on the global well-posedness of discrete boltzmann systems with chemical reaction

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2005; 28:1491–1506Published online 26 April 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mma.631MOS subject classi�cation: 76 P 05; 35A 05; 35 I 50

On the global well-posedness of discrete Boltzmannsystems with chemical reaction

Filipe Oliveira∗;† and A. J. Soares‡

Centro de Matem�atica; University of Minho; Campus of Gualtar; 4710-057 Braga; Portugal

Communicated by J. Banasiak

SUMMARY

We study an initial-boundary value problem in one-space dimension for the discrete Boltzmann equationextended to a diatomic gas undergoing both elastic multiple collisions and chemical reactions. Byintegration of conservation equations, we prove a global existence result in the half-space for smallinitial data N0 ∈B1

+ ∩L1. Copyright ? 2005 John Wiley & Sons, Ltd.

KEY WORDS: discrete Boltzmann equation; chemically reacting gases; hyperbolic PDEs

1. INTRODUCTION

In recent years, increasing attention has been devoted to Boltzmann-type models for gaseswith inelastic molecular interactions, due to the presence of chemically reacting mixtures inseveral applications as combustion engines, chemical reactors and plasma physics. Let usquote, for example, References [1–3] in the context of the full Boltzmann equation (BE),and References [4–7] in the context of the discrete Boltzmann equation (DBE). This discretemodel consists of a system of hyperbolic semi-linear partial di�erential equations describ-ing the time-space evolution of a gas whose particles can attain a �nite number of prese-lected velocities only [8]. For what concerns the initial-boundary value problem within theDBE, a rather vast bibliography is available in the context of inert gases (see for instanceReferences [9–11]). However, only few results are available in the literature concerning re-acting gases. In this context, in Reference [12], an existence result is proven for a particularmodel concerning a chemically active gas [5].

∗Correspondence to: Filipe Oliveira, Departamento de Matem�atica, Faculdade de Ciencias e Tecnologia,Universidade Nova de Lisboa, Monte da Caparica, Portugal.

†E-mail: [email protected]‡E-mail: [email protected]

Copyright ? 2005 John Wiley & Sons, Ltd. Received 31 August 2004

1492 F. OLIVEIRA AND A. J. SOARES

As pointed out in Reference [13], in spite of the simple mathematical structure of the DBE,the developments of the last decades have shown that the results on existence and uniquenessof solutions are, generally, weaker than the corresponding ones for the BE. Nevertheless, whendealing with �uid dynamical applications and chemistry and physics of moderately densegases, models of DBE provide an accurate description of the �ow pattern, quite suitable fornumerical implementations. This justi�es the renewed interest in models of DBE extended tochemically reacting gases, several years after the pioneering papers of Broadwell [14] andCabannes [15] that constituted a breakthrough in the study of the propagation of shock waveswithin one component inert gases.In fact, due to the simple structure of the discrete collision operator and the increasing

computer capacities, several numerical simulations have been implemented with the aim ofstudying rather complex processes. In particular, in Reference [16], the steady detonationproblem has been solved by means of the kinetic model of Reference [6] for gases undergo-ing autocatalytic reversible reactions. In this model, the particles can experience binary andtriple elastic collisions and chemical reactions. The associated initial-value problem has beensuccessfully studied from a numerical point of view in Reference [17], although no existencetheorem is available.In the present paper, we prove the global well-posedness of the initial-boundary value

problem in one-space dimension for the discrete model of Reference [6].The inclusion of multiple collisions with chemical reactions represents an additional di�-

culty, since cubic terms appear in the collision operator and the conservation laws di�er fromthe ones of classical inert gases.We continue this introduction by presenting the model.

1.1. Kinetic equations⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

@@t

NAi (x; t) + vi

@@x

NAi (x; t) =FA

i (N (x; t)) (i∈ {1; 2; 3})

@@t

NA2i (x; t) +

vi2

@@x

NA2i (x; t)=FA2

i (N (x; t)) (i∈ {1; 2; 3})

@@t

NA∗i (x; t) + vi

@@x

NA∗i (x; t)=FA∗

i (N (x; t)) (i∈ {1; 3})

(1)

where (x; t)∈ [0;+∞[×[0;+∞[, and the other terms have the following meaning:• N =(NA

