Some existence results for conservation laws with source-term

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2002; 25:1149–1160 (DOI: 10.1002/mma.332)MOS subject classi�cation: 35 L 65

Some existence results for conservation laws with source-term

Joao-Paulo Dias1;∗;† and Philippe G. LeFloch2;‡

1CMAF; University Lisboa; Av. Prof. Gama Pinto; 2; 1649-003 Lisboa; Portugal2Centre de Math�ematiques Appliqu�ees & Centre National de la Recherche Scienti�que; U.M.R. 7641;

Ecole Polytechnique; 91128 Palaiseau Cedex; France

Communicated by J. C. Nedelec

SUMMARY

We prove some new results concerning the existence and asymptotic behaviour of entropy solutionsof hyperbolic conservation laws containing non-smooth source-terms. Copyright ? 2002 John Wiley &Sons, Ltd.

1. INTRODUCTION

In this paper, we discuss the existence and asymptotic behaviour of entropy solutions for theCauchy problem associated with an hyperbolic conservation law containing a source-term

9tu+ 9xf(u)= g(x)h(u); u(x; t)∈R; x∈R; t¿0 (1)

u(x; 0)= u0(x); x∈R (2)

Following a pioneering work on the subject by Liu [1], we are interested in the followingsituation, that we assume through the paper:

• The �ux function is genuinely non-linear, say, after normalization,

f∈C2(R); f′′(u)¿0; f(0)=f′(0)=0 (3)

• The source-term is bounded and integrable in x and keeps a constant sign, say

06g(x)6G; g∈L1(R) (4)

h∈C1(R); h(u)¿0; h′(u)60 (5)

∗ Correspondence to: J.-P. Dias, CMAF, University Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal† E-mail: [email protected]‡ E-mail: le�[email protected]

Contract=grant sponsor: FCT, PRAXIS XXI, FEDER; contract=grant number: PRAXIS=2=2.1=MAT=125=94.

Copyright ? 2002 John Wiley & Sons, Ltd. Received 7 February 2001

1150 J.-P. DIAS AND P. G. LEFLOCH

The present paper supplements results established by Liu in Reference [1] concerning theexistence and asymptotic behaviour of solutions of (1)–(2). Our results are the followingones.In Section 2, under a mild additional assumption on f, we prove, for u0 ∈L2(R), the

existence of a weak solution u∈L∞loc(R+;L2(R)) of Cauchy problem (1), (2), by applying

Tartar’s compensated compactness method [2], along the lines of a generalization due toSchonbek in Reference [3].In Section 3, �rst assuming u0 and g in BV(R) (where BV(R) denotes the space of

Lebesgue integrable functions with bounded total variation on R, cf. for instance Refer-ence [4]), we prove the existence of an entropy solution in the sense of Kruzkov [5],u∈L∞(R×(0; T ))∩C([0; T ];L1(R))∩BV(R×(0; T )), for each T¿0. The proof is based onregularizing g and u0 and justifying the corresponding passage to the limit. We also prove anexistence result when u0 ∈L∞(R) and a stability result when the �ux function f is replacedwith �f, �¿0, and �→0. In the proofs we apply and further develop several techniques fromthe works of Kruzkov [5] and Tartar [2].Finally, in Section 4, we compare the asymptotic behaviour of an entropy solution u of (1)

with the solution v of a Riemann problem in the homogeneous case, following an idea fromChen and Frid [6].

2. EXISTENCE THEORY IN L2

In this section, we show that the existence of an entropy solution to (1)–(2) follows fromTartar’s compensated compactness method. In the proof we rely on the Lp framework intro-duced by Schonbek [3].Concerning the behaviour at in�nity of the data, we require

|f′(u)|6k(1 + |u|) for a constant k¿0 (6)

As usual, u∈L∞loc(R+;L2(R)) is called a weak solution of Cauchy problem (1), (2) for

u0 ∈L2(R) if and only if for all ’∈C∞0 (R×[0;+∞])∫

R+

∫Ru’t dx dt +

∫R+

∫Rf(u)’x dx dt +

∫Ru0(x)’(x; 0) dx +

∫R+

∫Rgh(u)’ dx dt=0 (7)

The existence theory below is based on the following entropy inequality:

9tu2

2+ 9xF(u)6ug(x)h(u) with F ′(u)= uf′(u)

Theorem 2.1Let the initial data u0 be in L2(R). Under the additional assumption (6), there exists a weaksolution u∈L∞

loc(R+;L2(R)) of Cauchy problem (1)–(2).

