte.fr

Jos É-Antonio Carrillo
email: carrillo@mat.uab.es

GLOBAL SOLUTIONS FOR THE ONE DIMENSIONAL WATER-BAG MODEL *

Keywords: Vlasov-Maxwell equations, Water-Bag model, Conservation laws Subject classifications 35A05, 78A35, 82D10

In this paper we study a special type of solution for the one dimensional Vlasov-Maxwell equations. We assume that initially the particle density is constant on its support in the phase space and we are looking for solutions with particle density having the same property at any time t > 0. More precisely, for each x the support of the density is assumed to be an interval [p -,p + ] with end-points varying in space and time. We analyze here the case of weak and strong solutions for the effective equations verified by the end-points and the electric field (water-bag model) in the relativistic setting.

Introduction

The Vlasov-Maxwell system governs the evolution of an ensemble of charged particles subject to electro-magnetic fields created by themselves and possibly external sources in which collisions are typically neglected. Given f the density number of charged particles at time t ∈ R + , position x ∈ R 3 and momentum p ∈ R 3 , the dynamics of the particles is described by the Vlasov equation

∂ t f + v(p) • ∇ x f + q(E(t,x) + v(p) ∧ B(t,x)) • ∇ p f = 0, (t,x,p) ∈ R + × R 3 × R 3 , (1.1)
where the electro-magnetic field (E,B) is defined in a self-consistent way by the Maxwell equations

∂ t E -c 2 0 curl x B = - j(t,x) ε 0 , j(t,x) = q R 3 v(p)f (t,x,p) dp, (t,x) ∈ R + × R 3 , (1.2)
∂ t B + curl x E = 0, (t,x) ∈ R + × R 3 , (1.3)
div x E = ρ(t,x) ε 0 , ρ(t,x) = q R 3 f (t,x,p) dp, div x B = 0, (t,x) ∈ R + × R 3 , (1.4)
where q,m are the charge and the mass of the particles, ε 0 is the electric permittivity of the vacuum and v(p) is the relativistic velocity associated to the momentum p

v(p) = p m 1 + |p| 2 m 2 c 2 0 - 1 2
, where c 0 is the light speed in the vacuum. Suitable initial conditions for the particle density and the electro-magnetic field have to be prescribed verifying certain compatibility conditions. The existence of global weak solution was obtained in [START_REF] Diperna | Global weak solutions of the Vlasov-Maxwell system[END_REF] and the existence of strong solutions has been investigated by different approaches in [START_REF] Glassey | Singularity formation in a collisionless plasma could only occur at high velocities[END_REF][START_REF] Bouchut | Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system[END_REF][START_REF] Klainerman | A new approach to study the Vlasov-Maxwell system[END_REF]. Despite of these advances in the existence theory for the Vlasov-Maxwell system, many questions concerning qualitative behavior, special solutions or regularity issues, to name a few, are completely open. Recently global existence and uniqueness results have been proved for reduced models for laser-plasma interaction [START_REF] Carrillo | Global solutions for the one-dimensional Vlasov-Maxwell system for laser-plasma interaction[END_REF][START_REF] Bostan | Mild solutions for the relativistic Vlasov-Maxwell system for laser-plasma interaction[END_REF] leading to particular global solutions of the Vlasov-Maxwell system. Neglecting the magnetic field and the relativistic corrections in the Vlasov equation leads to the Vlasov-Poisson model

∂ t f + p m • ∇ x f + qE(t,x) • ∇ p f = 0, (t,x,p) ∈ R + × R 3 × R 3 , curl x E = 0, div x E = ρ(t,x) ε 0 , (t,x) ∈ R + × R 3 ,
which is much better understood, see [START_REF] Arseneev | Global existence of a weak solution of the Vlasov system of equations[END_REF][START_REF] Pfaffelmoser | Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data[END_REF][START_REF] Lions | Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system[END_REF] for instance. The Vlasov-Poisson model can be justified as the limit of the relativistic Vlasov-Maxwell model when the characteristic speed of the particles remains small compared to the light speed [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] Bostan | Asymptotic behavior of weak solutions for the relativistic Vlasov-Maxwell equations with large light speed[END_REF].

In this work, we elaborate on some particular type of solutions of the onedimensional version of the Vlasov-Maxwell(Poisson) system which has received the attention in the plasma physics community [START_REF] Bertrand | Nonlinear electron plasma oscillations: the "water bag model[END_REF]. Let us assume that the initial density is proportional to the characteristic function of some region of the phase space between the graphs of two functions p ± 0 : R → R

f 0 (x,p) = α 1 {p - 0 (x)<p<p + 0 (x)} , (x,p) ∈ R 2 . (1.5)
We assume that p - 0 ≤ p + 0 . We are looking for a density function of the form

f (t,x,p) = α 1 {p -(t,x)<p<p + (t,x)} , (1.6)
where p ± : R + × R → R are unknown functions to be determined such that the above density f satisfies the Vlasov equation. We have the following immediate result: Proposition 1.1 (Smooth Water-bag Solutions). Let E : [0,T [×R → R be a given electric field which belongs to

L 1 loc ([0,T [×R), with 0 < T ≤ +∞. Assume that p ± : [0,T [×R → R are smooth functions p ± ∈ W 1,∞ loc ([0,T [×R) verifying ∂ t p ± + v(p ± )∂ x p ± = qE(t,x), (t,x) ∈]0,T [×R,
and p -≤ p + . Then the density f given by (1.6) is a weak solution (that is, a solution in distribution sense) of the Vlasov equation associated to the electric field E.

