AMMAR - Scalar Conservation With General Boundary

7/21/2019 AMMAR - Scalar Conservation With General Boundary

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J. Differential Equations 228 (2006) 111139

www.elsevier.com/locate/jde

Scalar conservation laws with general boundarycondition and continuous flux function

Kaouther Ammar a, Petra Wittbold a,, Jose Carrillo b

a TU Berlin, Institut fr Mathematik, MA 6-4, Strasse des 17. Juni 136, 10623 Berlin, Germanyb Departamento de Matemtica Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received 30 May 2005; revised 29 March 2006

Abstract

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with

nonhomogeneous boundary condition: ut +div (u)= f on Q= (0, T ) , u(0, )= u0 on and

u= a on some part of the boundary (0, T ) . Existence and uniqueness of the entropy solution isestablished for any C(R;RN), u0 L

(),f L(Q),a L((0, T ) ). In theL1-setting,

a corresponding result is proved for the more general notion of renormalised entropy solution.

2006 Elsevier Inc. All rights reserved.

Keywords:Conservation law; Nonhomogeneous boundary conditions; Continuous flux; Penalization; L1-Theory;

Renormalized entropy solution

1. Introduction

Let be a bounded open set in RN with Lipschitz boundary if N >1. We consider the

following initial boundary value problem for a scalar conservation law:

P (u0, a , f )

ut

+ div (u) = f onQ = (0, T ) ,

u = a on= (0, T ) ,

u(0, ) = u0 on ,

* Corresponding author. Fax: +4931421110.

E-mail addresses:[email protected] (K. Ammar), [email protected] (P. Wittbold),

[email protected] (J. Carrillo).

0022-0396/$ see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.jde.2006.05.002


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112 K. Ammar et al. / J. Differential Equations 228 (2006) 111139

where :R RN is a continuous vector field,u0 L(),f L(Q)and a L().

It is well known that the main difficulty, when dealing with hyperbolic first-order equations,

is to make precise the meaning of the boundary condition which may not be assumed pointwise,

but has to be read as an entropy condition on the boundary. In the BV-setting, for a smoothflux function and regular data u0,a ,f such an entropy boundary condition has been defined

in [1]. However, this condition involved the trace of the BV-solutionuand could therefore not

be extended to the L-setting. ForL-data u0,a, f =0 and a Lipschitz continuous flux , a

new integral formulation of the boundary condition has been given by Otto (cf. [9,12]) who also

proved well-posedness of the problemP (u0, a, 0)in this sense.

For a merely continuous flux function , a different formulation of an entropy solution of

P (u0, a , f ) has been proposed in [4] in the particular case of a homogeneous boundary condi-

tion, i.e.,a =0, and well-posedness has been shown in this setting for arbitraryL-datau0, f.

Following [4] an entropy solution ofP (u0, 0, f )is a function u L(Q)satisfying

{u>k}

(u k)t+

(u) (k)

+ f +

(u0 k)+(0, ) 0 (1)

for any (k,) R D([0, T[ RN) such that k 0 and 0, and for any (k,) R

D([0, T[ ), 0, and{k>u}

(k u)t+

(k) (u)

f +

(k u0)+(0, x ) d x 0 (2)

for any (k,) R D([0, T[ RN) such that k 0 and 0, and for any (k,) R

D([0, T[ ), 0.

In [14], an attempt has been made to extend the definition of entropy solution given by Otto

and to prove well-posedness of problemP (u0, a , f )with a merely continuous flux function .

As pointed out in [14], a main difficulty in this case is that BV-a priori estimates seem to be out of

reach even when the data u0, a , f is assumed to be smooth. Due to this lack of strong compact-

ness standard approximation techniques (e.g., by vanishing viscosity) seem to fail. Therefore it

seems to be necessary to apply Young measure techniques and to study measure valued entropy

solutions ofP (u0, a , f )(cf. [14]).

In this paper we propose a notion of entropy solution of problem P (u0, a , f ) which is a

natural generalization of both notions of entropy solutions introduced by Otto and in [4], respec-

tively (cf. Section 2). We prove existence and uniqueness of this entropy solution of problem

P (u0, a , f )for continuous flux and general L-datau0,a ,f, without using Young measure

techniques. Instead we apply a very particular approximation technique using penalization which

ensures strong compactness in L1(Q) of the approximate solutions via monotonicity (cf. Sec-

tions 3 and 4).

In a quite recent work [13], Vovelle and Porretta have studied problem P (u0, a , f ) in the

general L1-setting. In order to deal with unbounded solutions, they have defined a notion of

renormalized entropy solution which generalizes the definition of entropy solutions introduced

by Otto in [12] in the L frame work. They have proved existence and uniqueness of such

generalized solution in the case when is locally Lipschitz and the boundary data a verifies the

following condition: max(a)L1(), wheremax is the maximal effective flux defined by

max(s) = {sup |f(t)|, t [s, s+]}.


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K. Ammar et al. / J. Differential Equations 228 (2006) 111139 113

In this paper, for a merely continuous flux and a measurable boundary data a : R

for which the maximal local flux is integrable on , i.e., (a,x) L1() where :R

R is defined by (s,x):= sup{|(r) (x)|, r [s, s+]}, we propose a notion of

renormalized entropy solution ofP (u0, a , f )which slightly generalizes the notion of solutionintroduced in [13]. Existence and uniqueness of this solution is proved for arbitrary L1-data u0andf(cf. Section 5).

2. Entropy solutions and main results in the case ofL-data

Let

+(x,k,a) := maxkr, sak

(r) (s)

(x)

and

(x,k,a) := maxakr, sk

(r) (s) (x),for anyk, a R, a.e.x , denoting the unit outer normal to . We propose the following

definition of an entropy solution ofP (u0, a , f ).

Definition 2.1.Let a L(),u0 L(),fL(Q). Anentropy solutionofP (u0, a , f )

is a functionu L(Q)satisfying

+x,k,a(t,x) Q(u k)+t+ {u>k}(u) (k) + {u>k}f

+

(u0 k)+(0, ) and (3)

x,k,a(t,x)

Q

(k u)+t+ {k>u}

(k) (u)

{k>u}f

+

(k u0)+

(0, ) (4)

for any D([0, T ) RN), 0, for all k R.

Remark 2.2.