ProofLet u� be (smooth) approximate solutions constructed by the vanishing di�usion method, i.e.

9tu� + 9xf(u�) = g(x) h(u�) + �9xxu� (8)

Copyright ? 2002 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2002; 25:1149–1160

SOME EXISTENCE RESULTS FOR CONSERVATION LAWS 1151

u�(x; 0) = u�0(x); x∈R (9)

where u�0 is a smooth (say H 2(R)) function converging to u0 in L2(R). We apply the com-

pensated compactness method (cf. Reference [3]) in the following way. First of all, we derivea uniform L2 bound for the approximate solutions. Multiplying (8) by u� and integrating inx∈R, we arrive at

ddt

∫R

|u�|22dx + �

∫R|9xu�|2 dx=

∫Rg(x)h(u�)u� dx

Thanks to the monotonicity of h it follows that

ddt

∫R

|u�|22dx + �

∫R|9xu�|2 dx +

∫Rg(x)|u�(h(u�)− h(0))| dx

6∫Rg(x)h(0)|u�| dx61

2

∫R|u�0|2 dx +

Gh(0)2

2‖g‖L1

By Gronwall’s inequality, this implies that, for all t ∈ [0; T ]∫R

|u�(t)|22

dx + �∫ t

0

∫R|9xu�|2 dx ds+

∫ t

0

∫Rg(x)|u�(h(u�)− h(0))| dx ds6C(T ) (10)

Now, we want to apply Theorem 3.2 and Corollary 3.2 in Schonbek [3]. Let �∈C2(R) bean entropy for the Equation (1), � having compact support, q being the corresponding entropy�ux q(u)=

∫ u0 �′(s)f′(s) ds. We easily obtain

�(u�)t + q(u�)x= �(�(u�))xx − ��′′(u�)(u�x)2 + �′(u�)g(x)h(u�) (11)

Now, let us �x T¿0. In view of (10), there exists u= uT ∈L∞((0; T );L2(R)) and a subse-quence of {u�}�, still denoted by {u�}�, such that

u� *uT in L∞((0; T );L2(R)) weak∗

Furthermore we have, by (4), since |�′|6c, supp �⊂{�∈R||�|6��},∫ T

0

∫R|�′(u�)g(x)h(u�)| dx dt6 c

∫ T

0

∫|u�(x; t)|6��

g(x)h(u�) dx dt6c�T‖g‖L1

Henceforth, by (10) and proceeding as done in Reference [2] in the homogeneous case,we deduce that �(u�)t + q(u�)x lies in a compact set of H−1

loc (R×[0; T ]). By Theorem 3.2 andCorollary 3.2 in Reference [3] (with p=2), there exists a subsequence of {u�}�, still denotedby {u�}�, such that, since f is a genuinely non-linear function (it is strictly convex),

u�→uT a:e: in R×(0; T ); strongly in Lqloc(R×[0; T )); 16q¡2

and for all ’∈C∞c (R×[0; T ))∫ T

0

∫Rf(u�)’x dx dt→

∫ T

0

∫Rf(uT )’x dx dt



It remains to pass to the limit in the source term. First, we deduce

g(:)(h(u�)− h(0))→ g(:)(h(uT )− h(0)) a:e: in R×(0; T ) (12)

Next, if E is a measurable subset of R×(0; T ) and R¿1 it follows from (10) that

∫Eg(x)|h(u�)− h(0)| dx ds6

∫{|u�|6R}∩E

g(x)|h(u�)− h(0)| dx ds

+∫{|u�|¿R}

g(x)|u�| |u

�(h(u�)− h(0))| dx ds

6 2‖h‖L∞(−R;R)

∫Eg(x) dx ds+

C(T )R

As g belongs to L1(R) we �rst deduce that

lim supmeas(E)→0

∫Eg(x)|h(u�)− h(0)| dx ds6C(T )

R

As the above inequality is valid for each R¿1 we �nally obtain

limmeas(E)→0

∫Eg(x)|h(u�)− h(0)| dx ds=0 (13)

