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 convergence of a subsequence in the strong topology. c1999 Elsevier Science B.V. All rights reserved.
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∈C2(R); f(0) =g(0) = 0; g′(0) =f′(0) = 0; satisfying
g′()¿0 and g′′()¿0 for
6
= 0;
f′′()¿2 ∀∈R: (2)
A system of this type, when f(u) = 3u2=2 andg(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].
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; v) =
u+f′(u)∓√G(u; v)
2 ;
and using (2) we obtain that the problem is strictly hyperbolic ∀U6= 0, while in the origin (u; v) = (0;0) there is an umbilical point.
If we call
h∓(u; v) =
u−f′(u)∓√G(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
3∓:r∓=˙ {[(2 +f′′(u))Q±(f′′(u)−2)(u−f′(u))]g′(v) + [Q∓(u−f′(u))]vg′′(v)}=(2Q);
where
Q≡pG(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] dxdt+
Z +∞
−∞
U0(x;0) dx= 0; (4)
for any =(x; t)∈C1
0(R×[0;+∞)):
The functions ; q:R2→R are an entropy–entropy ux pair for the problem (3) if and only if
; q∈C1 and
We say that a solution U veries the entropy inequality in the sense of Lax [8] if for any convex entropy (with a corresponding ux q) we have
Z +∞
∀(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:
• TheR+(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; ±u ¡0;
w∓(u;0) = 0; ±u ¿0; then w−(u; v)606w+(u; v):
The Riemann invariants (w−; w+) constructed as before, satisfy:
@u
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.
If we denote by {Uk} 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)∈
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) =u
which is also a convex entropy.
T: (u; v)→(w−; w+) is a one to one map which denes a change of coordinate from the upper
half plane {(u; v):v¿0} to the region {(w−; w+):w−606w+}. Using this change of variableT, the
same equation can be written with respect to the Riemann invariants in the form
@2
Let us consider the Goursat problem for Eq. (10), with characteristic data
(w−;0) =−(w−);
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:
is given by
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 {Uk
} are approximating solutions for the problem (1). Then for any smooth entropy–entropy ux pair (; q); the quantity k
t +qkx lies in a relatively compact set of
Hloc−1.
Recall that for any sequence {Uk} in L∞ we can associate a family of probability measures
{(x; t)} so that for any continuous function f;{f(Uk)} converges weakly ∗ to R
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); 1q2−2q1i =h(x; t); 1i h(x; t); q2i − h(x; t); 2i h(x; t); q1i: (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
{Uk} constructed by using the viscosity method or the Lax–Friedrichs scheme converges strongly
References
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[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 equations a une dimension d’espace, J. Math. Pures et Appliquees 65 (1986) 423 – 468.
