拟一维欧拉流系统解的适定性问题

This project proposes to study the problems on the quasi-one dimensional Euler flow, which is an important model to describe the ideal flow in a duct. Due to the nonlinear inhomogeneous term which depends on the location and is caused by the geometric effects of the duct, many powerful methods and techniques designed for hyperbolic conservation laws cannot be directly applied to this system. The project plans to study the subjects including the formation of shocks for periodic classical solutions in periodic ducts, global small variational entropy solutions with small periodic initial data and stability of steady periodic solutions in periodic ducts, and global L infinity entropy solutions. It is hoped that the difficulties caused by the nonlinear inhomogeneous term, that comes from the geometric effect of the duct, and the resonance coming from the periodicity and the inhomogeneous term can be solved by the methods of characteristics, approximate characteristics, approximate conservation laws, invariant domains depending on location and compensate compactness, and several results would be given to enrich the mathematical theories on hyperbolic conservation laws, and to be applied on other problems of mechanics and physics.

本项目主要研究欧拉方程组的拟一维流问题。拟一维欧拉流是用于描述管道中理想流体的重要模型。由于方程组中带有由管道几何效应产生的依赖于位置的非线性非齐次项，原本适用于双曲守恒律系统的很多方法和技巧都无法直接应用于此系统。本项目将研究周期管道中周期初值经典解的有限时间激波形成、周期管道中周期小变差熵弱解的整体存在性与定常解稳定性、本质有界熵弱解的整体存在性等问题。期望通过特征线方法、近似特征线与近似守恒律方法、依赖于位置的不变区域方法、补偿紧致方法等来处理由管道效应非齐次项及其与周期效应产生的共振现象所带来的困难，从而在上述这几个问题上取得突破，进一步丰富双曲守恒律系统的数学理论，并可望应用于其他有关的一些重要的力学和物理问题。

本项目对以拟一维欧拉流系统为代表的偏微分方程系统解的性质，特别是其适定性进行了系列研究，获得了包括一般情形下经典解的有限时间奇性，非齐次项的稳定性作用，周期初值下解的长时间行为等一系列结果，完善了相关偏微分方程理论。相关工作发表于 Adv. Math., J. Math. Pures Appl., SIAM J. Math. Anal. 等重要学术刊物。