Decay of Entropy Solutions of Nonlinear Conservation Laws

Abstract

We are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws. The main objective of this paper is to present an analytical approach and to explore its applications to studying the large-time behavior of periodic entropy solutions of hyperbolic conservation laws. The asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon. It is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other. The insightful analysis of Glimm-Lax [GL], for scalar equations and 2 × 2 systems, has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions. Such a result was first shown by Lax [L1] in 1957 for one-dimensional convex scalar conservation laws. Dafermos [D1], applying his uniform processes, proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point. The Glimm-Lax theory [GL] indicates that, for 2× 2 strictly hyperbolic and genuinely nonlinear systems, any periodic Glimm solution decays like O(1/t). This result was proved by using the approximate characteristic method in the Glimm difference solutions, provided that the oscillation of the corresponding initial data is small. Recently, using the method of generalized characteristics, Dafermos [D3] showed that any periodic solution with local bounded variation and small oscillation for 2 × 2 systems decays asymptotically, with the same detailed structure pas found by Lax [L1] for the