Observe that the charge and current densities of the distribution in (1.6) are given by ρ(t,x) = qα (p

+ (t,x) -p -(t,x)), j(t,x) = qα(E(p + (t,x)) -E(p -(t,x)))
, where the kinetic energy function is given by

E(p) = mc 2 0 1 + p 2 m 2 c 2 0 1 2 -1 .
Notice that we have E (p) = v(p). Thus for the initial condition in (1.5) the one dimensional Vlasov-Maxwell equations reduce to the system

∂ t p ± + ∂ x E(p ± ) = qE(t,x), (t,x) ∈]0,T [×R, (1.7)
∂ t E = -α q ε 0 (E(p + (t,x)) -E(p -(t,x))), (1.8)
∂ x E = α q ε 0 (p + (t,x) -p -(t,x)), (t,x) ∈]0,T [×R, (1.9)
with the initial conditions

p ± (0,x) = p ± 0 (x), E(0,x) = E 0 (x), x ∈ R, (1.10)
satisfying

E 0 (x) = α q ε 0 (p + 0 (x) -p - 0 (x)), p - 0 (x) ≤ p + 0 (x), x ∈ R. (1.11)
Let us remark that (1.9) is a consequence of (1.7), (1.8) and the equality in (1.11).

The problem (1.7), (1.8), (1.9), (1.10) is called the water-bag model and has been introduced in [START_REF] Bertrand | Nonlinear electron plasma oscillations: the "water bag model[END_REF]. The idea is to reduce the Vlasov equation to a set of hydrodynamic equations while keeping its kinetic character. Besides the transport of charged particles, such models arise in various domains. In [START_REF] Brenier | A kinetic formulation for multi-branch entropy solutions of scalar conservation laws[END_REF] the inviscid Burgers equation is reduced to a closed system of moment equations, using a suitable concept of entropy multivalued solution. Similar techniques for the reconstruction of a function from a finite number of moments apply in geometric optics computations [START_REF] Gosse | Using K-branch entropy solutions for multivalued geometric optics computations[END_REF], [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF]. The main goal of this paper is to establish existence and uniqueness results for the water-bag model. In Section 2 we analyze the weak solutions: we study entropy solutions of the scalar conservation laws (1.7) coupled to the equations (1.8), (1.9) for the electric field. Smooth solutions are constructed as well for certain class of initial conditions in Section 3.

Weak solutions

For simplicity we assume that all the physical constants q,m,ε 0 ,c 0 ,α are equal to the unity. We remind the reader the standard existence and uniqueness results concerning the entropy solution for scalar conservation laws. We refer to [START_REF] Godlewski | Hyperbolic systems of conservation laws, Mathématiques et Applications[END_REF][START_REF] Evans | Partial Differential Equations[END_REF] for details on this topic. We consider here conservation laws with right hand side terms

∂ t u + ∂ x F (u) = G(t,x), (t,x) ∈ R + × R, (2.1)
u(0,x) = u 0 (x), x ∈ R. (2.2)
Theorem 2.1 (Entropy Solutions for Scalar Conservation Laws). Let us assume that F : R → R is a smooth function and G belongs to

L 1 loc (R + ;L ∞ (R)). Then for any initial condition u 0 ∈ L ∞ (R) there is a unique entropy solution u ∈ C(R + ;L 1 loc (R)) ∩ L ∞ loc (R + ;L ∞ (R)) for (2.1), (2.2) satisfying u(t) L ∞ (R) ≤ u 0 L ∞ (R) + t 0 G(s) L ∞ (R) ds, t ∈ R + . (2.3)
Moreover if v is the entropy solution associated to the initial condition v 0 ∈ L ∞ (R), the source term H ∈ L 1 loc (R + ;L ∞ (R)
) and the same smooth function F then we have the inequality

R |u(t,x) -v(t,x)|1 {|x|<R} dx ≤ R |u 0 (x) -v 0 (x)|1 {|x|<R+tM (t)} dx (2.4) + t 0 R |G(s,x) -H(s,x)|1 {|x|<R+(t-s)M (t)} dx ds,
where

M (t) = max{M u (t),M v (t)}, M u (t) = sup{|F (ξ)| : |ξ| ≤ u 0 L ∞ (R) + t 0 G(s) L ∞ (R) ds} and M v (t) = sup{|F (ξ)| : |ξ| ≤ v 0 L ∞ (R) + t 0 H(s) L ∞ (R) ds}.
If u 0 ∈ BV (R) and G ∈ L 1 loc (R + ;BV (R)) then the entropy solution has bounded variation and its total variation verifies

