Haar方法、平均矩阵、波抵消以及双曲系统的L1稳定性
Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems
AI总结：
针对非线性双曲系统的间断解，通过Haar-Holmgren型方法利用平均矩阵的几何性质处理不连续系数，证明Glimm或前沿追踪格式的解在L1范数下连续依赖于初值。
摘要：
我们发展了Haar和Holmgren方法的一个版本，适用于非线性双曲系统的间断解，并允许控制两个熵解之间的L1距离。主要困难在于处理具有不连续系数的线性双曲系统。我们的主要观察是，虽然熵解只包含压缩激波，但与两个这样的解相关的平均矩阵具有压缩或欠压缩激波，但没有稀疏-激波——这被认为是非唯一性和不稳定性的来源。我们的Haar-Holmgren型方法基于与平均矩阵相关的几何结构，并考虑了伴随问题和沿广义特征的波抵消。它扩展了LeFloch等人先前针对真正非线性系统提出的方法。在本文中，我们涵盖了具有小总变差的解和一类通量一般、不必真正非线性且包括例如流体动力学方程的系统。我们通过展示一个控制两个解之间距离的L1泛函，证明了由Glimm或前沿追踪格式生成的解在L1范数下连续依赖于其初值。
英文摘要：
We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks -- which are recognized as a source for non-uniqueness and instability. Our Haar-Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It extends the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controling the distance between two solutions.
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