Owing to (12)–(13), we may apply the Vitali theorem to deduce that

g(:)(h(u�)− h(0))→ g(:)(h(uT )− h(0)) in L1loc(R×(0; T ))Consequently, for every ’∈C∞

c (R×[0; T )),∫ T

0

∫Rg(x)h(u�)’ dx dt→

∫ T

0

∫Rg(x)h(uT )’ dx dt

Finally, using a standard diagonalization procedure, we see that there exists a measurablefunction u de�ned on the whole domain R×(0;+∞) such that u∈L∞((0; T );L2(R)) for allT¿0, and a subsequence still denoted by {u�}�, such that

u�→u a:e: in R×(0;+∞) and in Lqloc(R×[0;+∞)); 16q¡2

and for all ’∈C∞c (R×[0;+∞))

∫ +∞

0

∫Rf(u�)’x dx dt →

∫ +∞

0

∫Rf(u)’x dx dt

∫ +∞

0

∫Rgh(u�)’ dx dt →

∫ +∞

0

∫Rgh(u)’ dx dt

Hence, u is a weak solution of Cauchy problem (1)–(2).



3. EXISTENCE THEORY IN L∞

Recall that Kruzkov’s theory [5] applies under hypotheses (3)–(5) and some smoothnessassumptions on g, and provided the initial data satis�es u0 ∈L∞(R). By de�nition, a functionu∈L∞

loc(R×[0;+∞)) is an entropy solution of (1)–(2) if (7) holds together with the entropyinequality

9t�(u) + 9xq(u)− �′(u) gh(u)60

in D′(R×]0;∞[), for each entropy–entropy �ux pair (�; q), with � convex. Here by de�nitionq(u)=

∫ u0 �′(s)f′(s) ds.

For data u0 ∈L∞ ∩L1, Crandall [7] established an existence theorem for the Cauchy prob-lem (1)–(2). In the present section we begin by assuming u0 and g in BV(R), where BV(R)denotes the space of all Lebesgue integrable functions with bounded variation de�ned on R(cf. for instance Reference [4]). We now prove, under the assumptions made here, the fol-lowing improvement of Kruzkov’s classical theory [5].

Theorem 3.1Assume that u0; g∈BV(R). Then problem (1)–(2) admits an entropy solution u such that foreach T¿0, and with ST =R×(0; T ), u∈L∞(ST )∩C([0; T ];L1(R))∩BV(ST ) and

‖u(·; t)‖L∞ 6 ‖u0‖L∞ + ‖g‖L∞h(0)t a:e: (14)

TV u(·; t)6 TV(u0) + c1(t)TV(g); t¿0 (15)

∫R|u(x; t2)− u(x; t1)| dx6[(c+ f′(‖u0‖L∞))TV(u0) + h(−‖u0‖L∞)‖g‖L1 ]|t1 − t2|; t1; t2¿0 (16)

TVST u6 [(c+ 1+ f′(‖u0‖L∞))TV(u0) + h(−‖u0‖L∞)‖g‖L1 ]T + c2(T )TV(g); T¿0 (17)

where c¿0 is an universal constant and TV denotes the total variation.

ProofLet ��, �¿0, be a standard molli�er de�ned on R and set g�=�� ∗ g and u0�=�� ∗ u0. Considerthe solution u� of the viscous problem{

u� t + f(u�)x= g� h(u�) + ��u�

u�(0)= u0�(18)

It is well-known (for instance Reference [5], (4.6)) that, since g�h′(u�)60 and h¿0, thefollowing maximum principle holds:

|u�(x; t)|6‖u0�‖L∞ + ‖g�‖L∞h(0)t6‖u0‖L∞ + ‖g‖L∞ h(0) t (19)

On the other hand, it is a standard matter to derive (for a proof see, for instance, Theorem 2.3of Chapter 2 in Reference [8])

99t |u� x|+ 9

9x (f′(u�)|u� x|)− ��(u� x) sgn u� x

= g′�h(u�) sgn u�x + g�h′(u�)|u� x|6g′�h(u�) sgn u� x (20)