1 ; NA2 ; N

A3 ; N

A∗1 ; NA∗

3 ; NA21 ; NA2

2 ; NA23 ) is the vector number density associated to par-

ticles of a diatomic gas constituted by atoms A, accelerated atoms A∗ and molecules A2,subjected to autocatalytic chemical reaction of type

A2 +M�A+ A∗ +M; M =A; A∗; A2 (2)

M being the catalyst of the reaction;• (v1; v2; v3)= (c; 0;−c), where c is a positive constant de�ned in terms of the atom mass

m and molecular bond energy � of the diatomic molecule A2, through the relationc=

√(4�)=(5m); for sake of simplicity, we will assume m=1;

Copyright ? 2005 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2005; 28:1491–1506

DISCRETE BOLTZMANN EQUATION 1493

• the collision terms FMi are de�ned by polynomials of the form

FMi (N )= (P

(1)i;M (N ) + P(2)i;M (N ))− NM

i (Q(1)i;M (N ) +Q(2)

i;M (N ))

which describe the elastic scattering and chemical mechanism among particles; the polynomialsP(k)i;M , Q

(k)i;M are homogeneous with positive coe�cients, P(1)i;M and Q(1)

i;M of degree 2, P(2)i;M andQ(2)

i;M of degree 3 and 1, respectively. In particular, P(1)i;M and P(2)i;M represent the rate of creationof particles of species M with velocity along i-direction due to inert or reactive collisionsof binary and triple type, respectively. On the other hand, the terms NM

i Q(1)i;M and NM

i Q(2)i;M

represent the rate of disappearance of particles of species M with velocity along i-directiondue to inert or reactive collisions of triple and binary type, respectively.Let us observe that inert collisions involve the same number of incoming and outgoing

particles, i.e. two or three particles in correspondence of binary or triple collisions, respec-tively. Conversely, collisions with chemical reaction (2) lead to the formation of two atomsA, A∗ and one M -particle when a molecule A2 dissociates in the presence of the catalystM . Therefore, a triple reactive source term corresponds to a binary reactive sink term, andvice versa. The detailed evaluation of the terms FM

i (N ) is provided in Reference [6].The particular form of FM

i (N ) considered in the present paper is appropriate for all thoseelementary reactions of catalytic-type occurring, for example, in the brain chain of thehydrogen–oxygen and methane combustion mechanisms.The other case of elementary bimolecular reaction, including only binary encounters and

then only quadratic collision terms, has been considered in Reference [12], but the existenceresult is much weaker than the one of the present paper.Global chemical reactions involving a greater number of gas particles can be considered

but, in general, they decompose in a brain chain of elementary reactions of bimolecular andautocatalytic type. In this case, the collision terms obviously become rather cumbersome, butstill polynomials, in the unknown number densities, whose degree is equal to the maximumnumber of the colliding particles. The technique exploited here can also be applied, providedthat the reactions are two-way.

1.2. Conservation equations

We assume that (1) possesses the following independent conservation laws (see Reference [6]for details), corresponding, respectively, to the conservation of partial number densities ofspecies A − A2, A∗ − A2 and the total momentum of the system in the x-direction.

@@t

(3∑

i=1(NA

i + NA2i ))+

@@x

(3∑

i=1vi

(NA

i +12NA2

i

))= 0 (3)

@@t

(3∑

i=1NA2

i +∑

i=1;3NA∗

i

)+

@@x

(3∑

i=1

12viNA2

i +∑

i=1;3viNA∗

i

)= 0 (4)

@@t

(3∑

i=1vi(NA

i + NA2i ) +

∑i=1;3

viNA∗i

)+

@@x

(3∑

i=1v2i

(NA

i +12NA2

i

)+∑

i=1;3v2i N

A∗i

)= 0 (5)



1.3. Initial-boundary value problem

We consider the initial-boundary value problem associated to Equations (1) with initial data

NMi (x; 0)=NM

i0 (x) (6)

where Ni0 are prescribed functions of x¿0.We complete the initial-boundary value problem (1)–(6) with the following boundary

conditions:

⎛⎜⎜⎜⎝

NA1 (0; t)

NA∗1 (0; t)

NA21 (0; t)

⎞⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎝

�AA �A

A∗ �AA2

�A∗A �A∗

A∗ �A∗A2

�A2A �A2

A∗ �A2A2

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

NA3 (0; t)

NA∗3 (0; t)

NA23 (0; t)

⎞⎟⎟⎟⎟⎠ ∀t¿0 (7)

with∑M ′

�MM ′61; �A

M + �A∗M + 1

2�A2M6�M (8)

where �M =1 if M =A; A∗ and �A2 =12 .

