可压缩非守恒两相流模型的若干数学问题

Many of the fluids encountered in nature are multiphase flow, such as the compressible non-conservative two-phase flow model, which are common in many industries, like power, nuclear, chemical-process, petroleum and natural gas, cryogenics, aerospace, biomedical, micro technology. This model not only has a deep physical meaning, but also has important value of the theory of mathematics. The investigation of it has been a hot issues in this area at recent 10 years, there are many numerical results about the model or the related model, but the results about the well-posedness and physical parameters vanishing limits of the solution are few. This project will study the existence and decay of solution for the Cauchy problem to the compressible non-conservative two-phase flow model, for the two cases: with the capillary effect and without capillary effect, the optimal decay rate of the solutions will be established respectively; Also, we study some physical parameters vanishing limits of related problems to the compressible non-conservative two-phase flow model, such as the incompressible limit, we will obtain the global existence of solution to the corresponding incompressible model, vanishing capillary coefficient limit, get the global existence of classical solution and decay rate with respect to the capillary coefficient, vanishing viscosity coefficient limit, consider the global existence of strong solution and boundary layer theory. Furthermore, we consider the free boundary problem of compressible non-conservative two-phase flow model, we will prove the local well-posedness of the solution.

在自然界中遇到的许多流体都是多相流，例如可压缩非守恒两相流模型，它在许多行业，如电力，核，化学过程，石油和天然气，低温，航天，生物医药，显微技术等中都很常见。该模型不仅具有深刻的物理意义，而且也具有重要的数学理论价值。关于其研究近十多年来一直是本领域的热点问题，有许多关于该模型及其相关模型的数值结果，但关于解的适定性及物理参数消失极限的结果却很少。本项目将系统地研究可压缩非守恒两相流模型柯西问题解的存在性和衰减性，对于带有毛细管效应和不带毛细管效应两种情形，分别建立解的最优衰减率；研究可压缩非守恒两相流模型相关问题的某些物理参数消失极限，如不可压缩极限，得到相应不可压缩模型解的整体存在性，毛细管系数消失极限，得到整体经典解的存在性和关于毛细管系数的衰减率，粘性系数消失极限，考虑强解的整体存在性和边界层理论等；进一步考虑可压缩非守恒两相流模型的自由边界问题，证明解的局部适定性。

本研究项目的结果包括可压缩非守恒两相流模型及相关模型的适定性理论。首先对于可压缩非守恒两相流模型，研究了相关问题解的整体存在性和大时间行为(衰减)、毛细管系数消失极限和不可压缩极限以及简化模型自由边界问题解的局部存在性。另外研究了粘性系数依赖于密度的可压缩等熵Navier-Stokes方程组粘性激波解的稳定性；两层半(gravity two and half layer)模型解的适定性；不可压缩Navier-Stokes-Vlasov-Boltzmann方程组弱解的整体存在性；液体-气体两相流模型解的适定性和衰减；三维不可压弹性无粘动力学模型具有物理真空的自由边界问题的局部适定性等。取得了一系列研究成果，在SCI源期刊上发表和接受发表论文21篇，发表和接受发表论文的杂志包括：“SIAM J. Math. Anal.”、“Discrete Contin. Dyn. Syst.”、“J. Differential Equations”、“Commun. Math. Sci.”、“Proc.Roy.Soc.Edinburgh Sect.A”、“Z. Angew. Math. Phys.”等。