(where sgn x= |x|=x for x �=0). Since, by (5) and (19),

h(u�)6c3(t)= h(−‖u0‖L∞ − ‖g‖L∞h(0) t)

we arrive at ∫R|g′�h(u�) sgn u� x| R dx6c3(t)

∫R|g′�| R dx (21)

where for R¿0,

R(�)= �( |�|

R

); �∈R; �∈C2(R)

�(�)¿0

�(�)=1 if 06�6 12

� is a polynomial if 126�61 and �(�)= e−� if �¿1

Fix T¿0. From (20) and (21) (with, say, �61 and R¿1), it is a standard matter to obtain(see, for instance, Lemmas 2.5 and 2.6 in Reference [8]) to derive for t6T

ddt

∫R|u� x| R dx6

c(T )R

∫R|u� x| R dx + c3(T )

∫R|g′�| R dx

thus, ∫R|u� x| R dx6 exp

(c(T )tR

)[∫R|u0� x| R dx + c3(T )T

∫R|g′�| R dx

]

Hence letting R→∞, we see that for all t6T

∫R|u� x|du6

∫R|u0� x| dx + c3(T )T

∫R|g′�|dx

6 TV(u0) + c3(T )T TV(g) (22)

In a similar way we can derive (note that (9=9t)g=0)

‖u� t‖L16‖u� t(0)‖L1 ; where L1=L1(R)

On the other hand, at the initial time, setting M=f′(‖u0‖L∞), we have easily

‖u� t(0)‖L16M‖u0� x‖L1 + �‖�u0�‖L1 + ‖g�h(u0�)‖L1

We have �‖�u0�‖L16cTV(u0) (c¿0 independent of �) and ‖g� h(u0�)‖L16c0 ‖g‖L1 , wherec0 = h(−‖u0‖L∞). Hence,

‖u�t‖L16(M + C)TV(u0) + c0‖g‖L1 (23)



Concerning the L1 norm we also have

ddt

∫R|u�| R dx6M

∫R|u� x| R dx + �

∫R|u�|� R dx +

∫Rg�h(u�) sgn u� R dx

6M∫R|u� x| R dx +

�R2

∫R|u�| R dx + h(0)

∫Rg� R dx

Hence combining with (22) and letting R→∞ we �nd

∫R|u�| dx6

∫R|u0| dx +M

∫ T

0

∫R|u� x| dx d�+ h(0)‖g‖L1

6∫R|u0| dx +Mt TV(u0) + c4(t) TV(g) + h(0) ‖g‖L1 (24)

The proof is concluded via standard compactness arguments, as follows. First of all we�x a �nite time T¿0. Based on the a priori estimates derived above, there exists a subse-quence of {u�}� still denoted by {u�}� and a function uT ∈L∞(ST )∩C([0; T ];L1(R)) such thatu� →� uT a.e. in ST and in C([0; T ];L1loc(R)), such that uT is an entropy solution in ST ofCauchy problem (1)–(2) and satis�es, in ST , the estimates (14)–(16). Now, by a standarddiagonalization procedure, we can construct a subsequence of {u�}�, still denoted by {u�}�,and a u∈L∞

loc(R×(0;+∞)) such that u∈L∞(ST )∩C([0; T ];L1(R)) for each T¿0, u� →� ua.e. in R×(0;+∞) and in C([0; T ]; L1loc(R)) and u is an entropy solution of problem (1)–(2),satisfying estimates (14)–(16).Finally, for ’; ∈D(R×(0; T )), we deduce from (22) and (23) that

∫ST

u(9’9x +

9 9t

)dx dt = − lim

�→0

∫ST

(9u�

9x ’+9u�

9t )dx dt

6 ‖(’; )‖L∞(ST )

{∫ T

0[TV(u0) + c3(t)t TV(g)] dt

+∫ T

0[(M + C)TV(u0) + c0 ‖g‖L1 ] dt

}

and so u∈BV(ST ) and (17) holds. This completes the proof of Theorem 3.1.

Remark 3.2To apply Kruzkov’s uniqueness result (cf. the statement in Reference [5], Section 3, The-orem 2), one needs that g∈C1(R) (or at least g be locally lipschitz). Within Crandall’sapproach (cf. [7]), uniqueness is also established for solutions constructed via the non-linearsemi-group method with initial data in L1(R)∩L∞(R).

