输运方程的空间平均正则化
A regularization for the transport equations using spatial-averaging
摘要：
本文探讨了平均技术应用于运输方程作为替代消失粘度。这种方法已被证明是Burgers方程和Euler方程的有效激波正则化，但还没有应用到类似的输运方程。然而，对于这个系统，弱解的经典概念并不总是足够的，因此，一个更一般的概念，分布解包含Dirac-delta函数必须引入。此外，分布解的黎曼问题是已知的弱$*$极限的粘性扰动输运方程的粘性为零。在比较的经典方法的消失粘度，黎曼问题的平均输运方程，它表明，相同的三角洲激波分布解被捕获的过滤消失。数学和物理的动机提供的平均技术考虑，包括一个存在性和唯一性的解决方案在类光滑的初始条件。
英文摘要：
This paper examines an averaging technique applied to the transport equations as an alternative to vanishing viscosity. Such techniques have been shown to be valid shock-regularizations of the Burgers equation and the Euler equations, but has yet to be applied to the similar transport equations. However, for this system, the classical notion of weak solutions is not always sufficient thus a more general notion of a distribution solution containing Dirac-delta functions must be introduced. Moreover, the distribution solution to the Riemann problem is known to be the weak-$*$ limit of the viscous perturbed transport equations as viscosity vanishes. In comparison to the classical method of vanishing viscosity, the Riemann problem is examined for the averaged transport equation and it is shown that the same delta-shock distribution solution is captured as filtering vanishes. Both mathematical and physical motivation are provided for the averaging techniques considered including an existence and uniqueness result of solutions within the class of smooth initial conditions.
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