可压缩Euler方程组及其相关模型的定常解的适定性

本项目旨在研究可压缩Euler方程组及其相关的流体动力学模型的定常解的适定性。我们着重研究三维管道中的有旋亚音速流的适定性、绕过不光滑物体的亚音速流的适定性、可压缩Euler方程组的定常解的不唯一性、Euler-Poisson方程组的光滑跨音速解的结构稳定性及其音速线的几何结构等。我们将寻找Euler方程组及其相关模型的一些重要的内蕴结构，并结合椭圆方程的极值原理与爆破技巧、混合型方程组的能量估计、偏微分方程解的几何性质研究、以及凸积分方法等来处理这些问题。本项目的研究既能丰富和发展偏微分方程的数学理论、研究方法和研究技巧，又能为其他学科和实际问题的研究提供强有力的分析工具和理论参考。

The aim of this proposal is to study the well-posedness of the steady solutions for the compressible Euler system and related hydrodynamic models. Our emphases will be the well-posedness for the subsonic flows with vorticity in three dimensional nozzles, the well-posedness for subsonic flows past non-smooth bodies, non-uniqueness for steady solutions of the compressible Euler system, the structural stability of smooth transonic solutions for the Euler-Poisson system and geometric structure of the sonic curves for the associated solutions, etc. We will look for some important intrinsic structures for the Euler system and related models and solve these problems via combining various methods such as the maximum principle and blowup techniques for the elliptic equations, the energy estimate for the mixed type system, the study on the geometric properties of solutions for partial differential equations, and the method of convex integration, etc. The research in this proposal will not only expand and develop the mathematical theory, methods, and techniques of partial differential equations, but also provide strong analysis tools and theoretical references for other subjects and practical problems.