一维Euler方程组在结点处的保熵耦合条件
Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions
摘要：
本文研究了具有不同截面积管道连接处的$3\times 3$一维Euler方程组的一组新的耦合条件。除了质量守恒外，我们还要求接头处的总焓相等，流出管道的比熵等于流入管道所有熵的凸组合。先前使用的耦合条件包括压力相等或动压。它们仅限于只有一个管道具有流出流向的接头的特殊情况。最近，Reigstad [SIAM J. Appl. Math.，75：679- 702，2015]表明，这种基于压力的耦合条件可以通过产生机械能来产生等温流的非物理解。我们的新的耦合条件，确保能量以及熵守恒，也适用于连接任意数量的管道灵活的流动方向的路口。我们证明了广义黎曼问题解的存在性和唯一性，在一个路口附近的恒定的定态属于亚音速区域。这就为齐次和非齐次柯西问题的适定性提供了基础，初始数据具有足够小的全变差。
英文摘要：
This paper is concerned with a set of novel coupling conditions for the $3\times 3$ one-dimensional Euler system with source terms at a junction of pipes with possibly different cross-sectional areas. Beside conservation of mass, we require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. Previously used coupling conditions include equality of pressure or dynamic pressure. They are restricted to the special case of a junction having only one pipe with outgoing flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679--702, 2015] showed that such pressure-based coupling conditions can produce non-physical solutions for isothermal flows through the production of mechanical energy. Our new coupling conditions ensure energy as well as entropy conservation and also apply to junctions connecting an arbitrary number of pipes with flexible flow directions. We prove the existence and uniqueness of solutions to the generalised Riemann problem at a junction in the neighbourhood of constant stationary states which belong to the subsonic region. This provides the basis for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for initial data with sufficiently small total variation.
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