(i) Note that an entropy solution ofP (u0, a , f )is, in particular, a weak solution of the equation

ut+div (u)= f. In fact, choosing D((0, T ) ), 0, k uL(Q) in (4), we find u

t

+ div (u)=f in D (Q). Moreover,u satisfies the initial

condition in the following sense: ess-limt0u(t, ) u01= 0.

(ii) Definition 2.1 is a natural extension of the definition of entropy solution given by Otto

(cf. [9,12]) in the case of a Lipschitz continuous flux function (cf. [12, Proposition 2]).

The boundary condition u = a is integrated in the integral entropy formulation.


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The main result of this paper is the following theorem.

Theorem 2.3.For any (u0, a , f )L()L()L(Q), there exists a unique entropy

solution of P(u0, a , f ).

Uniqueness of entropy solutions follows as a consequence of the following L1-comparison

principle for entropy solutions. Here, sign+ :R P(R) denotes the multi-valued function de-

fined by

sign+(r) =

0 ifr < 0,

[0, 1] ifr = 0,

1 ifr > 0.

Theorem 2.4.Fori = 1, 2, let(u0i , ai , fi )L() () L(Q)andui L

(Q)be

an entropy solution ofP (u0i , ai , fi ). Then there exists L

(Q)with sign+

(u1 u2)a.e.in Q such that, for any D([0, T[ RN), 0,

(x,a1, a2)

Q

(u1 u2)+t+ {u1>u2}

(u1) (u2)

+

Q

(f1 f2)+

(u01 u02)+(0, ). (5)

Remark 2.5. If u1

L(Q) satisfies (3) and (4) for data f1

, u01

, a1

and flux , then u1satisfies (3) and (4) with dataf1, u01, a1 and flux function(). In the same way, ifu2

is an entropy solution ofP (u02, a2, f2), thenu2 is an entropy solution ofP (u02, a2, f2)

with replaced by(). Consequently, under the assumptions of Theorem 2.4, one also has

existence of L(Q)with sign+(u2 u1)such that, for all D([0, T[ RN), 0,

such that

+(x,a1, a2)

Q

(u2 u1)+t+ {u2>u1}

(u2) (u1)

+ Q

(f2 f1)+

(u02 u01)+

(0, ).

Corollary 2.6.Fori =1, 2, let(u0i , ai , fi ) L() L() L(Q)andui L

(Q)be

an entropy solution ofP (u0i , ai , fi ). Then,

|u1 u2|(t)

t0

max{min(a1,a2)r, smax(a1,a2)}

(r) (s) (x)+

t0

|f1 f2| +

|u01 u02| (6)

for a.e.t (0, T ).


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Proof. The result is a direct consequence of Theorem 2.4, Remark 2.5 and the following equality

w+(x,a1, a2) + (x,a1, a2) = max

{min(a1,a2)r,smax(a1,a2)}(r) (s)

(x)

.

Remark 2.7. The contraction principle, i.e., inequality (6), is up to replacing || by | |

precisely the statement of [13, Theorem 3.1].

The proofs of Theorems 2.3 and 2.4 will be given in the following sections.

3. Smooth boundary data and a comparison result

As pointed out in the previous section, Definition 2.1 is a natural extension of the notion of

entropy solution defined in [12] in the case of Lipschitz continuous flux . A natural extension of

the entropy conditions (1), (2), proposed in [4] in the case of a homogeneous boundary condition,

would rather be of the following type:

for all(t, x) Q, for anyr > 0, for all D(B((t,x); r)), 0:

0

Q

(u k)+t+ {u>k}

(u) (k)

+ {u>k}f

+

(u0 k)+(0, ); (7)

for allk ess-sup{B((t,x);r)}a, and

0 Q

(k u)+t+ {k>u}(k) (u) {k>u}f +

(k u0)+(0, ); (8)

for all k ess-inf{B((t,x);r)}a, where the usual convention is used that max = ,

min= +.

Remark 3.1.(i) Note that (7), (8) imply the local semi-Kruzhkov inequalities in Q:

0

Q

(u k)+t+ {u>k}

(u) (k)

+ {u>k}f

+

(u0 k)+(0, ) and

0 Q

(k u)+t+ {k>u}

(k) (u)

{k>u}f

+

(k u0)+(0, )

for all k R, for any D([0, T[ ), 0. Moreover, conditions (7), (8) contain a family

of boundary entropy inequalities where the set of admissible test constants is restricted in terms

of the boundary data.

(ii) In the particular casea =0, a functionuL(Q)is an entropy solution ofP (u0, 0, f ),

u0 L(),f L(Q), in the sense of [4], i.e., usatisfies entropy conditions (1) and (2), if

and only ifu satisfies the entropy conditions (7) and (8).

(iii) Note that an entropy solution in the sense of Definition 2.1 always also satisfies the

family of entropy inequalities (7), (8). However, the converse implication is not true in general.

In particular, a function u L(Q)satisfying the weaker conditions (7) and (8) is, in general,

not unique.


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Indeed, it is well known that the problem without boundary condition: ut+div (u)= f

onQ,u(0, ) = u0,(u0, f ) L() L(Q), in general, admits more than one entropy solu-

tion, i.e., a function u L(Q) satisfying the differential equation in D(Q), the initial condition

ess-limt0 u(t, ) u01= 0 and, moreover, the local semi-Kruzhkov inequalities inside ofQ.Let u1, u2L(Q) be two different solutions of this type. Then, for an appropriately chosen

functiona L(),u1, u2 also satisfy the weak boundary entropy conditions (7), (8). Indeed,

if we choosea L()such that

ess-supB

a u1L(Q) u2L(Q),

ess-infB

a

u1L(Q) u2L(Q)

for any ball B RN+1, then, for u= u1, u2, the right-hand side of inequalities (7) and (8)

vanishes for the set of admissible test constantsk, and therefore the family of entropy inequalities(7) and (8) is trivially satisfied.

The existence of a function a L() with the described properties follows from simple

measure theoretical arguments (just look at the 1-dimensional case, e.g., =(0, 1), where the

boundary reduces to =(0, T ) {0} (0, T ) {1}. It is an elementary exercise in measure

theory to construct a Lebesgue measurable set A (0, T ) such that 0


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Remark 3.3. Under the assumptions of Theorem 3.2, one also has existence (compare with

Remark 2.5) of L(Q) with sign+(u2 u1) such that, for all D([0, T[ RN),

0, such that

+(x,a1, a2)Q

(u2 u1)+t+ {u2>u1}

(u2) (u1)

+

Q

(f2 f1)+

(u02 u01)+(0, ).