Our second result in this section combines Kruzkov’s theory and the compensated com-pactness method:



Theorem 3.3Assume that (4) and (5) hold and u0 ∈L∞(R). Then there exists a function u de�ned inR×(0;+∞) such that u∈L∞(ST ) for all T¿0, with

‖u(·; t)‖L∞6‖u0‖L∞ + ‖g‖L∞ h(0)t; a:e: in t¿0 (25)

and u is an entropy solution of Cauchy problem (1)–(2).

ProofBy Kruzkov’s results in Reference [5], letting g�=�� ∗ g ({��}� being a molli�er), for each�¿0 there exists an (actually unique) entropy solution of the Cauchy problem{

u� t + f(u�)x= g�h(u�); x∈R; t¿0

u�(x; 0)= u0(x); x∈R(26)

satisfying the bound

|u�(x; t)|6 ‖u0‖L∞ + ‖g�‖L∞h(0)t

6 ‖u0‖L∞ + ‖g‖L∞ h(0)t a:e: in R×(0;+∞) (27)

We now let �→ 0. For each entropy–entropy �ux pair (�; q) of the equation ut+(f(u))x=0

��= �(u�)t + q(u�)x − �′(u�)g�h(u�)60 in D′(R× (0;+∞))

Let us �x T¿0. By (27) and (4), we have that {��}� bounded in W−1;∞(ST ). Since ��60we conclude, by a famous result of Murat [9], that {��}� is compact in H−1

loc (R×[0; T )).Moreover, ∫

R|�′(u�)g�h(u�)| dx6c(t)‖g‖L1

and so {�′(u�) g� h(u�)}� remains in a bounded set of M(R×(0; T )) (signed Radon measureswith �nite mass). We see then that

{�(u�)t + q(u�)x}�= {�� + �′(u�)g�h(u�)}�is compact in H−1

loc (R×[0; T )). We conclude by applying Tartar’s compensated compactnessmethod [2]. By a standard diagonalization procedure, there exists a subsequence, still de-noted {u�}�, and a measurable function u de�ned on R×(0;+∞), such that u� →� u a.e. inR×(0;+∞), u∈L∞(ST ), ∀T¿0 and veri�es (25). Finally, it is clear that u is a entropysolution of Cauchy problem (1)–(2).

Now, let us replace the �ux function f in (1) by f�= �f, �¿0, and assume the sameassumptions as in Theorem 3.1 above. For each �¿0 (�61) let u� be the entropy solutionof the corresponding Cauchy problem obtained by approximating the method in the proofof Theorem 3.1, for �xed g, h and u0. For each T¿0 uniform (in �) estimates for u� inL∞(ST )∩L1(ST )∩BV(ST ) hold.



By compactness argument and using a diagonalization procedure, we extract a subsequencestill denoted by {u�}� and a function u∈L1loc(R×(0;+∞)) such that

u� →�u in L1loc(R×(0;+∞)) and a:e: in R×(0;+∞)

u∈L∞(ST )∩L1(ST ) for each T¿0. By semi-continuity, we also have u∈BV(ST ). Moreover,u is a global solution of the following (limit) Cauchy problem{

ut = g h(u) in R×(0;+∞) (28)

u(x; 0)= u0(x) in R (29)

in the following sense: u∈L∞loc(R× [0;+∞)) and∫

R×(0;+∞)u’t dx dt +

∫R×(0;+∞)

h(u)g’ dx dt +∫Ru0(x)’(x; 0) dx=0;

for each ’∈C∞c (R× [0;+∞)) (30)

Note that, since u∈L∞(ST )∩L1(ST )∩BV(ST ) for all T¿0, we deduce from (28) inD′ : ut ∈L∞(ST )∩L1(ST ) for all T¿0. Moreover, we deduce that �(u)t − �′(u)gh(u)60 inD′(R×(0;+∞)), for each convex entropy � of the Equation (1).Now, assume that

h(�)¿0; �∈R (31)

and let (�) be such that ′(�)=1=h(�), �∈R. If we de�ne, for (x; t)∈R× [0;+∞),u∗(x; t)= −1[ (u0(x)) + g(x)t] (32)