Conditions (7) mean that the contribution to the incoming �ux of each �xed species A; A∗; A2on the boundary at x=0 is given by a fraction of outgoing molecules of the same speciesthat have been re�ected back from the boundary, and a fraction of molecules of other speciesthat have been reacted.Moreover, condition (8) assures that the macroscopic �ow of the gas is not inward to the

boundary, since

Net�ux =∑

i=1;3vi[NA

i (0; t) + NA∗i (0; t) +

12 N

A2i (0; t)

]

=∑M

(�AM + �A∗

M + 12�

A2M − �M

)NM3 (0; t)

6 0

1.4. Compatibility conditions

Finally, we suppose that the initial data satisfy the following compatibility conditions:⎛⎜⎜⎜⎜⎝

NA10(0)

NA∗10 (0)

NA210 (0)

⎞⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎝

�AA �A

A∗ �AA2

�A∗A �A∗

A∗ �A∗A2

�A2A �A2

A∗ �A2A2

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

NA30(0)

NA∗30 (0)

NA230 (0)

⎞⎟⎟⎟⎟⎠ (9)



and ⎛⎜⎜⎜⎜⎜⎜⎝

ddx

NA10(0)

ddx

NA∗10 (0)

ddx

NA210 (0)

⎞⎟⎟⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎝

�AA �A

A∗ �AA2

�A∗A �A∗

A∗ �A∗A2

�A2A �A2

A∗ �A2A2

⎞⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎝

ddx

NA30(0)

ddx

NA∗30 (0)

ddx

NA30(0)

⎞⎟⎟⎟⎟⎟⎟⎠

(10)

We end this introduction with a few notations:For N (x; t)= (NA

1 (x; t); : : : ; NA23 (x; t)) and X ⊂R+ ×R+, we put

L∞(X )=

{N=‖N‖X;∞= max

i;Mess sup(x; t)∈X

|NMi (x; t)|¡+∞

}

Also, we de�ne the space of bounded continuously di�erentiable functions with boundedderivatives:

B1(X )={f∈C1(X )=f;

@f@x

;@f@t

∈L∞(X )}

and we denote by B1+(X ) its positive cone, composed of all positive functions of B

1(X ).As usual, for N0(x)= (NA

10(x); : : : ; NA230 (x)), we introduce the space

L1(R+)={N0=‖N0‖1 =

∑i;M

∫ +∞

0|NM

i0 (x)| d�(x)¡+∞}

Finally, for T¿0 and N ∈L∞([0;+∞[×[0;T ]), we introduce the upper boundE(T )= max

i;Msup

(x; t)∈R+×[0;T ]|NM

i (x; t)|; E0 =E(0)

and for N0 ∈L1(R+), we put

m0 =∫ +∞

0

(3∑

i=1(NA

i0 + 2NA2i0 ) +

∑i=1;3

NA∗i0

)(x) d�(x)

corresponding to the initial mass of the system.

2. LOCAL SOLUTIONS

Theorem 2.1Let N0 ∈B1

+([0;+∞[) satisfying the compatibility conditions (9), (10). Then there exists aunique solution

N ∈B1+([0;+∞[×[0;T0])

for the mixed-problem (1), (6), (7), where the life-span T0¿0 depends exclusively on

E0 = maxi;M

supx∈R+

|NMi0 (x)|



ProofThe proof of this theorem follows the lines of Theorem 3.1 in Reference [10], where theexistence of local solutions for a discrete Boltzmann model with binary collisions is treated.

For self-completeness, we sketch here the above-mentioned Proof:For convenience, we put

vMi = vi if M =A; A∗ and vA2i =

vi2

For some parameter �¿0, we put (1) in the form

@@t

NMi (x; t) + vM

i@@x

NMi (x; t) + �NM

i (x; t)=FM;�i (N (x; t)) (11)

where

FM;�i (N (x; t))=FM

i (N (x; t)) + �NMi (x; t)

Let (x; t)∈ [0;+∞[×[0;+∞[.If x − vM

i t¿0, integrating (11) along the vMi -characteristic through (x; t) leads to

NMi (x; t)= e

−�tNi0 (x − vMi t) +

∫ t

0e−�(t−�)FM;�

i (N )(x − vMi (t − �); �) d� (12)