Corollary 3.4.Leta C(),u0 L(),f L(Q). ThenuL(Q)is an entropy solu-

tion ofP (u0, a , f )if and only ifu satisfies the family of entropy inequalities(7) and(8).

Proof. We only have to prove that (7) and (8) imply the entropy conditions (3) and (4). Tothis end note that, for any k R, the constant function (t,x) Q k is an entropy solution

of P (u0, a , f ) for data u0 =k , a =k , f =0. Therefore, if u L(Q) satisfies (7) and (8),

then, by the comparison result, Theorem 3.2, applied with u1= u and u2 k , there exists

sign+(u k) such that

(x,k,a)

Q

(u k)+t+ {u>k}

(u) (k)

+ f

+

(u0 k)

+

(0, ). (10)

For k R fixed, choosing (kn)n R with kn kas n , passing to the limit in inequality (10)

corresponding to kn, using the fact that, for any n sign+(u kn), one has limn n =

sign+0(u k) a.e. inQ, we obtain (3). In the same way one can prove that (8) implies (4).

Proof of Theorem 3.2. As usual we use Kruzhkovs technique of doubling variables (cf. [7,8])

in order to prove the comparison result (see also [4]). We choose two pairs of variables (t,x)

and (s,y) and consider u1 as a function of (s,y) Q and u2 as a function of (t,x) Q. For

arbitrary > 0, let(B

i )i=0,...,m be a covering of satisfyingB

0 = , and such that, foreachi 1,B i is a ball of diameter , contained in some larger ball

Bi withBi is partof the graph of a Lipschitz function. Let ( i )i=0,...,m denote a partition of unity subordinate to

the covering(B i )i . Let D((0, T ) RN), 0.

Note that, due to the fact that both functions u1, u2 satisfy the classical semi-Kruzhkov

entropy inequalities for any k R in D([0, T[ ), one can prove exactly as in [4] (see

also [6]) that u1, u2 satisfy the following local comparison principle: there exists L(Q)

with sign+(u1 u2)a.e. inQ such that, for any D([0, T[ ), 0,

0 Q

(u1 u2)

+

t+ {u1>u2}(u1) (u2) +

Q

(f1 f2)+

(u01 u02)+ (0, ). (11)


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In particular, (11) holds with= 0 . Now, leti {1, . . . , m }be fixed in the following. For

simplicity, we omit the dependence on and i and simply set = i , B=Bi . As in [4], we

choose a sequence of mollifiers (n)n in RN such that x n(x y) D(), for all y B ,

n(x) = n(x y)dyis an increasing sequence for all x B, and n(x) = 1 for all x Bwithd(x,RN \ ) > c/nfor somec=c(i, )depending onB = B i . Let (m)m denote a sequence

of mollifiers in R with supp m (2/m, 0). Define the test function

m,n(t,x,s,y) = (t,x)(x)n(x y)m(t s).

Note that, form, nsufficiently large,

(s,y) m,n(t,x,s,y) D]0, T[ RN

, for any(t, x) Q,(t,x) m,n(t,x,s,y) D[0, T[ , for any(s,y) Q.Moreover, the function

n(t,x) =

Q

m,n(t,x,s,y)d(s,y) = (t,x)(x)

n(x y)dy

T0

m(t s)ds

= (t,x)(x)n(x) (12)

satisfies n D([0, T[ ), 0 n ,n.

Let ki :=maxBa1. Then, as u1 satisfies (7), choosing k=u2(t,x) ki and (s,y)=

m,n(t,x,s,y)in (7), for a.e.(t, x) Q, we get

0

Q

u1 u2 k

i

+(m,n)s+ {u1>u2ki }

(u1)

u2 k

i

y m,n

+ Q {u1>u2ki } f1m,n.As u2 is an entropy solution of P (u02, a2, f2), choosing k = u1(s,y) k

i , (t,x) =

m,n(t,x,s,y)in (4), for a.e.(s,y) Q, we find

0

Q

u1 k

i u2

+(m,n)t+ {u1ki >u2}

u1 ki

(u2)

x m,n

Q {u1ki >u2}f2m,n+ u1 ki u02+m,n(0, x , s , y )

=

Q

u1 k

i u2 k

i

+(m,n)t+

Q

{u1ki >u2ki }

u1 ki

u2 k

i

x m,n


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Q

{u1ki >u2ki }

{u2ki } f2m,n+

u1 k

i u02 k

i

+m,n(0, x , s , y )

+ Q

ki u2+(m,n)t+ {ki >u2}ki (u2) x m,n

Q

{ki >u2}f2m,n+

ki u02

+m,n(0,x,s,y),

where the last equality follows from the fact that (r ks)+ =(r k s k)+ +(k s)+,

and{rk>s }= {rk>sk}{sk}+ {su2ki }

u1 ki

u2 k

i

x ()n(x y)m(t s)

+

QQ

u1 k

i u2 k

i

+tn(x y)m(t s)

+ QQ {u1ki >u2ki } {u1>ki } (f1 {u2ki }f2)m,n+

Q

u1 k

i u02 k

i

+m,n(0, x , s , y )

+

QQ

ki u2

+(m,n)t+ {k

i>u2}

ki

(u2)

x m,n

QQ {ki >u2}f2m,n+ Q ki u02

+m,n(0,x,s,y). (13)

Denote the seven integrals on the right-hand side of the preceding inequality by I1, . . . , I 7successively. There is no difficulty in passing to the limit with m and n suc-

cessively in I1, I2. As to I3, one proves as in [4] (cf. also [5,6]) that limsup m,n I3 Q

1({u1>ki }(f1 {u2ki} f2))(t,x)(x) for some 1 L

(Q) with 1 sign+(u1

u2 ki )a.e. inQ. IntegralI4 can also be estimated as in [4]. To this end, define

m,n(x,s,y) = (0,x)(x)n(x y)

T

s

m(r ) d r= (0,x)(x)n(x y)

2/m

s2/m

m(r)dr.

Note thatm,n(x, ,) D([0, T[ RN),m,n 0, for anyx . Asu1 satisfies (7),


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I4=

Q

u1 k

i u02 k

i

+(m,n)s

2/m0

{u1>ki u02}(u1) ki u02 y m,n+

2/m0

{u1>ki u02}f1m,n+

u01 k

i u02

+m,n(x, 0,y).