T V (u(t)) ≤ T V (u 0 ) + t 0 T V (G(s)) ds, t ∈ R + .
(2.5)

Furthermore, for any t,R > 0 we have

R |u(t,x) -u 0 (x)|1 {|x|<R} dx ≤ tM u (t) T V (u 0 ) + t 0 T V (G(s)) ds + 2R t 0 G(s) L ∞ (R) ds. (2.6)
It is well known that for conservation laws without source term (G = 0) the solution operator

S(t)u 0 = u(t,•) is order preserving on L 1 (R) ∩ L ∞ (R) that is, for any u 0 ,v 0 ∈ L 1 (R) ∩ L ∞ (R) such that u 0 ≤ v 0 a.e.
we have S(t)u 0 ≤ S(t)v 0 a.e., for any t ∈ R + . This is a direct consequence of the Crandall-Tartar lemma [13, page 81]. The same result holds true for conservation laws with source terms

G ∈ L 1 loc (R + ;L ∞ (R)) and for initial conditions u 0 ∈ L ∞ (R).
Lemma 2.1 (Comparison Principle with Sources). Assume that the source G ∈ L 1 loc (R + ;L ∞ (R)) and denote by S G (t) : L ∞ (R) → L ∞ (R) the solution operator given by S G (t)u 0 = u(t,•) for any u 0 ∈ L ∞ (R), t ∈ R + where u is the entropy solution of (2.1), (2.2). For any t ∈ R + the operator S G (t) is order preserving.

Proof. Since the solutions of (2.1) with bounded initial conditions propagate with finite speed (cf. (2.3), (2.4)), it is sufficient to prove the result for

G ∈ L 1 loc (R + ;L 1 (R)) ∩ L 1 loc (R + ;L ∞ (R)) and initial conditions in L ∞ (R) ∩ L 1 (R). There- fore consider u 0 ,v 0 ∈ L ∞ (R) ∩ L 1 (R) such that u 0 ≤ v 0 . We claim that R {S G (t)u 0 -S G (t)v 0 } dx = R {u 0 -v 0 } dx, t ∈ R + .
(2.7) Indeed, by (2.4) it is sufficient to prove it for compactly supported functions u 0 ,v 0 and this comes easily by interpreting S G (t)u 0 ,S G (t)v 0 as the limit of smooth solutions for approximating viscous problems, as the viscosity vanishes. We denote by (•)

+ the positive part function. Combining (2.4), (2.7) yields 2 R (S G (t)u 0 -S G (t)v 0 ) + dx= R (S G (t)u 0 -S G (t)v 0 ) dx + R |S G (t)u 0 -S G (t)v 0 | dx ≤ R (u 0 -v 0 ) dx + R |u 0 -v 0 | dx = 2 R (u 0 -v 0 ) + dx = 0, implying that S G (t)u 0 ≤ S G (t)v 0 a.e. x ∈ R, ∀ t ∈ R + . Consider p ± 0 ∈ L ∞ (R),E 0 ∈ L ∞ (R) satisfying p - 0 ≤ p + 0 and E 0 = p + 0 -p - 0 . We define the application F on L 1 loc (R + ;L ∞ (R))
given by FE = Ẽ where

Ẽ(t,x) = E 0 (x) - t 0 {E(p + (s,x)) -E(p -(s,x))} ds,
and p ± are the entropy solutions of

∂ t p ± + ∂ x E(p ± ) = E(t,x), (t,x) ∈ R + × R,
with the initial conditions p ± 0 . It is easily seen by ( 2.3) that

Ẽ(t) L ∞ (R) ≤ E 0 L ∞ (R) + t 0 { p + (s) L ∞ (R) + p -(s) L ∞ (R) } ds (2.8) ≤ E 0 L ∞ (R) + t( p + 0 L ∞ (R) + p - 0 L ∞ (R) ) + 2t t 0 E(s) L ∞ (R) ds.
For any t ∈ R + we denote by e T : [0,T ] → R the function given by

e T (t) = E 0 L ∞ (R) + T ( p + 0 L ∞ (R) + p - 0 L ∞ (R) ) e 2T t .
We check immediately that the set

D T = {E ∈ L 1 (]0,T [;L ∞ (R)) : E(t) L ∞ (R) ≤ e T (t), ∀ t ∈ [0,T ]} is left invariant by the application F T defined by F T E = FE| [0,T ]×R for any E ∈ L 1 loc (R + ;L ∞ (R)). A straightforward computation based on the contraction inequality (2.4) shows that F T is continuous on C([0,T ];L 1 loc (R)).
We denote by M T the constant given by

M T = sup{|E (ξ)| : |ξ| ≤ max{ p - 0 L ∞ (R) , p + 0 L ∞ (R) } + T 0 e T (t) dt} < 1.
Proposition 2.2 (Continuity of the Map). Assume that p ± 0 ,E 0 ∈ L ∞ (R). For any T ∈ R + we have the inequality

