Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol103.Issue1.1999:

Journal of Computational and Applied Mathematics 103 (1999) 139–144

Convergence of approximate solutions of the Cauchy problem
for a 2 × 2 nonstrictly hyperbolic system of conservation laws
N. Luis E. Leon

Departamento de Matematicas, Ftad. de Ciencias, Univ. Central de Venezuela, Caracas, Venezuela
Received 17 May 1997; revised 6 March 1998

Abstract
A convergence theorem for the vanishing viscosity method and for the Lax–Friedrichs schemes, applied to a nonstrictly
hyperbolic and nongenuinely nonlinear system is established. Using the theory of compensated compactness we prove
c 1999 Elsevier Science B.V. All rights reserved.
convergence of a subsequence in the strong topology.
Keywords: Cauchy problem; Hyperbolic system; Conservation laws

1. Introduction
We are concerned here with the study of the Cauchy problem for the following 2 × 2 system of
conservation laws:
ut + (f(u) + g(v))x = 0;
vt + (uv)x = 0;

(u(x; 0); v(x; 0)) = (u0 ; v0 );

(1)

where f; g ∈ C 2 (R); f(0) = g(0) = 0; g′ (0) = f′ (0) = 0; satisfying
g′ () ¿ 0
′′

and

g′′ ()¿0

f () ¿ 2 ∀ ∈ R:

for  6= 0;

(2)

A system of this type, when f(u) = 3u2 =2 and g(v) = v2 =2 has been investigated by Kan [7]. Systems
of this type have been widely investigated in connection with oil reservoir engineering models. In

order to use the compensated compactness method developed by Murat [9] and Tartar [11], we will
need a priori estimates in L∞ , independent of the viscosity. To this purpose, we shall make use of
the theory of invariant regions due to Chueh et al. [1].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
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N.L.E. Leon / Journal of Computational and Applied Mathematics 103 (1999) 139–144

2. Approximate solutions and convergence
Given a 2 × 2 system of conservation laws:
Ut + F(U )x = 0;
U (x; 0) = U0 (x);

(3)

we say that it is strictly hyperbolic in a region
if ∀U ∈

the eigenvalues of 3F are real and
verify the inequality − ¡ + : In our case, if
G(u; v) = (u − f′ (u))2 + 4vg′ (v);
we have
u + f′ (u) ∓
∓ (u; v) =
2

√

G(u; v)

;

and using (2) we obtain that the problem is strictly hyperbolic ∀U 6= 0, while in the origin (u; v) =
(0; 0) there is an umbilical point.
If we call
√
u − f′ (u) ∓ G(u; v)
h∓ (u; v) =

;
2
then the eigenvectors are given by
r∓ (u; v) = (g′ (v); h∓ (u; v))T :
We say that the system (3) is genuinely nonlinear in
if 3∓ :r∓ 6= 0; ∀U ∈
. In this case we
obtain
˙ {[(2 + f′′ (u))Q ± (f′′ (u) − 2)(u − f′ (u))]g′ (v) + [Q ∓ (u − f′ (u))]vg′′ (v)}=(2Q);
3∓ :r∓ =
where
Q≡

p

G(u; v);

and the genuine nonlinearity fails only on {(u; v) ∈ R2 : v = 0}.
We say that a measurable function U = U (x; t) is a weak solution for the Cauchy problem (3) if
and only if

Z

0

+∞

Z

+∞

[Ut + F(U )x ] dx dt +

−∞

Z

+∞

U0 (x; 0) dx = 0;

(4)

−∞

for any  = (x; t) ∈ C01 (R × [0; +∞)):
The functions ; q : R2 → R are an entropy–entropy
ux pair for the problem (3) if and only if
; q ∈ C 1 and
3q = 33F:

(5)

N.L.E. Leon / Journal of Computational and Applied Mathematics 103 (1999) 139–144