Obviously, the first two integrals tend to 0 asm whereas the last integral converges to

(u01 ki u02)

+(0,x)(x)n(x y).

Next, note that, by the Fubini theorem and (12),

I5+ I6+ I7=

Q

ki u2

+(n)t+ {ki >u2}

ki

(u2)

xnd(t,x)

Q

{ki>u2}f2nd(t,x) +

ki u02

+n(0,x)dx.

Following [4], we define the functional L := L(u2)on D([0, T[ RN)by

L( ) = Q

ki u2+t+ {ki >u2}ki (u2) x f2+

ki u02+ (0,x)dx.Asu2 is an entropy solution, we have

L( ) +

x, ki , a2

0

for all D([0, T[ RN), 0, i.e., L is the sum of the positive linear functional

D([0, T[ RN) L( ) + (x,ki , a2) and the negative linear functional D([0, T[ RN)

(x,ki , a2). Sincen= n is an increasing sequence satisfying

0 , L(n)+

(x,ki , a2)n is a bounded and increasing sequence and thus con-

verges,

(x,ki , a2)n is a bounded decreasing sequence and therefore converges. As a

consequence,I5+ I6+I7= L( n)converges as n . Combining the preceding estimates

ofI1, . . . , I 7, passing to the limit in (13) with m and n to successively yields

0

Q

u1 ki u2 k

i

+

ti+

Q

{u1ki >u2ki}

u1 k

i

u2 k

i

x ( i )

+Q

1{u1>ki } (f1 {u2ki }f2) i+

u01 k

i u02 k

i

+(0,x)i (x)

+ limn

L( n). (14)


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This is half of the inequality to be proved. In order to prove the second half, we choose now

as a test function

m,n(t,x,s,y) = (s,y)(y)n(y x)m(s t).

Then, form, nsufficiently large,

(s,y) m,n(t,x,s,y) D

[0, T[

, for any(t, x) Q,

(t,x) m,n(t,x,s,y) D

]0, T[ RN

, for any(s,y) Q.

As u1= u1(s,y) satisfies (7), choosing k= u2(t,x)ki and =m,n(t,x, ,)in (7) (note

that, due to the new choice of the test function, this choice is admissible), for a.e.(t, x) Q, we

get

0

Q

u1 u2 k

i

+(m,n)s+ {u1>u2ki }

(u1)

u2 k

i

y m,n

+

Q

{u1>u2ki }f1m,n+

u01 u2 k

i

+m,n(t,x, 0, y)

=

Q

u1 ki u2 k

i

+

(m,n)s+ {u1ki >u2ki }

u1 k

i

u2 k

i

y m,n

+Q

{u1ki >u2ki }{u1ki } f1m,n+

u01 k

i u2 k

i

+m,n(t,x, 0, y)

+

Q

u1 k

i

+(m,n)s+ {u1>ki }

(u1)

ki

y m,n

+ f1m,n+

u01 k

i

+m,n(t,x, 0, y)

,

where for the last equality we have used the fact that(r s k)+ = (r k s k)+ + (r k)+,

{r>sk}= {rk>sk}{rk}+ {r>k}, for allr, s,k R.

As u2= u2(t,x) is an entropy solution, choosing k= u1(s,y)ki , =m,n in (4) yields,

for a.e.(s,y) Q,

x, u1(s,y) ki , a2

m,n

Qu1 ki u2+(m,n)t+

Q

{u1

k

i >u

2}u1 ki (u2) x m,n

Q

f2m,n.Note that the integral on the left is

(x,ki , a2)m,n. Moreover, obviously,

(r k s)+ = (r k s k)+ for all r,s,k R. Therefore, integrating the preceding


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inequalities in (t,x) respectively (s,y) over Q, summing up, using the same type of argu-

ments as above, passing to the limit with m, n successively, for some 2 L(Q) with

2 sign+(u1 k

i u2), we obtain

x, ki , a2

i

Q

u1 k

i u2 k

i

+ti+

Q

{u1ki u2ki}

u1 ki

u2 k

i

x ( i )

+

Q

2{u2k}{((u1) (k)) y +f1} +

(u01 ki )

+ (0, y).

Using the same arguments as above, we can prove thatL( n)converges (as L( n)) withn.Note also that(r k s k)+ + (r k s k)+ = (r s)+, for allr, s,k R. Moreover, if we

define

:= 1{u1>ki } + 2{u2ki }

+ 2{u2


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Q

(u1 u2)+

(1 m)ti

+ {u1u2}

(u1) (u2)

x

(1 m)i

+ Q

(f1 f2)(1 m)i+

(u01 u02)+

(0,x)(1 m)i (x)

x, ki , a2

i (1 m) limn

L

i (1 m)n

limn

L i (1 m)n=

x, ki , a2

i limn

L

i (n mn)

limn

L i (n mn).

Note that i nm= i m for n sufficiently large. Therefore, limm limnL

( i (nmn)) = limm limnL( i (n mn)) = 0, and thus, passing to the limit with m in the preceding inequality yields

Q

(u1 u2)+ti+ {u1u2}

(u1) (u2)

x ( i ) +

Q

(f1 f2) i

+

(u01 u02)+(0,x)i (x)

x, ki , a2

i

for all i=1, . . . , m . Summing over i =0, . . . , m , taking into account the local inequality

(11) fori = 0, we findQ

(u1 u2)+t+ {u1u2}

(u1) (u2)

x +

Q

(f1 f2)+

(u01 u02)+(0, x)

(17)

m

1 x, ki , a2 i (18)

for any > 0.

Now, let >0 and choose >0 such that, (t,x),(s,y) with d ((t, x),(s,y)) < ,

|a1(t,x) a1(s,y)| . Then, for any(t, x) Bi ,i,

ki = maxB

i a1 a1(t,x) + .

Therefore, (17) implies using the monotonicity of

ink ,Q

(u1 u2)+t+ {u1u2}

(u1) (u2)

x


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+

Q

(f1 f2)+

(u01 u02)+(0, x)

m1

(x,a1+ , a2) i =

(x,a1+ , a2),

for any > 0. By continuity of we deduce (9).

4. Existence of entropy solutions

Due to the comparison principle, Theorem 3.2, it is sufficient to prove the following existence

result for smooth boundary data:

Theorem 4.1. Let u0 L(), f L(Q), a C(). Then there exists a unique entropy

solutionu L(Q)ofP (u0,f,a).