R |F T E 1 -F T E 2 |(t,x)1 {|x|<R} dx ≤ 2T t 0 R |E 1 -E 2 |(s,x)1 {|x|<R+(t-s)M T } dx ds, for any E 1 ,E 2 ∈ D T , ∀ t ∈ [0,T ], R > 0.
Proof. Consider E 1 ,E 2 ∈ D T and let us denote by p ± 1 , p ± 2 the entropy solutions corresponding to the fields E 1 ,E 2 and the initial conditions p ± 0 . By the definitions of

F T E 1 , F T E 2 and (2.4) we deduce easily that R -R |(F T E 1 -F T E 2 )|(t,x) dx ≤ t 0 R -R {|p + 1 -p + 2 |(s,x) + |p - 1 -p - 2 |(s,x)} ds ≤ 2 t 0 s 0 R |E 1 -E 2 |(τ,x)1 {|x|<R+(s-τ )M T } dxdτds ≤ 2T t 0 R |E 1 -E 2 |(s,x)1 {|x|<R+(t-s)M T } dx ds. Theorem 2.3 (Global Entropy Solutions for the Water-bag model). Assume that p ± 0 ,E 0 ∈ L ∞ (R) satisfying E 0 = p + 0 -p - 0 .
Then there is a global unique weak solution (p (1.8), (1.9), (1.10). Moreover if p - 0 ≤ p + 0 then p -≤ p + . Proof. It is sufficient to prove the existence of a unique solution (p + ,p -,E) on [0,T ] × R for any T ∈ R + .

+ ,p -,E) ∈ L ∞ (]0,T [×R) 2 × W 1,∞ (]0,T [×R), ∀ T ∈ R + for the water-bag model (1.7),
We define the sequence (E n ) n≥0 given by

E 0 (t,x) = E 0 (x),∀ (t,x) ∈ [0,T ] × R and E n+1 = F T E n , ∀ n ∈ N. Observe that (E n ) n ⊂ D T . For any R > 0 we consider the sequence of functions z n R (t) = R |E n+1 - E n |(t,x)1 {|x|<R+(T -t)M T } dx, t ∈ [0,T ], n ∈ N. By Proposition 2.2 it is easily seen that z n R (t) ≤ 2T t 0 R |E n -E n-1 |(s,x)1 {|x|<R+(T -s)M T } dx ds = 2T t 0 z n-1 R (s) ds, t ∈ [0,T ], n ≥ 1, implying that z n R (t) ≤ (2T t) n n! z 0 R L ∞ (]0,T [) , ∀n ∈ N. We deduce that (E n ) n is a Cauchy sequence in C([0,T ];L 1 loc (R)) since R |E n+p -E n |(t,x)1 {|x|<R} dx ≤ z n (t) + z n+1 (t) + ... + z n+p-1 (t). It follows that (E n ) n converges in C([0,T ];L 1 loc (R)
) towards a fixed point E of F T . Moreover we check easily that E ∈ D T . Take now p + ,p -the unique entropy solutions of (1.7) corresponding to the limit field E and the initial conditions p + 0 ,p - 0 . By construction (p + ,p -,E) is a solution for the water-bag model (1.7), (1.8), (1.10). The equation (1.9) is a consequence of (1.7), (1.8) and the constraint E 0 = p + 0 -p - 0 . The bounds for the derivatives of E comes from the bounds of p ± (see (2.3)) and (1.8), (1.9). For the inequality p -≤ p + use Lemma 2.1. The uniqueness of the weak solution is obtained by a straightforward computation involving the Gronwall lemma.

Remark 2.1 (Vlasov-Maxwell Solutions with Defect Measures). A natural question related to the previous existence result is the following: given (p + ,p -,E) a weak solution for the water-bag model, is it true that f (t,x,p) = 1 {p -(t,x)<p<p + (t,x)} solves the Vlasov equation

∂ t f + v(p)∂ x f + E(t,x)∂ p f = 0, (t,x,p) ∈ R + × R × R ?
Generally the answer to this question is negative, but we can prove that f solves a Vlasov equation with a entropy defect measure. Of course we appeal here to the kinetic formulation of conservation laws [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF][START_REF] Perthame | Kinetic formulation of conservation laws[END_REF]. Indeed, observe that the function f represents as f (t,x,p) = χ(p,p + (t,x)) -χ(p,p -(t,x)) where the function χ is given by

χ(ξ,u) =      + 1, 0 < ξ < u, -1, u < ξ < 0, 0, otherwise.
Since p ± are entropy solutions we know that there are the non negative kinetic entropy defect measures m ± such that

∂ t χ(p,p ± ) + v(p)∂ x χ(p,p ± ) -E(t,x)δ 0 (p -p ± ) = ∂ p m ± , (t,x,p) ∈ R + × R 2 , χ(p,p ± (0,x)) = χ(p,p ± 0 (x)), (x,p) ∈ R 2 ,
where the notation δ 0 stands for the Dirac mass concentrated at the origin. Therefore we obtain