141

We say that a solution U veries the entropy inequality in the sense of Lax [8] if for any convex
entropy  (with a corresponding
ux q) we have
Z

+∞

Z

+∞

[(U )t + q(U )x ] dx dt¿0;

−∞

0

where (x; t) is nonnegative and  ∈ C01 :
A function w∓ ∈ C 1 satisfying
(3w∓ (u; v))T r± (u; v) = 0;

∀(u; v); is called the rst (respectively, second) Riemann invariant for the system (3). Let us denote
by R± the rst and the second rarefaction wave of our system.
Lemma. For the Cauchy problem (1) the following properties hold:

• The R+ (R− ) curves are in one to one correspondence with the points of the negative (positive)
u-semiaxis;
• The positive (negative) u axis is itself an R+ (R− ) curve;
• Every R+ (R− ) curve which does not stay on the u-axis is increasing (decreasing) and goes
to +∞ (−∞) on the right (left).
Now; since w∓ is constant along every R± curve; if we prescribe w∓ (u; 0) as follows:
w∓ (u; 0) = u;
w∓ (u; 0) = 0;

±u ¡ 0;

±u ¿ 0;

then w− (u; v)606w+ (u; v):
The Riemann invariants (w− ; w+ ) constructed as before, satisfy:
@u
@w−
@v
@w−
@u

@w+
@v
@w+

1
= ;
2
h−
= ′ ;
2g (v)
1
= ;
2
h+
= ′ :
2g (v)

(6)

Now, by using the results of [1, 6], we have:

Theorem. The invariant regions for the system (1) are given by:
X

c = {w− + c¿0} ∩ {w+ − c60} ∩ {v¿0};

c ¿ 0;

and these regions are bounded by the two families of rarefaction waves.
Furthermore, by using Lemma 1, the family of invariant regions is strictly increasing with respect
to c and it spans the whole state space as c → +∞.

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N.L.E. Leon / Journal of Computational and Applied Mathematics 103 (1999) 139–144

If we denote by {U k } the family of approximate solutions obtained by using either the vanishing
viscosity method or the Lax–Friedrichs scheme, then we have the following result:
Corollary. Let (u0 ; v0 ) ∈

P

c.

If we suppose u0 ; v0 ; u0x and v0x lying in L∞ ; ∀(x; t) it follows:

w → lim+ (u∈ ; v∈ ) ∈ {(u; v): v¿0}:

(7)

∈→0

We shall apply the compensated compactness method and use the convergence theory of Di Perna
[2] and Serre [10] for hyperbolic equations, to the solutions constructed by using either the vanishing
viscosity method or the numerical schemes. In order to do this, we need to construct a family of
weak entropies for our problem.
We shall use [7] to extend the methods in [10]. In our case, (5) is equivalent to
vvv − g′ (v)uu + (f′ (u) − u)uv = 0:

(8)

A rather important solution for this equation is given by the mechanical energy
∗ (u; v) =

u2
+
2

Z vZ
0

0



g′ ()
d d;


(9)

which is also a convex entropy.
T : (u; v) → (w− ; w+ ) is a one to one map which denes a change of coordinate from the upper
half plane {(u; v): v¿0} to the region {(w− ; w+ ): w− 606w+ }. Using this change of variable T , the
same equation can be written with respect to the Riemann invariants in the form
@2 
1
+
@w− @w+ + (w) − − (w)



@+ (w) @
@− (w) @
−
@w− @w+
@w+ @w−



= 0:

(10)

Let us consider the Goursat problem for Eq. (10), with characteristic data
(w− ; 0) = − (w− );
(w∗ ; w ) = 0;
−

(11)

+

∗ ¡ 0 and  (w ) = 0 when w 6w∗ . The solution will be as regular as the initial datum
where w−
−
−
−
−
− , and
(w− ; w+ ) = 0;
∗ . To nd a regular solution of the Goursat problems (10) and (11) we need to estifor w− 6w−
mate the coecients of the rst-order term in Eq. (10). Using (8), we will control this term in a
neighborhood of the umbilical point.
Lemma. A necessary and sucient condition for the continuity of the derivatives:
@+
;
@w−

