理想可压缩流体的运动规律可用欧拉方程来描述。两维欧拉方程组的黎曼问题、激波反射与衍射问题等所涉及到的双曲-椭圆混合型方程的定解问题是国内外的研究重点和热点。 本项目旨在研究两维欧拉方程组黎曼问题中跨音速激波的数学理论，主要内容包括：（1）两维黎曼问题中四个激波以及四个稀疏波相互作用时解的整体存在性；（2）作为对欧拉方程组的合理近似，研究非线性波动方程组和压差方程组相关问题；（3）研究具有狭窄截面的管道中跨音速流的适定性。要解决的关键问题是如何将双曲和椭圆区域内的解有效地“粘接”在一起，理解它们相互作用的机制。希望本项目能够促进非线性双曲-椭圆混合型方程理论的发展。

The ideal compressible flow can be described by the Euler equations. The well-posedness of the related hyperbolic-elliptic mixed typed equations for 2D Riemann problems for the Euler equations, shock reflection and diffraction, etc are the focus and hot issues in the world. The project aims at the mathematical theory of transonic flow in 2D Riemann problems for the Euler equations. The main contents include: (1) the global existence of solution for the interaction of four shocks as well as four rarefaction waves in 2D Riemann problems; (2) the related problems for nonlinear wave equations and pressure gradient equations, which can both approximate the Euler equations reasonably; (3) the well-posedness of transonic flow in an infinite channel with a smooth bump inward. The key point is how to “splice” the solutions in hyperbolic and elliptic regions together. We wish the project can push the theory of nonlinear hyperbolic-elliptic mixed typed equations a further step ahead.