If x − vMi t¡0, one gets

NMi (x; t)= e

−�(t−tMi )NMi (0; t

Mi ) +

∫ t

tMi

e−�(t−�)FM;�i (N )(x − vM

i (t − �); �) d� (13)

where tMi = t − (x=vMi ) is the intersection of the vM

i -characteristic with the boundary x=0.Noticing that in this case vM

i ¿0 (i=1), condition (7) yields

NMi (x; t)= e

−�(t−tMi )∑M ′

�MM ′NM ′

3 (0; tMi ) +∫ t

tMi

e−�(t−�)FM;�i (N )(x − vM

i (t − �); �) d� (14)

Finally, integrating along the vM ′3 -characteristic through (0; t

Mi ),

NMi (x; t) = e

−�t∑M ′

�MM ′e−�tMi NM ′

30 (−vM ′3 tMi )

+ e−�t∑M ′

�MM ′

∫ tMi

0e−�(tMi −�)FM ′; �

3 (N )(x − vM ′3 (t − �); �) d�

+∫ t

tMi

e−�(t−�)FM;�i (N )(x − vM

i (t − �); �) d�

Now, for T¿0, we put �T =[0;+∞[×[0;T ], �+M; i; T = {(x; t)∈�T ; x¿vMi t} and �−

M; i; T ={(x; t)∈�T ; x¡vM

i t}.



We introduce the Banach space

X 1(T )={N =(NM

i )i;M ∈C0(�T )∩L∞(�T );@NM

i

@x∈C0(�T )∩L∞(�T )

}

equipped with the norm

‖N‖1;T = ‖N‖∞;�T +∥∥∥∥@N@x

∥∥∥∥∞;�T

where ‖N‖∞;�T = maxi;Msup

(x; t)∈�T

|Ni;M (x; t)|

For E;G¿0 we consider the closed convex subset S(T; E;G) of X 1(T ) formed by all functionsN ∈X 1(T ) such that for all i;M ,

NMi (x; 0)=NM

i0 (x); x∈ [0;+∞[and

06NMi (x; t)6E;

∣∣∣∣@NMi

@x(x; t)

∣∣∣∣6G; (x; t)∈�T

Using the compatibility conditions (9), (10) and the fact that all the non-linear terms involvedin (1) are Lipschitz (see Reference [10] for details) one can prove that the map

� :N → �(N )

given by

�(N )Mi (x; t) = e−�tNi0 (x − vM

i t)

+∫ t

0e−�(t−�)FM;�

i (N )(x − vMi (t − �); �) d� if (x; t)∈�+M; i; T (15)

and

�(N )Mi (x; t) = e−�t∑

M ′�MM ′e−�tMi NM ′

30 (−vM ′3 tMi )

+ e−�t∑M ′

�MM ′

∫ tMi

0e−�(ti−�)FM ′; �

3 (N )(x − vM ′3 (t − �); �) d�

+∫ t

tMi

e−�(t−�)FM;�i (N )(x − vM

i (t − �)); �) d� if (x; t)∈�−M; i; T (16)

is a contraction of S(T; E;G) for adequate values of �, T , E, and G, where �, T and E dependexclusively on E0.Then, the unique �x-point N of � is a local solution of (1), with

N ∈C0(�T )∩L∞(�T ) and@N@x

∈C0(�T )∩L∞(�T )

It is a technicality to prove that N is continuously di�erentiable with respect to t and that Nsolves Equation (1). The reader can refer to Reference [10] (proof of Theorem 3.1).



3. GLOBAL SOLUTIONS

We consider the new set of variables

M1 =NA1 + NA∗

1

M2 =NA21

M3 =NA2 + NA2

2

M4 =NA23

M5 =NA3 + NA∗

3

This new set satis�es the system

@@t

Mi(x; t) + wi@@x

Mi(x; t)=Ri(N (x; t)); i∈ {1; : : : ; 5} (17)

where (w1; w2; w3; w4; w5)= (c; c2 ; 0;− c

2 ;−c) and

R1 = FA1 + FA∗

1

R2 = FA21

R3 = FA2 + FA2

2

R4 = FA23

R5 = FA3 + FA∗

3

We make the usual following assumption: for every j ∈ {1; : : : ; 5}, Rj can be put in the form

Rj(N )=NMi R(1)j − R(2)j (18)

where vMi �=wj and R(k), k=1; 2 are polynomials with positive coe�cients. Note that this

condition holds in the systems studied in Reference [17].Under this assumption, we prove in this section the following theorem.

