DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

THE RELATIONSHIP BETWEEN KINETIC SOLUTIONS
AND RENORMALIZED ENTROPY SOLUTIONS
OF SCALAR CONSERVATION LAWS
SATOMI ISHIKAWA AND KAZUO KOBAYASI
We consider L 1 solutions of Cauchy problem for scalar conservation laws. We study two
types of unbounded weak solutions: renormalized entropy solutions and kinetic solutions.
It is proved that if u is a kinetic solution, then it is indeed a renormalized entropy
solution. Conversely, we prove that if u is a renormalized entropy solution which satisfies
a certain additional condition, then it becomes a kinetic solution.
Copyright © 2006 S. Ishikawa and K. Kobayasi. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We consider the following Cauchy problem for the scalar conservation law:
∂ t u +divA(u) = 0, (t,x) ∈ Q ≡ (0,T) × R d , (1.1)
u(0,x) = u 0 (x), x ∈ R d , (1.2)
where A : R → R d is locally Lipschitz continuous, u 0 ∈ L 1 (R d ), T>0, d ≥ 1. It is well
known by Kružkov [7] thatifu 0 ∈ L ∞ (R d ), then there exists a unique bounded entropy
solution u of (1.1)-(1.2). By nonlinear semigroup theory (cf. [3, 4]) a generalized (mild)
solution u of (1.1)-(1.2) has been constructed in L 1 spaces for any u 0 ∈ L 1 (R d ). However,
since the mild solution u is, in general, unbounded and the flux A is assumed no growth
condition, the function A(u) may fail to be locally integrable. Consequently, divA(u)cannot
be defined even in the sense of distributions, so that it is not clear in which sense the
mild solution satisfies (1.1). In connection with this matter, Bénilan et al. [1]introduced
the notion of renormalized entropy solutions in order to characterize the mild solutions
constructed via nonlinear semigroup theory in the L 1 framework.
On the other hand, Chen and Perthame [2] (also see [8]) introduced the notion of
kinetic solutions and established a well-posedness theory for L 1 solutions of (1.1)-(1.2)
by developing a kinetic formulation and using the regularization by convolution.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 433–440