(∂ t + v(p)∂ x ){χ(p,p + ) -χ(p,p -)} -E(t,x){δ 0 (p -p + ) -δ 0 (p -p -)} = ∂ p {m + -m -},
and by taking into account that ∂ p {χ(p,p + ) -χ(p,p -)} = -{δ 0 (p -p + ) -δ 0 (p -p -)} finally we can write

∂ t f + v(p)∂ x f + E(t,x)∂ p f = ∂ p {m + -m -}, (t,x,p) ∈ R + × R 2 , f (0,x,p) = χ(p,p + 0 (x)) -χ(p,p - 0 (x)) = 1 {p - 0 (x)<p<p + 0 (x)} , (x,p) ∈ R 2 .
Remark 2.2 (Total Energy Balance). Another interesting question concerns the behavior of the total energy given by

W (t) = R 2 E(p)f dxdp + 1 2 R E(t,x) 2 dx, t ∈ R + .
For example if p -(t,x) ≤ 0 ≤ p + (t,x), (t,x) ∈ [0,T ] × R and the initial energy is finite we can prove that the total energy is not increasing on [0,T ]. Multiplying the above Vlasov equation by E(p) one gets after integration

d dt R 2 E(p)f dxdp - R E{E(p + (t,x)) -E(p -(t,x))} dx = R 2 v(p)m -(t,x,p) dxdp - R 2 v(p)m + (t,x,p) dxdp.
Using (1.8) we deduce also that

1 2 d dt R E(t,x) 2 dx + R E(t,x){E(p + (t,x)) -E(p -(t,x))} dx = 0, implying that d dt R 2 E(p)f dxdp + 1 2 R E(t,x) 2 dx = R 2 v(p)m -(t,x,p) dxdp - R 2 v(p)m + (t,x,p) dxdp.
Therefore we are done if we check that m ± = 0 on [0,T ] × R × R ∓ . Take p 0 < 0 and let us multiply the kinetic formulation of χ(p,p + ) by the derivative of the convex function S p0 (p) = (p -p 0 ) -. After standard computations involving the usual formula

R χ(ξ,u)S (ξ) dξ = S(u) -S(0), ∀ S(•), ∀ u ∈ R we obtain d dt R S p0 (p + (t,x)) dx = - R m + (t,x,p 0 ) dx.
Therefore one gets for any t ∈ [0,T ]

R S p0 (p + (T,x)) dx + T 0 R m + (t,x,p 0 ) dxdt = R S p0 (p + 0 (x)) dx = 0, saying that m + = 0 on [0,T ] × R × R -.
In a similar way we check that m -= 0 on [0,T ] × R × R + .

Remark 2.3 (Non-relativistic setting). This case is a little bit more difficult since the non relativistic energy function E(p) = p 2 2 is only locally Lipschitz. This time the analogous of the estimate (2.8) becomes quadratic

Ẽ(t) L ∞≤ E 0 L ∞ + 1 2 t 0 max{ p + (s) 2 L ∞ , p -(s) 2 L ∞ } ds ≤ E 0 L ∞ + t 0 max{ p + 0 2 L ∞ , p - 0 2 L ∞ } + s s 0 E(τ ) 2 L ∞ dτ ds ≤ E 0 L ∞ + tmax{ p + 0 2 L ∞ , p - 0 2 L ∞ } + t 2 t 0 E(s) 2 L ∞ ds.
In this case for T > 0 small enough we denote by e T (•) the unique solution of

d dt e T = T 2 (e T (t)) 2 , 0 < t < T,
with the initial condition e T (0

) = E 0 L ∞ + T max{ p + 0 2 L ∞ , p - 0 2 L ∞ }. It is easily seen that the set D T = {E ∈ L 1 (]0,T [;L ∞ (R)) : E(t) L ∞ ≤ e T (t), ∀ t ∈ [0,T ]} is left invari-
ant by the application F T and following the same arguments as in the relativistic setting we construct a local unique weak solution (p

+ ,p -,E) ∈ L ∞ (]0,T [×R) 2 × W 1,∞ (]0,T [×R)
for the non relativistic water-bag model. For results on the multiwater-bag model in this setting see [START_REF] Besse | The multi-water-bag model for collisionless kinetic equations[END_REF].

Strong solutions

This section is devoted to the analysis of smooth solutions for the relativistic water-bag model. We show that smooth non decreasing initial conditions launch global smooth solutions.

Proposition 3.1 (Non-decreasing initial data for Scalar CL's). Assume that

F ∈ W 2,∞ (R), G ∈ L ∞ loc (R + ;W 1,∞ (R)) such that F ≥ 0, ∂ x G ≥ 0.
Then for any non decreasing initial condition u 0 ∈ W 1,∞ (R) the problem (2.1), (2.2) has a unique strong solution u ∈ W 1,∞ (]0,T [×R), ∀ T ∈ R + which is non decreasing with respect to x.