Theorem 3.1Let N0 ∈B1

+([0;+∞[)∩L1([0;+∞[) satisfying the compatibility conditions (9)–(10). Thenthere exists �; �′¿0 such that if

m0 =∫ +∞

0

(3∑

i=1(NA

i0 + 2NA2i0 ) +

∑i=1;3

NA∗i0

)(x) dx¡� and E0 = max

i;Msupx∈R+

|NMi0 (x)|¡�′

the mixed-problem (1)–(6)–(7) has a unique solution

N ∈B1+([0;+∞[×[0;+∞[)



ProofLet N0 ∈B1

+([0;+∞[)∩L1(]0;+∞[) satisfying (9)–(10) and N ∈B1+([0;+∞[×[0;T0]) the

solution to (1)–(6)–(7) given by Theorem 2.1.

In order to establish Theorem 3.1, we prove the following key lemma.

Lemma 3.2For all t ∈ [0;T0],

E(t)6CE0 + C ′m0(E(t) + E(t)2) (19)

where the constants C, C ′ are independent of T0 and N0.

ProofFollowing the ideas in References [9,11], the proof of this lemma is based on the integrationof the conservation laws (3)–(5) along closed curves formed by the characteristic lines of (1).

In what follows we will denote indi�erently by C and C ′ some positive constants.Let X =(x; t)∈R+ ×R+. We begin by considering the case ct − x¿0.In this case the c-characteristic through X meets the boundary x=0 at a time

tc= t − (x=c)¿0.Let j ∈ {1; : : : ; 5}.We �rst assume that x − wjt¡0.In this case wj ∈ {c=2; c} (i.e. j ∈ {1; 2}) and the characteristic corresponding to the speed

wj meets the boundary x=0 at the time tj= t − (x=wj). Integrating (17) along this line,

Mj(x; t)=Mj(0; tj) +∫ t

tjRj(N (x − wj(t − �); �)) d� (20)

Also, by (7),

Mj(0; tj)6∑

M;M ′�M ′M NM

3 (0; tj)6C(M4(0; tj) +M5(0; tj))

Hence,

Mj(x; t)6C(M4(0; tj) +M5(0; tj)) +∫ t

tjRj(N (x − wj(t − �); �)) d�

6C(M4(0; tj) +M5(0; tj)) + C ′(E(t) + E(t)2)∫ t

tj

∑vMi �=wj

NMi (x − wj(t − �); �) d�

(21)

where we have used condition (18).We now estimate I =

∫ ttj

∑vMi �=wj

NMi (x − wj(t − �); �) d�.



Let us consider the di�erential form

w=−(

3∑i=1(NA

i + 2NA2i ) +

∑i=1;3

NA∗i

)dx +

(3∑

i=1vi(NA

i + NA2i ) +

∑i=1;3

viNA∗i

)dt

By making use of the conservation laws (3), (4) we easily check that w is an exact form,hence by putting A=(0; tc) and B=(x + ct; 0),∫

[OA]w +

∫[AX ]

w +∫[XB]

w +∫[BO]

w=0 (22)

�x

�t

��

��

Ctj = t − x=wj

B

x + ct

AtC = t − x=c

D

x + c(t − tj)x − wjt 0

X (x; t)

(c)

(wj) (−c)

Furthermore,∫[OA]

w=∫ tc

0

(3∑

i=1vi(NA

i + NA2i ) +

∑i=1;3

viNA∗i

)(0; �) d�60 (23)

by the boundary condition (7).Also,∫

[AX ]w =

∫ t

tc

(3∑

i=1((vi − c)NA

i + (vi − 2c)NA2i )+

∑i=1;3

(vi − c)NA∗i

)(x−c(t − �); �) d�

(24)

∫[XB]

w = −∫ t

0

(3∑

i=1((vi + c)NA

i +(vi + 2c)NA2i )+

∑i=1;3

(vi + c)NA∗i

)(x+c(t − �); �) d�

(25)

and ∫[BO]

w=∫ x+(t=c)

0

(3∑

i=1(NA

i + 2NA2i ) +

∑i=1;3

NA∗i

)(x +

tc

− �; 0)d�6m0 (26)