the function u∗ is measurable in R×(0;+∞) and satis�es u∗ ∈L∞(ST )∩L1(ST )∩BV(ST )∩C([0; T ];L1loc(R)) for all T¿0. Moreover, setting u0�=�� ∗u0, g�=�� ∗g (�� being a molli�er),u∗� (x; t)= −1[ (u0�(x)) + g�(x)t] is a strong solution of the corresponding problem. Letting�→ 0 we easily deduce that u∗ is a global solution of (28)–(29). Since the function u de-�ned above belongs to L∞

loc(R×[0;+∞))∩L∞(ST )∩BV(ST ) for all T¿0, we can replace uby its symmetric mean �u (cf. References [4,10]). As in Reference [10], we conclude that�u∈C([0; T ];L1loc(R)) for all T¿0, �u= u a.e. in R×(0;+∞) and �u(x; 0)= u0(x), x∈R. Hence,if in addition we assume g∈C1(R) (or at least g locally lipschitz continuous), Kruzkov’suniqueness result [5, Section 3, Theorem 2] applies which implies �u= u∗ a.e.

4. ASYMPTOTIC BEHAVIOUR

In this section we assume g∈C1(R) and u0 ∈L∞(R). Let u∈L∞loc(R+;L∞(R)) be the unique

entropy solution of (1)–(2) satisfying its initial data in the sense (cf. [5]): There exists a setMu ⊂ R+, Mu with Lebesgue measure zero such that, for each R¿0∫

|x|6R|u(x; t)− u0(x)| dx→ 0 if t→ 0+; t =∈Mu (33)



Given v0 ∈L∞(R), let v be the unique entropy solution of the Cauchy problem

9tv+ 9xf(v)=0; v(x; t)∈R; x∈R; t¿0

v(x; 0)= v0(x); x∈R(34)

satisfying v∈L∞(R×R+) and the corresponding condition (33).Recall from Reference [5] that if ’∈D(R×(0;+∞)) and ’¿0, k, ‘∈R, we have∫

R+

∫R{|u− k|’t + sgn (u− k)(f(u)− f(k))’x + sgn (u− k)gh(u)’} dx dt¿0 (35)

and ∫R+

∫R{|v− ‘|’t + sgn (v− ‘)(f(v)− f(‘))’x} dx dt¿0

Since gh(v) + sgn (v− ‘)gh(v)¿0 a.e. we derive∫R+

∫R{|v− ‘|’t + sgn (v− ‘)(f(v)− f(‘))’x + sgn (v− ‘)gh(v)’+ gh(v)’} dx dt¿0 (36)

In view of (35) and (36), it is not di�cult to generalize the classical Kruzkov argument [5]and check that

∫R+

∫R{|u− v|’t + sgn (u− v)(f(u)− f(v))’x

+sgn (u− v)g(h(u)− h(v))’+ gh(v)’} dx dt¿0 (37)

that is in D′(R×(0;+∞))9t |u− v|+ 9x(sgn (u− v)(f(u)− f(v)))6 sgn (u− v)g(h(u)− h(v)) + gh(v)

Since h′60, we have actually obtained

9t |u− v|+ 9x(sgn (u− v)(f(u)− f(v)))6gh(v) (38)

in D′(R×(0;+∞)).Now, if u0 − v0 ∈L1(R) we deduce by integration from (38)∫

R|u(x; t)− v(x; t)| dx6

∫R|u0 − v0| dx +

∫ t

0

∫Rg(x)h(v) dx ds; t¿0 (39)

Assume that

support g ⊂ [−1; 1] and h(0)=0 (40)

and let

v0(x)=R0(x)=

{uL¡0 if x¡0

uR¿0 if x¿0(41)



Let a(�)=f′(�) (recall that f′(0)=0 and f′′(�)¿0). The corresponding solution of the(Riemann) problem (34) is given by (t¿0)

v(x; t)=R(x=t)=

uL if x=t6a(uL)

a−1(x=t) if a(uL)6x=t6a(uR)

uR if x=t¿a(uR)

(42)

We will prove the following asymptotic result based on a paper by Chen and Frid [6]. (Forrelated results see also Dafermos [11].)