Proof. We define the sequence of functions (u n ) n≥0 where u 0 (t,x) = u 0 (x) ∀ (t,x) ∈ R + × R and for any n ∈ N, u n+1 solves the problem

∂ t u n+1 + F (u n (t,x))∂ x u n+1 = G(t,x), (t,x) ∈ R + × R, (3.1)
u n+1 (0,x) = u 0 (x), x ∈ R. (3.2)
Actually we will prove that (u n ) n are smooth and therefore the above problem is understood in the classical sense. Assume that u n belongs to L ∞ loc (R + ;W 1,∞ (R)), ∂ x u n ≥ 0 which is true for n = 0, and let us show that the same holds for u n+1 . We denote by X n (s;t,x) the characteristics associated to

F (u n ) d ds X n (s;t,x) = F (u n (s,X n (s;t,x))), X n (t;t,x) = x.
Therefore we have

u n+1 (t,x) = u 0 (X n (0;t,x)) + t 0 G(s,X n (s;t,x)) ds, (t,x) ∈ R + × R. (3.3)
We check easily that u n+1 ∈ L ∞ loc (R + ;W 1,∞ (R)) and since x → X n (s;t,x), u 0 and G(s,•) are non decreasing we deduce that ∂ x u n+1 ≥ 0. Moreover we can find bounds for the time and space derivatives uniformly with respect to n. For any h > 0 we have

∂ t {u n+1 (t,x + h) -u n+1 (t,x)}+{F (u n (t,x + h)) -F (u n (t,x))}∂ x u n+1 (t,x + h) + F (u n (t,x))∂ x {u n+1 (t,x + h) -u n+1 (t,x)} = G(t,x + h) -G(t,x). Since ∂ x u n ≥ 0, ∂ x u n+1 ≥ 0, F ≥ 0 we have {F (u n (t,x + h)) -F (u n (t,x))}∂ x u n+1 (t,x + h) ≥ 0,
and therefore

∂ t D h u n+1 + F (u n (t,x))∂ x D h u n+1 ≤ D h G(t,x),
where the notation D h z(x) stands for z(x + h) -z(x) for any function z. Integrating along the characteristics one gets

D h u n+1 (t,x) ≤ (D h u 0 )(X n (0;t,x)) + t 0 D h G(s,X n (s;t,x)) ds, implying that D h u n+1 (t,x) h ≤ u 0 L ∞ (R) + t 0 ∂ x G(s) L ∞ (R) ds.
Since we know that ∂ x u n+1 ≥ 0 finally we obtain

∂ x u n+1 (t) L ∞ (R) ≤ u 0 L ∞ (R) + t 0 ∂ x G(s) L ∞ (R) ds, and
∂ t u n+1 (t) L ∞ (R) ≤ G(t) L ∞ (R) + F L ∞ (R) u 0 L ∞ (R) + t 0 ∂ x G(s) L ∞ (R) ds .
We claim that the sequence (

u n ) n converges in C([0,T ] × R), ∀ T ∈ R + . Indeed, since (∂ x u n ) n is bounded in L ∞ (]0,T [×R), there is a constant C T depending on u 0 L ∞ (R) , T 0 ∂ x G(s) L ∞ (R) ds and F L ∞ such that |X n+1 (s;t,x) -X n (s;t,x)| ≤ C T t s u n+1 (τ ) -u n (τ ) L ∞ dτ, (3.4)
for any (s,t,x)

∈ [0,T ] 2 × R. Combining (3.3), (3.4) yields u n+2 (t) -u n+1 (t) L ∞ (R) ≤ CT t 0 u n+1 (s) -u n (s) L ∞ (R) ds, n ∈ N, implying that the sequence (u n ) n converges in C([0,T ] × R) towards some function u. Since (∂ t u n ) n , (∂ x u n ) n are bounded we deduce that u ∈ W 1,∞ (]0,T [×R). It remains to prove that u solves (3.1), (3.2). There is a subsequence (n k ) k , lim k→+∞ n k = +∞ such that lim k→+∞ (∂ t u n k ,∂ x u n k ) = (∂ t u,∂ x u), weakly in L ∞ (]0,T [×R) 2 .
Obviously we have also the convergence lim

k→+∞ u n k -1 = u in C([0,T ] × R). Multi- plying (3.1) by a test function ϕ ∈ C 0 c ([0,T ] × R) one gets T 0 R ∂ t u n k ϕ dxdt + T 0 R F (u n k -1 (t,x))∂ x u n k ϕ dxdt = T 0 R G(t,x)ϕ(t,x) dxdt.
We can pass easily to the limit for k → +∞ and we obtain