By (22),∫ t

0

(3∑

i=1((vi + c)NA

i + (vi + 2c)NA2i ) +

∑i=1;3

(vi + c)NA∗i

)(x + c(t − �); �) d�

+∫ t

tc

(3∑

i=1((c − vi)NA

i + (2c − vi)NA2i ) +

∑i=1;3

(c − vi)NA∗i

)(x − c(t − �); �) d�6m0

(27)

and ∫ t

0

( ∑i=1;2

NAi +

3∑i=1

NA2i + NA∗

1

)(x + c(t − �); �) d�6Cm0 (28)

We now consider C=(0; tj) and D=(x + c(t − tj); tj).As before, ∫

[OC]w +

∫[CD]

w +∫[DB]

w +∫[BO]

w=0 (29)

An elementary computation leads to∫[OC]

w =∫ tj

0

(3∑

i=1vi(NA

i + NA2i ) +

∑i=1;3

viNA∗i

)(0; �) d�60 (30)

∫[CD]

w = −∫ x+c(t−tj)

0

(3∑

i=1(NA

i + 2NA2i ) +

∑i=1;3

NA∗i

)(�; tj) d� (31)

∫[DB]

w = −∫ t

tj

(3∑

i=1((vi + c)NA

i + (vi + 2c)NA2i ) +

∑i=1;3

(vi + c)NA∗i

)(x + c(t − �); �) d�

6 0 (32)

By (29), ∫ x+c(t−tj)

0

(3∑

i=1(NA

i + 2NA2i ) +

∑i=1;3

NA∗i

)(�; tj) d�6m0 (33)

Next, we consider the di�erential form w′ de�ned by

w′= −(

3∑i=1((wj − vi)NA

i + (2wj − vi)NA2i ) +

∑i=1;3

(wj − vi)NA∗i

)dx

+

(3∑

i=1vi((wj − vi)NA

i +(wj − 1

2vi

)NA2

i ) +∑

i=1;3vi(wj − vi)NA∗

i

)dt (34)



By (5), w′ is an exact form∫[XC]

w′ =∫ t

tj

(3∑

i=1

((wj − vi)2NA

i + 2(wj − 1

2vi

)2NA2

i

)

+∑

i=1;3(wj − vi)2NA∗

i

)(x − wj(t − �); �) d� (35)

∫[CD]

w′=−∫ x+c(t−tj)

0

(3∑

i=1((wj − vi)NA

i + (2wj − vi)NA2i ) +

∑i=1;3

(wj − vi)NA∗i

)(�; tj) d�

(36)

and∫[DX ]

w′=∫ t

tjc

(3∑

i=1((wj−vi)NA

i +(2wj−vi)NA2i )+

∑i=1;3

(wj−vi)NA∗i

)(x+c(t−�); �) d�

+∫ t

tj

(3∑

i=1vi((wj − vi)NA

i + (wj − 2vi)NA2i ) +

∑i=1;3

vi(wj − vi)NA∗i

)(x + c(t − �); �) d�

=∫ t

tj

(3∑

i=1

((vi + c)(wj − vi)NA

i + 2(wj − vi

2

)(c+

vi2

)NA2

i

))((x + c(t − �); �))

+∑

i=1;3((vi + c)(wj − vi)NA∗

i )((x + c(t − �); �)) d� (37)

Hence, ∫[XC]

w′ = −∫[CD]

w′ −∫[DX ]

w′

6∣∣∣∣∫[XC]

w′∣∣∣∣+

∣∣∣∣∫[DX ]

w′∣∣∣∣

6C∫ x+c(t−tj)

0

(3∑

i=1(NA

i + 2NA2i ) +

∑i=1;3

NA∗i

)(�; tj) d�

+C∫ t

0

( ∑i=1;2

NAi +

3∑i=1

NA2i + NA∗

1

)(x + c(t − �); �) d�

6Cm0

by (28) and (33).



Finally, by (35),

I =∫ t

tj

∑vMi �=wj

NMi (x − wj(t − �); �) d�6Cm0 (38)

Next, we consider the case where xj= x − wjt¿0.