Theorem 4.1Assume that g∈C1(R), (40), u0 ∈L∞(R), and u0 − R0 ∈L1(R). Then we have

1T

∫ T

0|u(t�; t)− R(�)| dt→ 0 if T→+∞; a:e: in �∈R (43)

Moreover, if u∈L∞(R×R+), the convergence in (43) holds in L1loc(R�).

ProofFrom (39) we get for all t¿0∫

R|u(x; t)− R(x=t)| dx6

∫R|u0 − R0| dx +

∫ t

0

∫Rg(x)h(R(x=s)) dx ds

6 c0 +∫[−1;1]×]0; t]

g(x)h(R(x=s)) dx ds (44)

Now, for t¿max(1; [a(uR)]−1; [−a(uL)]−1),∫[−1;1]×]0; t]

g(x)h(R(x=s)) dx ds

=∫1=s¿x=s¿a(uR)

g(x)h(uR) dx ds+∫ t

0

∫|x|61

a(uL)6x=s6a(uR)

g(x)h(a−1(x=s)) dx ds

+∫−1=s6x=s6a(uL)

g(x)h(uL) dx ds

6c1 +∫ t

0

∫|x|61

a(uL)6x=s6a(uR)

g(x)h(a−1(x=s)) dx ds (45)

Moreover, since h(0)=0, we have

h(a−1(x=s))=0 +[h′(a−1(�))

1a′(�)

]x=s

x=s; �∈ [a(uL); a(uR)] and hence, with x=s∈ [a(uL); a(uR)],

h(a−1(x=s))6c2|x=s|6c2; |x=s|; if 0¡¡1



We then �nd ∫ t

0

∫|x|61

a(uL)6x=s6a(uR)

g(x)h(a−1(x=s)) dx ds6c3; t1− (46)

From (44)–(46) and for t¿max(1; [a(uR)]−1; [−a(uL)]−1), we deduce that∫R|u(x; t)− R(x=t)| dx6c4; t1−; with ∈]0; 1[ (47)

The desired result is thus a consequence of Theorem 2.2 in Reference [6]; where it is enoughto assume u∈L∞

loc(R+; L∞(R)) to deduce (43). The second part follows from Lebesgue’stheorem.

ACKNOWLEDGEMENTS

The authors are indebted to the referee for valuable suggestions and improvements, namely in Theo-rem 2.1 where he has proposed the suppression of an additional assumption on h and has given anadaptation of the proof under the weaker hypothesis that we include. JPD was partially supported byFCT, PRAXIS XXI, FEDER and Project PRAXIS=2=2.1=MAT=125=94.

REFERENCES

1. Liu T-P. Nonlinear resonance for quasilinear hyperbolic equation. Journal of Mathematical Physics 1987; 28:2593–2602.

2. Tartar L. Compensated compactness and applications to partial di�erential equations. Heriot-Watt Symposium,vol. 4. Pitman: New York, 1979.

3. Schonbek ME. Convergence of solutions to nonlinear dispersive equations. Communications in PartialDi�erential Equations 1982; 8:959–1000.

4. Vol’pert AI. The spaces BV and quasilinear equations. Mathematics of the USSR-Sbornik 1967; 2:225–267.5. Kruzkov SN. First order quasilinear equations in several independent variables. Mathematics of the USSR-Sbornik 1970; 10:217–243.

6. Chen G-Q, Frid H. Large-time behavior of entropy solutions of conservation laws. Journal of Di�erentialEquations 1999; 152:308–357.

7. Crandall MG. The semigroup approach to �rst order quasilinear equations in several space variables. IsraelJournal of Mathematics 1972; 12:108–132.

8. Godlewski E, Raviart PA. Hyperbolic Systems of Conservations Laws. Math�ematiques et Applications, vol no.3=4, Ellipses, Paris, 1991.

9. Murat F. L’injection du cone positif de H−1 dans W−1; q est compacte pour tout q¡2. Journal deMathematiques Pures et Appliquees 1981; 60:309–322.

10. DiPerna RJ. Uniqueness of solutions to hyperbolic conservation laws. Indiana University Mathematics Journal1979; 28:137–188.

11. Dafermos CM. Asymptotic behavior of solutions of hyperbolic balance laws. In Bifurcation Phenomena inMathematical Physics and Related Topics, Bardos C, Bessis D (eds). Reidel: Dordrecht, 1980; 521–533.


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