The main difficulty in proving existence of an entropy solution of the nonhomogeneous scalar

conservation law with only continuous flux is that, even for smooth data, for the standard ap-

proximation procedures (e.g., regularization of the flux, vanishing viscosity method), BV-a priori

estimates do not seem to be available (cf. [14]). For Lipschitz continuous flux functions , this

difficulty does not occur. In order to overcome the lack of BV-a priori estimates in the general

case, in the literature (cf., e.g., [14]) Young measure techniques have been applied and measure-

valued entropy solutions had to be introduced.We show that, when choosing a different approximation procedure, one can prove strong

L1-compactness of the sequence of the approximate solutions without using Young measure

techniques, thus without being obliged to deal with a concept of measure-valued entropy solu-

tions. We stress that, even in our particular approximation procedure, we are still not able to

prove BV-a priori estimates if the data is smooth. The strong L1-compactness of the sequence

of approximate solutions is a consequence of the monotonicity of the penalization procedure we

use. The idea is to approximate problem P (u0,f,a)via a sequence of homogeneous Dirichlet

problems for the scalar conservation law on some larger domain

Q=(0, T )

,

, and

to introduce an appropriate penalization term onQ\Q which formally forces the approximatesolution to converge to (a continuous extension of)aonQ \ Q. Details are given in the following.Proof of Theorem 4.1. Let denote some Lipschitz domain strictly larger than ,Q=(0, T ) . We define the trivial extension by 0 of the data u0,fon the larger domain:

u0:=

u0 on ,

0 on\ , f :=

f onQ,

0 onQ \ Q.Let a denote a continuous extension of a onto

Q satisfying aL(

Q) aL(). Let

m, n N (the penalization parameters) and define the penalization term m,n(t,x,r) :=

Q\Q(m(r a(t,x)))+ n(a(t,x)r )+, r R, a.e. (t,x)Q. Note that m,n is Lipschitzcontinuous inr , uniformly in(t, x):m,n(t,x,r) m,n(t,x,s) (m + n)|r s|


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for anyr, s R, a.e.(t, x) Q. Moreover, for allm, m,n,n N withn n,m m,m,n(t,x,r) m,n (t,x,r) =

Q\Q(n n)a(t,x) r

+ 0 and

m,n(t,x,r) m,n(t,x,r) = Q\Q(m m)r a(t,x)+ 0,for allr R, a.e.(t, x) Q, and

limm,n

m,n(t,x,r) =

0 ifr R,(t, x) Q,

R ifr = a(t,x),(t,x) Q \ Q, otherwise.

Consider the extended, penalized problem with homogeneous boundary condition:

Pm,n(u0, f )

ut

+ div (u) + m,n(u) = f onQ,u = 0 on= (0, T ) ,u(0, ) = u0 on.

In [4], existence of an entropy solution u C ([0, T]; L1(Q)) L(Q) (obtained via non-linear semi-group theory, cf. [3]) has been shown for problem P0,0(u0, f ), i.e., the problem

without penalization term. Due to the Lipschitz continuity ofm,n, using Banachs fixed point

theorem, we immediately deduce existence of an entropy solution um,nC ([0, T]; L1(Q))L(Q) of the penalized problem Pm,n(u0, f ), um,n being the unique semi-group and entropy

solution of problem

P (u0, 0,g)

ut

+ div (u) = g onQ,u = 0 on,u(0, ) = u0 on

without penalization and right-hand side g= f m,n(um,n).

By [4], a comparison principle holds for these entropy solutions. In particular, entropy solu-

tions for different penalization parameters can be compared: for any m, m, n N withm m,

there exists L(Q)with sign+(um,n um,n)a.e. onQsuch that, for a.e.t (0, T ),

um,n(t, ) um,n(t, )+

t0

f m,n(um,n)

f m,n(um,n)

=

t

0

m,n(um,n) m,n(um,n)

Q\Q(m

m)(um,n a)+

t0

m,n(um,n) m,n(um,n)+

0.


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Consequently,um,n um,na.e.(t, x) Q.In the same way, one can prove that, for all m, n, n N withn n,

um,n um,n a.e.(t, x) Q.This comparison result already ensures the a.e. convergence of the solutionsum,nas, succes-

sively, m and n .

In order to get anL-bound on the approximate solutions, letc:= fL(Q) + u0L() +

aL( )+ 1. Note thatv : (t,x) Q c(t+ 1)is a classical solution of

vt+ div (v) = c onQ,v= c(t+ 1) on

,

v(0, ) = c on,and thus, of course, also an entropy solution ofP(c,c(t+1),c). Therefore, by the comparisonprinciple, Theorem 3.2, there exists sign+(um,n v), such that, D([0, T[)

+,

0 =

+

x,c(t+ 1), 0

Q

um,n c(t+ 1)+

t+

f m,n(um,n) c

.

By the choice ofc, it follows that, for a.e.t (0, T ),

um,n(t ) c(t+ 1)+

t0

m,n(um,n) n

t0

(a um,n)+ = 0,

and thus

um,n c(t+ 1) a.e. onQ, for anym, n N. (19)In the same way one can prove that c(t+1) um,n a.e. onQ, i.e., (um,n)m,n is uniformlybounded in L(Q). As a consequence, passing to a subsequence if necessary and using thediagonal principle, there exists a sequence un= um(n),n which converges in L

1(Q) as n to some function u L(Q). In order to prove that u is an entropy solution of P (u0, a , f ),by Corollary 3.4, it is sufficient to prove that u satisfies the family of inequalities (7), (8).

To this end, let (t,x) Q, r >0, D([0, T[ RN)+ with supp( ) B((t, x); r). Note

that, ifB((t,x); r) = , using the convergence and L-boundedness of un and the fact

that the penalization term m(n),n(un) = 0 on Q, there is no problem to pass to the limit

with n in inequalities (3), (4) for any k R, and it follows that u satisfies the semi-

Kruzhkov inequalities locally in Q. Now, suppose that B((t,x); r) = , and let us prove

the boundary entropy inequalities. To this end, let > 0, k maxB((t,x);r)

a + . As

a is a continuous extension of a, there exists > 0 such that maxQ a k + /2, where

Q := {(t,x) [0, T[ RN; dist((t,x),Q) }. Replacing the test function , if necessary,

by with D([0, T[ RN)+ satisfying =1 on supp and = 0 outside Q , we may

assume that= 0 outsideQ . By (10), we have


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0

Q

(un k)+t+ {un>k}

(un) (k)

+ {un>k}f

+ QQ \Q (un k)+

t+ {un>k}(un) (k) + {un>k}f

QQ \Q

{un>k}m(n),n(un). (20)

Denote the integrals on the right-hand side byI1,I2,I3 successively. Note that(a un)+ = 0

on{un> k} Q , thus

I3= m(n) QQ \Q {un>k}(un a)+ ( 0).

