Decay of Entropy Solutions of Nonlinear Conservation Laws
Decay of Entropy Solutions of Nonlinear Conservation Laws
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DOI：
10.1007/s002050050138
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发表时间：
1999-05
影响因子：
2.5
通讯作者：
Gui-Qiang G. Chen;H. Frid
中科院分区：
文献类型：
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作者：
Gui-Qiang G. Chen;H. Frid
本文研究非线性守恒律熵解的渐近性态。本文的主要目的是提出一种分析方法，并探讨其应用研究的大时间行为的周期熵双曲守恒律。非线性双曲型守恒律方程周期解的渐近衰减性是一种重要的非线性现象。据观察，方程的真正非线性迫使每个特征族的波强烈相互作用并相互抵消。Glimm-Lax [GL]对标量方程和2× 2系统的深刻分析表明，同一族的相互作用激波和稀疏波的相互抵消导致了周期解的衰减。Lax [L1]在1957年首次对一维凸标量守恒律给出了这样的结果。Dafermos [D1]应用他的一致过程证明了通量拐点集没有聚点的情况下的衰减结果。Glimm-Lax理论[GL]指出，对于2× 2严格双曲和真非线性系统，任何周期Glimm解都以O（1/t）的形式衰减.在Glimm差分解中，只要相应的初始数据的振荡较小，用近似特征线方法证明了这一结果。最近，Dafermos [D3]利用广义特征线方法证明了2× 2系统的任何具有局部有界变差和小振动的周期解都是渐近衰减的，其详细结构与Lax [L1]对
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