@−
;
@w+

@(+ − − )
@w−

and

@(+ − − )
;
@w+

N.L.E. Leon / Journal of Computational and Applied Mathematics 103 (1999) 139–144

143

is given by
lim

|w|→0



g′ (v) + vg′′ (v)
f (u) − 1 −
g′ (v)
′′



= 0;

(u;v) = T −1 (w)

i.e.; there exist k ∈ R+ and n ∈ N ∗ such that
g(v) = kv2n + o(|v|2n );
f(u) = (n + 21 )u2 + o(|u|2 );

(12)

when |u| + |v| → 0; and in this case
@+ (w) 1
= ;
|w|→0 @w−
2
lim

@− (w) 1
= ;
|w|→0
@w+
2
lim

+ (w) − − (w) = n(w+ − w− ) + o(|w|):
We conclude with the following existence result.
Theorem. For any  ¿ 0 there exists an initial datum − ; vanishing when −6w− ¡ 0; such that
the solution  of the Goursat problem (10) and (11) and its derivatives up to the second order;
are bounded on any bounded subset.
These entropies can be used to reduce the support of the Young measure to a single point. Now,
using (9), it is possible to obtain suitable estimates and apply the compensated compactness method
to conclude with the following.
Theorem. Suppose that {U k } are approximating solutions for the problem (1). Then for any
smooth entropy–entropy
ux pair (; q); the quantity kt + qxk lies in a relatively compact set of
−1
Hloc
.
measures
Recall that for any sequence {U k } in L∞ we can associate a family of probability
R
{(x; t)} so that for any continuous function f; {f(U k )} converges weakly ∗ to f()(x; t) d.
Now, we apply the div-curl lemma [11] to get
Corollary. There exists a family of Young measures (x; t); such that; for any entropy–entropy
ux (j ; qj ); j = 1; 2 the following relation holds:
h(x; t); 1 q2 − 2 q1 i = h(x; t); 1 i h(x; t); q2 i − h(x; t); 2 i h(x; t); q1 i :

(13)

Finally, we can prove the following strong convergence result:
Theorem. Assume that u0 ; v0 ; (u0 )x ; (v0 )x ∈ L∞ (R); v0 (x)¿0; ∀x; then the approximate sequence
{U k } constructed by using the viscosity method or the Lax–Friedrichs scheme converges strongly
in Lploc ; p ¡ + ∞; to a weak solution of the system (1) which satises the entropy inequality.

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N.L.E. Leon / Journal of Computational and Applied Mathematics 103 (1999) 139–144

References
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Indiana Univ. Math. J. 26 (1977) 372 – 411.
[2] R.J. Di Perna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82 (1983) 27 – 70.
[3] R.J. Di Perna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983)
1– 30.
[4] R.J. Di Perna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292
(1985) 383 – 420.
[5] R.J. Di Perna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88 (1985) 223 – 270.
[6] D. Hop, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc. 289 (1985) 591– 610.
[7] P.T. Kan, On the Cauchy problem of a 2 × 2 system of non-strictly hyperbolic conservation laws, Ph.D. Thesis,
Courant Institute of Math. Sciences, NY University, 1989.
[8] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM,
Philadelphia, 1973.
[9] F. Murat, Compacite par compensation, Ann. Sc. Normale Superior Pisa 5 (1978) 489 – 507.
[10] D. Serre, La compacite par compensation pour les systemes hyperboliques non lineaires de deux e quations a une
dimension d’espace, J. Math. Pures et Appliquees 65 (1986) 423 – 468.
[11] L. Tartar, Compensated compactness and applications to partial dierential equations, Nonlinear Anal. Mech. Res.
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