Then, due to the L-boundedness of(un)n, (20) implies

limsupn

m(n)

QQ \Q

{un>k}(un a)+ C

for some constant C . It follows thatu ka.e. on supp( ). As a consequence, we have

limn

I2=

QQ \Q

(u k)+t+ {u>k}

(u) (k)

= 0.

Therefore, neglecting the negative term I3, passing to the limit in (20) yields inequality (7), for

k maxB((t,x);r)a + . As >0 is arbitrary, (7) holds for any k maxB((t,x);r)a . In the

same way one can prove that the family of entropy inequalities (8) holds. We have thus proved

thatu is an entropy solution ofP (u0, a , f ).

Proof of Theorem 2.3. Let (u0, a , f )L() L() L(Q). Let (an)n C()with

an ain L1() as n and such that anL() Constfor some constantConst, for all n.

By Theorem 4.1, for anyn, there exists an entropy solutionun ofP (u0, an, f). By Theorem 3.2

and Corollary 3.4, for anym, n N, for a.e.t (0, T ),

un(t) um(t) t0

max{min(an,am)r, smax(an,am)}

(r) (s) (x).Therefore, un converges, as n , in L

(0, T; L1()) to some function u L(0, T;

L1()). Moreover, by (19), (un)n remains uniformly bounded in L(Q), and thus also

u L(Q). Passing to the limit in inequalities (3), (4) corresponding to un yields that u is

an entropy solution ofP (u0, a , f ).


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Now, let v L(Q) be an arbitrary entropy solution ofP (u0, a , f ). By Theorem 3.2 and

Remark 3.3, for a.e. t (0, T ),

un(t) v(t) t0

max{min(an,a)r,smax(an,a)}

(r) (s) (x),for any n N. Therefore, un converges to v as n in L

(0, T; L1()), thus v= u, the

entropy solution obtained by approximation, which completes the proof of uniqueness of an

entropy solution ofP (u0, a , f ).

Remark 4.2.In the L-setting, the entropy solution can be equivalently defined as follows.

Definition 2.1.Let a L((0, T ) ), u0 L(),f L(Q). Anentropy solution ofP (u0, a , f )is a functionu L

(Q)satisfying

a(t,x) k+

Q

(u k)+t+ {u>k}

(u) (k)

+ {u>k}f

+

(u0 k)+(0, ) and (21)

k a(t,x)+ Q(k u)+t+ {k>u}(k) (u) {k>u}f

+

(k u0)+(0, ) (22)

for any D([0, T ) RN), 0, for allk R, where :R+ R+ is a modulus of continuity

of .

However Definition 2.1 is more advantageous in theL1-setting as we will see in the following

section.

5. Renormalized entropy solutions

In the preceding sections we have only considered problem P (u0, a , f ) for L-data. It is

also possible to extend the results to theL1-setting. In fact, in [2], we have introduced the notion

of a renormalized entropy solution for the Cauchy problemut+ div (u)=f on(0, T ) RN,

u(0, )= u0 on RN and proved existence and uniqueness of this solution for arbitrary L1-data

u0, f, for any locally Lipschitz continuous flux function . In [5] we have extended the def-

inition of a renormalized entropy solution to the problem P (u0

, a , f ) in the particular case of

a homogeneous boundary condition, i.e., a =0, and proved its existence and uniqueness for

any(u0, f )L1() L1(Q)for any continuous flux function . In [13], again only for a lo-

cally Lipschitz continuous flux , existence and uniqueness of a renormalized entropy solution

ofP (u0, a , f ) has been proved for arbitrary L1-data u0, f, and for any measurable boundary


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dataa M()withmax(a) L1(), wheremax :R R is the so-called maximal effective

flux defined bymax(s) := sup{|(r)|; r [s, s+]}, for anys R.

In our setting, for a merely continuous flux function :R RN and for any measur-

able boundary data a : R

with (a,x) L

1

() where :R

R

is defined by(s,x) := sup{|(r) (x)|, r [s, s+]}, we define a renormalized entropy solution of

P (u0, a , f )in theL1-setting as follows:

Definition 5.1.Leta M()with(a,x) L1(),u0 L1()andf L1(Q). A functionu

ofL1(Q)is said to be a renormalized entropy solutionofP (u0, a , f )if there exist some families

of non-negative bounded measuresl:= l (u)and l := l (u)on Q such that

l ,l l+

0,

and the following entropy inequalities are satisfied: for all k R, for all l k, for any D([0, T ) RN), 0,

+(x,k,a l) +

Q

(u l k)+t+ {ul>k}

(u l) (k)

+ {ul>k }f

+

(u0 l k)+(0, ) l , , (23)

and for all k R, for all l k, for any D([0, T ) RN), 0,

(x,k,a l) +

Q

(k u l)+t+ {k>ul}

(k) (u l)

{k>ul}f

+

(k u0 l)+(0, ) l , . (24)

Remark 5.2.The preceding definition of renormalized entropy solution generalises the one in-

troduced in [6] for homogeneous boundary data and the definition introduced in [13] for moregeneral boundary data.

Proposition 5.3. Letu0 L(), f L(Q), aL(). Then u L(Q) is an entropy

solution ofP (u0, a , f )iffu is a renormalized entropy solution ofP (u0, a , f ).

Proof. Let u be an entropy solution ofP (u0, a , f ). Then for all k R, for all l k, for any

D([0, T ) RN), 0

+

(x,k,a l) + Q(u l k)+t+ {ul>k }(u l) (k) + {ul>k }f

+

(u0 l k)+(0, )


20/29


=

+(x,k,a) +

Q

(u k)+t+ {u>k}

(u) (k)

+ {u>k}f

+

(u0 k)+

(0, )

+

(x,l,a)

(u0 l)+

(0, )

Q

(u l)+t+ {u>l}

(u) (l)

+ {u>l}f

+

+(x,k,a l)

+(x,k,a)+

+(x,l,a)+

Q

{u>l}f

+(x,l,a) Q (u l)

+t+ {u>l}(u) (l) + {u>l}f

(u0 l)+(0, )

Q

{u>l}f+

+(x,k,a l) +(x,k,a) + +(x,l,a)

=:

T(k,l,a) .

