为什么需要许多激波理论。非保守系统的动力学关系
Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems
AI总结：
针对具有严格凸数学熵的非保守双曲型偏微分方程组，通过补充跨激波熵耗散率的动力学关系来制定初值问题，并证明该关系等价于基于Lipschitz连续路径族的定义，且可通过增广系统的行波解相平面分析导出。
摘要：
对于一类具有严格凸数学熵的非保守双曲型偏微分方程组，我们通过补充一个动力学关系来制定初值问题，该关系规定了跨激波的熵耗散率。我们的条件可以视为Abeyaratne、Knowles和Truskinovsky为亚声速相变以及LeFloch为非线性双曲系统的非经典欠压缩激波引入的类似概念对非保守系统的推广。对于所考虑的系统类别，所提出的非保守系统动力学关系等价于Dal Maso、LeFloch和Murat基于给定Lipschitz连续路径族的定义。与先前的理论一致，动力学关系应从与非保守系统增广版本相关的行波解的相平面分析中导出。我们通过几个例子说明，应用中出现的非保守系统符合我们的框架，并且对于典型的湍流流体动力学模型，我们提供了行波存在性和性质的详细分析，从而得到了相应的动力学函数。
英文摘要：
For a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by Abeyaratne, Knowles, and Truskinovsky for subsonic phase transitions and by LeFloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for nonconservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch, and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconservative system. We illustrate with several examples that nonconservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics, we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function.
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