�x

�t

��

��B

x + ct

A

E

xj = x − wjt

tC = t − x=c

0

X (x; t)

(c)

(wj)

(−c)

In this case the characteristic corresponding to a speed wj meets t=0 at xj= x − wjt. Weput E=(xj; 0).Here, for j ∈ {1; : : : 5},

Mj(x; t)=Mj(xj; 0) +∫ t

0Rj(N (x − wj(t − �); �)) d� (39)

and, as before,

Mj(x; t)6CE0 + C ′(E(t) + E(t)2)∫ t

0

∑vMi �=wj

NMi (x − wj(t − �); �) d� (40)

In order to estimate J =∫ t0

∑vMi �=wj

NMi (x−wj(t−�); �) d�, we integrate w′ on the triangle EXB:∫

[XE]w′ =

∫ t

0

(3∑

i=1

((wj − vi)2NA

i + 2(wj − v2i

2

)NA2

i

)

+∑

i=1;3(wj − vi)2NA∗

i

)(x − wj(t − �); �) d� (41)

and

∣∣∣∣∫[EB]

w′∣∣∣∣+

∣∣∣∣∫[BX ]

w′∣∣∣∣

6∫ x

xj

(3∑

i=1((wj − vi)NA

i + (2wj − vi)NA2i ) +

∑i=1;3

(wj − vi)NA∗i

)(�; 0) d�



+∫ t

0

(3∑

i=1

(((vi + c)(wj − vi)NA

i + 2(wj − vi

2

)(c+

vi2

)NA2

i

)))((x + c(t − �); �))

+∑

i=1;3((vi + c)(wj − vi)NA∗

i )((x + c(t − �); �)) d�

hence,

∣∣∣∣∫[EB]

w′∣∣∣∣+

∣∣∣∣∫[BX ]

w′∣∣∣∣6Cm0 + C ′

∫ t

0

( ∑i=1;2

NAi +

3∑i=1

NA2i + NA∗

1

)(x + c(t − �); �) d�6Cm0

by (28).Finally,

J =∫ t

0

∑vMi �=wj

NMi (x − wj(t − �); �) d�6C

∫[XE]

w′6Cm0 (42)

By combining this estimate with (40), we obtain, in the case where x − wjt¿0,

∀j; Mj(x; t)6CE0 + Cm0(E(t) + E(t)2) (43)

If x − wjt¡0, by (21) and (38),

Mj(x; t)6C(M4(0; tj) +M5(0; tj)) + Cm0(E(t) + E(t)2) (44)

For j ∈ {4; 5}, integration along the characteristic of speed wj leads to

Mj(0; tj) = Mj(wjtj; 0) +∫ tj

0Rj(N (−wj(� − tj); �)) d�

6CE0 + (E(t) + E(t)2)∫ tj

0

∑vMi �=wj

NMi (−wj(t − �); �) d�

6CE0 + C ′m0(E(t) + E(t)2)

by (42).Replacing in (44) leads to

Mj(x; t)6CE0 + Cm0(E(t) + E(t)2)

also in the case x − wjt¡0.



Finally, if the characteristic corresponding to a speed c does not meet the boundary x=0,we put xc= x − ct and F =(xc; 0).

�x

�t

��

��

��B

x + ct

F

x − ct

E

xj = x − wjt0

X (x; t)

(c)

(wj)

(−c)

Integrating w on the triangle XBF and w′ on the triangle XFE leads by analogous compu-tations to (43) which concludes the proof of Lemma 3.2.

We now prove the following lemma.

Lemma 3.3There exists �¿0, �′¿0 such that if m0¡� and E0¡�′ then there exists a constant K =K(N0)¿0 depending exclusively on the initial data N0 such that

∀06t6T0; E(t)6K(N0)

One can easily show by standard arguments that Theorem 3.1 can be obtained fromLemma 3.3.

ProofWe de�ne the polynomial

P :Y ∈R→C ′m0Y 2 + (C ′m0 − 1)Y + CE0

Let m0¡1=C ′ := � and �′¿0 small enough such that if E0¡�′,

(C ′m0 − 1)2 − 4CC ′m0E0¿0

Then P possesses the two roots

Y1(E0; m0) =1− C ′m0 −

√(C ′m0 − 1)2 − 4CC ′m0E0

C ′m0¿0

Y2(E0; m0) =1− C ′m0 +

√(C ′m0 − 1)2 − 4CC ′m0E0

C ′m0¿0



Since for all t ∈ [0;T0]; P(E(t))¿0 and t → E(t) is continuous,

∀06t6T0; E(t)6Y1 or ∀06t6T0; E(t)¿Y2

Clearly, Y26E0 cannot hold for E0 small enough, since

limE0→0

Y2(E0; m0)=21− C ′m0C ′m0

¿0

Hence, Lemma 3.3 holds with K(N0)=Y1(E0; m0).

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on the global well-posedness of discrete boltzmann systems with chemical reaction

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