Note that

T(k,l,a) =

max{kr,sl}

(r) (s) (x) max{kr,sa}

(r) (s) (x)+ max

{lr,sa}

(r) (s) (x){a>l} 0.Let

l, :=

+(x,l,a) +

Q (u l)+t+ {u>l}

(u) (l)

+ {u>l}f

+

(u0 l)+(0, ) +

Q

{u>l}f.

Then, l is a non-negative measure on Q and l 0 for l uL(Q) + u0L() +

aL( ).Moreover,

l

Q|f|{u>l}+

+(x,l,a) +

(u0 l)

+,

and by the preceding inequalities,l satisfies (23).

Working on the second entropy inequality, we construct a family of bounded non-negative

measures(l )l onQ


21/29


l , :=

(x,l,a) +

Q

(l u)+t+ {l>u}

(l) (u)

{l>u}f

+

(l u0)+(0, ) + Q

{l>u}f+,

which satisfy (24) andl

(u0 l)

dx+

(x,l,a) +

Q

|f|{u


22/29


+(x,k,a l) +

Q

(un l k)

+t+ {unl>k }

(un l) (k)

+ {unl>k}fn

+

un0 l k+(0, ) nl, , (25)for allk R, for all l k, for any D([0, T ) RN), 0,

(x,k,a l) +

Q

(k un l)

+t+ {k>unl}

(k) (un l)

{k>unl}fn

+ k un0 l

+(0, )

nl , , (26)

and, moreover, for anyn |l|,

nl Q

|fn|{un>l}+

+(x,l,an) +

un0 l

+

Q

|f| +

+(x,l,a) +

(u0 l)+, and (27)

nl Q

|fn|{unl}+

Q

{ul}|f|.

Arguing similarly, we prove that u satisfies the renormalized entropy inequality (23) with

l satisfying liml l =0 and it follows that u is a renormalized entropy solution of

P (u0, a , f ).

Uniqueness. Uniqueness of the renormalized entropy solution follows as a consequence of the

following comparison result.


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Proposition 5.7.Let(u10, a1, f1) L() C() L(Q),(u20, a2, f2) L

1() M()

L1(Q), with(a2, ) L1(). Letu1be the entropy solution of the problem P (u

10, a1, f1),u2be

the renormalized entropy solution of the problem P (u20, a2, f2). Then, there exists L(Q)

with sign+

(u1 u2l) a.e. in Q such that, for any D([0, T[ RN), 0, for anyl a1L(),

l ,

(x,a1, a2 l)

Q

{u1>u2l}

(u1) (u2 l)

+

Q

(u1 u2 l)+t+

Q

(f1 f2)

+

(u01 u02 l)+ (0, ). (29)

We postpone the proof of this result and show first how to deduce uniqueness of renormalized

entropy solution.

Letv be a renormalized entropy solution ofP (u0, a , f )and unbe defined as above. Then by

Proposition 5.7, there exists n L(Q)withn sign

+(un v ln) a.e. inQsuch that, for

any D([0, T[ RN), 0, for anyln max(n, anL()),

ln ,

(x,an, a ln)

Q

{un>vln}

(un) (v ln)

+

Q

(un v ln)+t

+

Q

n(fn f )+

(u0n u0 ln)+ (0, ). (30)

Similarly, we prove that there exists n L(Q)with n sign+(v ln u1)a.e. inQ suchthat, for any D([0, T[ RN), 0, for anyln max(n, anL()),

ln ,

+(x,an, a ln)

Q

{un


24/29


Proof of Proposition 5.7. Letki andm,n(t,x,s,y):= (t,x)(x)n(x y)m(t s) be de-

fined as in Section 3. Then, as u1= u1(s,y) satisfies (7), choosing k= u2(t,x)ki l , with

l > ki and(s, y) = m,n(t,x,s,y) in (7), for a.e.(t, x) Q, we get

0 Q

u1 u2 k

i l

+(m,n)s+

Q

{u1>u2ki l}

(u1)

u2 ki l

y m,n

+

Q

{u1>u2ki l}f1m,n.

As u2 is a renormalized entropy solution of P (u02, a2, f2), choosing k = u1(s,y) ki ,

(t,x) = m,n(t,x,s,y) in (4), for a.e.(s,y) Q, we find

l (u2), m,n

Q

u1 k

i (u2 l)

+(m,n)t+ {u1ki >u2l}

u1 ki

(u2 l)

x m,n

Q

{u1ki >u2l}f2m,n+

u1 k

i (u02 l)

+m,n(0, x , s , y )

= Q u1 ki u2 k

i

+(m,n)t+ Q {u1ki >u2ki }u1 k

i u2 ki x m,n

Q

{u1ki >u2ki }

{u2ki } f2m,n+

u1 k

i u02 k

i

+m,n(0, x , s , y )

+

Q

ki (u2 l)

+(m,n)t+ {ki >u2l}

ki

(u2 l)

x m,n

Q {ki >u2l}f2m,n+ ki (u02 l)

+m,n(0,x,s,y).

Integrating both inequalities in(t, x), respectively in(s,y)overQ, summing up, we obtain

l , m,n

QQ

{u1ki >u2ki}

u1 ki

u2 k

i

x ()n(x y)m(t s)

+ QQ

u1 ki u2 ki +tn(x y)m(t s)+

QQ

{u1ki >u2ki }

{u1>ki }(f1 {u2ki }

f2)m,n


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+

Q

u1 k

i u02 k

i

+m,n(0, x , s , y ) +

QQ

ki (u2 l)

+(m,n)t

+ QQ

{ki >u2l}ki (u2 l) x m,n

QQ

{kiu2l}f2m,n+

Q

ki u02 l

+m,n(0,x,s,y). (32)

Denote the seven integrals on the right-hand side of the preceding inequality by J1, . . . , J 7successively. There is no difficulty in passing to the limit with m , n successively

in J1, J2. As to J3, one proves that limsupm,n J3

Q1({u1>ki }(f1 {u2k

i} f2))

(t,x)(x) for some 1 L

(Q) with 1 sign+

(u1 > u2 k

i ) a.e. in Q. Integral J4 canalso be estimated as in Section 3, and we prove that

J4=

Q

u1 k

i u02 k

i

+(m,n)s

2/m0

{u1>ki u02}

(u1)

ki u02

y m,n

+

2/m0

{u1>ki u02}f1m,n+

u01 k

i u02

+m,n(x, 0,y).

