关于高维Euler方程组间断解适定性理论的一些研究

This project studies the well-posedness problems of the discontinuous solutions to multi-dimensional Euler equations which are closely connected to the problems in supersonic flow past an obstacle, the refraction of shock waves, etc., and the major problems to be investigated include the well-posedness of the transonic shocks in supersonic flow past a wedge, the stability of the nonlinear wave patterns of regular refraction of shocks, as well as the related free boundary problems of nonlinear hyperbolic, elliptic, mixed-type or/and composite-type partial differential equations. This project highlights the investigation for the discontinuous solutions to multi-dimensional Euler equations, and the problems to be studies are closely connected to physical backgrounds and pratical applications. Analytical tools, such as harmonic analysis, psudo-differential operators, para-differential operators, etc., will be employed to analyze the singularity of the solutions at discontinuities, asymptotic behaviors at infinity, etc.. The researches of this project are anticipated to promote the development of the theory and methods for problems of multi-dimensional nonliner partial differential equations, especially for mixed-type and composite-type equations. It is important both in theory and in applications.

本项目将结合超音速绕流问题、激波折射问题等对高维Euler方程组间断解的适定性问题进行研究，重点将考察超音速流绕楔体流动产生的跨音速激波的适定性、激波正规折射结构的稳定性、以及涉及的非线性双曲型、椭圆型、混合型、复合型等多种类型偏微分方程的自由边界问题等。本项目的研究工作突出对高维Euler方程组间断解的研究，所研究的问题紧密结合实际的物理背景与应用，将综合应用调和分析、拟微分算子、仿微分算子等工具对解在间断处的奇性、无穷远处的渐进性态等进行分析。本项目的研究工作对于发展关于高维非线性方程组，特别是混合型、复合型方程组定解问题的理论和方法有一定的促进作用，具有重要的理论意义与应用前景。

Euler方程组是空气动力学中的最基本、最重要的方程之一，作为拟线性双曲守恒律方程组的一个重要实例，对其弱解的数学理论的研究一直是国内外偏微分方程研究领域的前沿和热点之一。本项目结合空气动力学中的重要物理现象研究高维Euler方程组间断解的适定性理论及其所涉及的双曲型、混合型偏微分方程的自由边界问题。我们的研究工作取得了丰富的成果，在楔体超音速绕流产生的跨音速激波的稳定性问题、三维球对称正激波稳定性问题、二维管道流激波解的存在性和稳定性问题、激波正规折射结构稳定性问题等方面取得了重要进展，所得成果得到国内外同行的重视。本项目的研究成果促进了非线性、混合型方程的自由边界问题相关数学理论的研究，丰富和发展了关于非线性混合型方程定解问题的理论和方法，具有重要的理论意义与应用前景。