T 0 R ∂ t u ϕ dxdt + T 0 R F (u(t,x))∂ x u ϕ dxdt = T 0 R G(t,x)ϕ(t,x) dxdt,
saying that u is a strong solution of (3.1). Moreover u verifies the initial condition (3.2) since

u(0,x) = lim n→+∞ u n (0,x) = u 0 (x), x ∈ R.
Since any strong solution coincides with the entropy solution, we have also the uniqueness of the strong solution. Theorem 3.2 (Global Smooth Solutions). Assume that

p ± 0 ,E 0 ∈ W 1,∞ (R) sat- isfying d dx p ± 0 ≥ 0, d dx E 0 = p + 0 -p - 0 ≥ 0.
Then there is a global unique strong solution

(p + ,p -,E) ∈ W 1,∞ (]0,T [×R) 2 × W 2,∞ (]0,T [×R), ∀ T ∈ R + for the water-bag model.
Proof. By Theorem 2.3 we know that there is a global weak solution (p

+ ,p -,E) ∈ L ∞ (]0,T [×R) 2 × W 1,∞ (]0,T [×R), ∀ T ∈ R + for
the water-bag model satisfying p -≤ p + . By (1.9) we have ∂ x E ≥ 0 and by the definition the energy function E is convex. Thus applying Proposition 3.1 implies that the entropy solutions p ± belong to W 1,∞ (]0,T [×R) and are strong solutions for (1.7). The bounds for the second order derivatives of the electric fields follow immediately from the bounds of the first order derivatives for p ± and (1.8), (1.9). The uniqueness of the strong solution (p + ,p -,E) for the water-bag model is a direct consequence of the uniqueness of the weak solution. The previous theorem states the existence of smooth solutions (p ± ,E) for the water-bag model, when the initial conditions (p ± 0 ,E 0 ) are non decreasing Lipschitz continuous functions. In this case we know by Proposition 1.1 that the density f (t,x,p) = 1 {p -(t,x)<p<p + (t,x)} is a weak solution of the Vlasov equation and therefore (f,E) solves the Vlasov-Maxwell problem corresponding to the initial conditions f 0 (x,p) = 1 {p - 0 (x)<p<p + 0 (x)} and E 0 . An interesting question is what happens when the initial conditions p ± 0 are not smooth. Is it true that the solutions of the Vlasov-Maxwell system and water-bag model satisfy f (t,x,p) = 1 {p -(t,x)<p<p + (t,x)} ? The next theorem gives a partial affirmative answer to this question at least for non decreasing initial conditions. Theorem 3.3 (Weak Water-Bag Solutions). Assume that p ± 0 ,E 0 ∈ L ∞ (R) are non decreasing and satisfy

E 0 = p + 0 -p - 0 ∈ L ∞ (R).
We denote by f 0 : R 2 → R the function f 0 (x,p) = 1 {p - 0 (x)<p<p + 0 (x)} , (x,p) ∈ R 2 . Then there is a global unique weak solution (f,E) ∈ L ∞ (]0,T [×R x ;L 1 (R p )) × W 1,∞ (]0,T [×R), ∀ T ∈ R + for the Vlasov-Maxwell system corresponding to the initial conditions (f 0 ,E 0 ). Moreover we have f (t,x,p) = 1 {p -(t,x)<p<p + (t,x)} where p ± are weak solutions for the water-bag model (1.7), (1.8), (1.9), (1.10).

Proof. We have f 0 ∈ L ∞ (R x ;L 1 (R p )) and E 0 ∈ W 1,∞ (R) and thus there is a unique weak solution (f,E) ∈ L ∞ (]0,T [×R x ;L 1 (R p )) × W 1,∞ (]0,T [×R), ∀ T ∈ R + for the Vlasov-Maxwell system. The difficult point is to check that the density f remains of the form 1 {p -(t,x)<p<p + (t,x)} , where p ± solve weakly the conservation laws (1.7). Since this holds true for smooth initial conditions, let us proceed by regularization. For any ε > 0 consider p + 0,ε ,p - 0,ε ,E 0,ε ∈ W 1,∞ (R), non decreasing satisfying

E 0,ε = p + 0,ε -p - 0,ε , p + 0,ε L ∞ ≤ p + 0 L ∞ , p - 0,ε L ∞ ≤ p - 0 L ∞ , E 0,ε W 1,∞ ≤ E 0 W 1,∞ . lim ε 0 (p + 0,ε ,p - 0,ε ) = (p + 0 ,p - 0 ) in L 1 loc (R) 2 , lim ε 0 E 0,ε = E 0 in C 0 (R).
As before we know that there is a global unique weak solution for the Vlasov-