Obviously, the first two integrals tend to 0 as m whereas the last integral converges to

(u01 ki u02)

+(0,x)(x)n(x y).

Combining the preceding estimates ofJ1, . . . , J 7, passing to the limit in (32) with m and n

to successively yields

l (u2), m,n

Q

u1 k

i u2 k

i

+ti+

Q

{u1ki >u2ki }

u1 ki

u2 k

i

x ( i )

+

Q

1{u1>ki } (f1 {u2ki }

f2) i+

u01 k

i u02 k

i

+(0,x)i (x)

+ limn

L(u2 l)(n). (33)

In order to prove the second half of the inequality, we choose now as a test function

(s,y)(y)n(y x)m(s t ).

As u1= u1(s,y) satisfies (7), choosingk= u2(t,x)ki l and =m,n(t,x, ,) in (7)

we get,


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0

Q

u1 u2 k

i l

+(m,n)s

+ {u1>u2ki

l}(u1) u2 ki l y m,n+Q

{u1>u2ki l}f1m,n+

u01 u2 k

i l

+m,n(t,x, 0, y)

=

Q

u1 k

i u2 k

i l

+(m,n)s+

Q

{u1ki >u2ki l}

{u1ki } f1m,n

+

u01 k

i u2 k

i l

+m,n(t,x, 0, y)

+

u01 ki +m,n(t,x, 0, y) + Q

f1m,n

+

Q

u1 k

i

+(m,n)s+ {u1>ki }

(u1)

ki

y m,n

+

Q

{u1ki >u2kil}

u1 ki

u2 k

i l

y m,n.

As u2 =u2(t,x) is a renormalized entropy solution, choosing k= u1(s,y) ki , =m,nin (4) yields, for a.e.(s,y) Q,

l , m,n

x, u1(s,y) ki , a2 l

m,n

Q

{u1ki >u2l}

u1 ki

(u2 l)

x m,n f2m,n

+ Qu1 ki u2 l+(m,n)t.

Note that the integral on the left is l , m,n

(x,ki , a2 l)m,n. As (r

k s )+ =(r k s k)+ for all r, s,k R, integrating the preceding inequalities in (t,x),

respectively in(s, y)overQ, summing up, using the same type of arguments as above, passing to

the limit with m, n successively, for some 2 L(Q) with 2sign

+(u1 ki u2 l),

we obtain

l , m,n

x, ki , a2 l i

Q

u1 k

i u2 k

i l

+ti


27/29


+

Q

{u1ki u2ki l}

u1 ki

u2 k

i l

x ( i )

+ Q

2{u2l


28/29


l , i

x, ki , a2 l

i (1 m) limn

L(u1) i (1 m)n lim

nL(u2 l) i (1 m)n= l , i

x, ki , a2 l

i limn

L(u1) i (n mn) lim

n

L(u2 l)

i (n mn)

.

Passing to the limit withm in the preceding inequality yields

Q (u1 u2 l)+ti+ {u1u2l}(u1) (u2 l) x ( i )

+

Q

(f1 f2) i+

(u01 u02 l)+(0,x)i (x)

x, ki , a2 l

i l , i ,

for all i=1, . . . , m . Summing over i=0, . . . , m , taking into account the local inequality

fori = 0, we findQ

(u1 u2 l)+t+ {u1u2l}

(u1) (u2 l)

x

+

Q

(f1 f2)+

(u01 u02 l)+(0, x)

m

1 x, ki , a2 l l , i (36)

for any > 0.

Proceeding again as in Section 3, we deduce thatQ

(u1 u2 l)+t+ {u1u2l}

(u1) (u2 l)

x

+ Q (f1 f2)+ (u01 u02 l)+(0, x)

m1

(x,a1+ , a2 l)i l, i


29/29


=

(x,a1+ , a2 l) l , ,

for any > 0. By continuity of the result follows as 0.

Extensions and further remarks

Using similar arguments, one can prove the existence of renormalized entropy solutions for

triply degenerate problems of type:

b(v)t g(v) + div (u) = f

with nonhomogeneous boundary conditions andL1 data.

In the special case b IdR, this type of problem is already treated in [10,11,15]. We willconsider the general case in a forthcoming paper.

References

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Differential Equations 4 (9) (1979) 10171034.

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Super. Pisa Cl. Sci. 29 (2000) 313329.

[3] Ph. Bnilan, M.G. Crandall, A. Pazy, Nonlinear Evolution Equations in Banach Spaces, book in preparation.

[4] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999) 269361.

[5] J. Carrillo, P. Wittbold, Uniqueness of renormalized solutions of degenerate ellipticparabolic problems, J. Differ-ential Equations 156 (1999) 93121.

[6] J. Carrillo, P. Wittbold, Renormalized entropy solutions of scalar conservation laws with boundary condition, J. Dif-

ferential Equations 185 (2002) 137160.

[7] S.N. Kruzhkov, Generalized solutions of the Cauchy problem in the large for first-order nonlinear equations, Soviet

Math. Dokl. 10 (1969) 785788.

[8] S.N. Kruzhkov, First-order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970) 217

243.

[9] J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDEs, Appl.

Math. Math. Comput., vol. 13, Chapman & Hall, London, 1996.

[10] C. Mascia, A. Porretta, A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic

equations, Arch. Ration. Mech. Anal. 163 (2002) 87124.

[11] A. Michel, J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundaryconditions and application to the convergence of FV methods, SIAM J. Numer. Anal. 41 (6) (2003) 22622293.

[12] F. Otto, Initial boundary-value problem for a scalar conservation law, C. R. Acad. Sci. Paris 322 (1996) 729734.

[13] A. Porretta, J. Vovelle,L1-Solutions to first-order hyperbolic equations in bounded domains, Comm. Partial Differ-

ential Equations 28 (2003) 381408.

[14] G. Vallet, Dirichlet problem for a nonlinear conservation law, Rev. Mat. Complut. XIII (1) (2000) 231250.

[15] J. Vovelle, Prise en compte des conditions aux limites dans les quations hyperboliques non-linaires, Mmoire de

thse, Universit Aix-Marseille 1, Dcembre 2002.

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