两流相互作用下的可压缩双流体模型的数学问题研究

非守恒形式的双流体模型来源于流体力学等应用学科，是目前描述液体-气体等双流体运动的重要模型之一，在化学工程和石油工业等领域中有重要的应用。与单流体模型相比，双流体模型具有压力为二元函数、非守恒性以及两流体之间的相互作用等本质区别和困难。关于该模型的研究已经成为了近年来国内外相关领域的研究前沿热点之一，特别对两流体速度相等的简化情形的研究已经有一些深刻的分析结果。然而对具有两流体相互作用且速度可不相等的情形的研究才刚刚起步，目前的结果仅限于忽略两流体之间的阻力等情形且两流体压力相等情形目前只有关于一维的结果。本项目拟综合运用精细的能量方法、弱收敛方法、谱分析以及渐近展开等方法，研究可压缩非守恒形式的双流体模型解的整体适定性、解关于时间的衰减估计以及粘性消失极限等问题，特别考虑具有两流体之间的阻力以及压力相等的高维情形。

Compressible non-conservative two-fluid model, arising in fluid dynamics, is one of the important models describing the motion of two fluids such as gas and liquid. It has important applications in chemical engineering, petroleum industry and some other related fields. Compared with the single-fluid model, the two-fluid model has some essential differences and chanllenges such as two-component pressure functions, non-conservative form and fluid-fluid interaction. It has attracted more and more mathematicians' attention during the past few years. In particular, there have been some seminal works on the mathematical analysis of the simplified model with equal velocity. However, mathematical analysis of the full model began quite recently. The known works focus on the case without considering the fluid-fluid interaction, and in particular they are only about one-dimensional model for the case of equal pressure functions. In this proposal, we will use delicate energy method, weak compactness method, spectral analysis, asymptotic expansion and some other relevant technique to study the global well-posedness of the solutions and associated time-decay estimates as well as vanishing viscosity limits, especially for the cases of considering the fluid-fluid interaction and equal pressure in multi-dimensions.