Maxwell system (f ε ,E ε ) ∈ L ∞ (]0,T [×R x ;L 1 (R p )) × W 1,∞ (]0,T [×R), ∀ T ∈ R + associ- ated to the initial conditions f 0,ε (x,p) = 1 {p - 0,ε (x)<p<p + 0,ε (x)} and E 0,ε . By Theorem 3.2 it exists also a global unique strong solution (p + ε ,p - ε , Ẽε ) ∈ W 1,∞ (]0,T [×R) 2 × W 2,∞ (]0,T [×R), ∀ T ∈ R + for the water-bag model corresponding to the initial condi- tions (p + 0,ε ,p - 0,ε ,E 0,ε ). Proposition 1.1 implies that 1 {p - ε (t,x)<p<p + ε (t,x
)} solves weakly the Vlasov equation associated to the electric field Ẽε . Combining with the uniqueness of the weak solution for the Vlasov-Maxwell system yields the equalities

f ε (t,x,p) = 1 {p - ε (t,x)<p<p + ε (t,x)} , E ε (t,x) = Ẽε (t,x). We claim that (p - ε ) ε>0 ,(p + ε ) ε>0 are relatively compact in L 1 loc (]0,T [×R) and (f ε ) ε>0 is relatively compact in L 1 loc (]0,T [×R x ;L 1 (R p ))
. As p - ε ,p + ε are smooth, the considerations for entropy solutions apply and thus we deduce the uniform estimates with respect to ε > 0

E ε (t) L ∞ ≤ E 0,ε L ∞ + T ( p + 0,ε L ∞ + p - 0,ε L ∞ ) e 2T t ≤ E 0 L ∞ + T ( p + 0 L ∞ + p - 0 L ∞ ) e 2T t = e T (t) p ± ε (t) L ∞ ≤ p ± 0,ε L ∞ + t 0 E ε (s) L ∞ ds ≤ p ± 0 L ∞ + t 0 e T (s) ds max{ ∂ t E ε (t) L ∞ , ∂ x E ε (t) L ∞ } ≤ p - ε (t) L ∞ + p + ε (t) L ∞ ≤ p - 0 L ∞ + p + 0 L ∞ + 2
T 0 e T (s) ds.

By the contraction property (2.4) we have for any h,R > 0 Since p ± 0,ε are non decreasing and bounded, they have bounded variation

T V (p ± 0,ε ) ≤ 2 p ± 0,ε L ∞ ≤ 2 p ± 0 L ∞ .
Similarly, since E ε (t,•) is non decreasing and bounded we deduce that Combining (3.5), (3.6), (3.7) and taking into account that sup ε>0 E ε W 1,∞ (]0,T [×R) < +∞, it is easily seen that (p ± ε ) ε>0 are relatively compact in L 1 loc (]0,T [×R). Therefore, up to a sequence extraction, we have lim ε 0 p ± ε = p ± in L 1 loc (]0,T [×R) for some bounded non decreasing functions p ± . We denote by f the density f (t,x,p) = 1 {p -(t,x)<p<p + (t,x)} . Obviously we have implying that lim ε 0 f ε = f in L 1 loc (]0,T [×R x ;L 1 (R p )) (up to the same sequence extraction). Moreover, performing eventually a new extraction, we deduce by Arzela-Ascoli theorem that (E ε ) ε>0 converges uniformly on compact sets towards some electric field E ∈ W 1,∞ (]0,T [×R). Combining all these convergences it is easily seen that the limit functions (p ± ,f = 1 {p -<p<p + } ),E) solve weakly both the water-bag model and the Vlasov-Maxwell system. Therefore we have proved that the unique weak solution of the Vlasov-Maxwell system is (f = 1 {p -<p<p + } ,E) where (p ± ,E) is a weak solution for the water-bag model.

T V (E ε (t,•)) ≤ 2 E ε (t) L ∞ ≤

R

  |p ± ε (t,x + h) -p ± ε (t,x)|1 {|x|<R} dx≤ R |p ± 0,ε (x + h) -p ± 0,ε (x)|1 {|x|<R+t} dx (|E ε (s,x + h) -E ε (s,x)|1 {|x|<R+t-s} dxds.

R

  |f ε -f | dp = R |1 {p - ε <p} 1 {p<p + ε } -1 {p -<p} 1 {p<p + } | dp ≤ R {|1 {p - ε <p} -1 {p -<p} | + |1 {p<p + ε } -1 {p<p + } |} dp = |p - ε (t,x) -p -(t,x)| + |p + ε (t,x) -p + (t,x)|

  2e T (t), t ∈ [0,T ], ε > 0

	and therefore by (2.5), (2.6) one gets
				t
	sup ε>0	T V (p ± ε (t)) ≤ sup ε>0 {T V (p ± 0,ε ) +	0	T V (E ε (s)) ds} < +∞
	and
					h
	|p ± ε (h,x) -p ± 0,ε (x)|1 {|x|<R} dx ≤ h T V (p ± 0,ε ) +	T V (E ε (s)) ds
	R				0
				h
		+ 2R		E ε (s) L ∞ ds.	(3.6)
				0
	Applying one more time the contraction property (2.4) we obtain
		|p ± ε (h,x) -p ± 0,ε (x)|1 {|x|<R+t} dx	(3.7)
		R
		t
		+	|E
		0 R

R |p ± ε (t + h,x) -p ± ε (t,x)|1 {|x|<R} dx≤ ε (s + h,x) -E ε (s,x)|1 {|x|<R+t-s} dxds.

Acknowledgement. JAC is partially supported by the project MTM2008-06349-C03-03 from DGI-MICINN (Spain) and 2005SGR00611.