二维定常Euler方程超音速解的存在性

Existence of Supersonic Solution for Two-Dimensional Steady Euler Equations

本文研究广义压力下跨音速流动问题在音速曲线附近的解。给定音速曲线和正特征线上的条件,构造了二维等熵Euler方程的超音速解。由于该系统是退化的,在音速曲线上失去双曲性并产生奇点。因此通过引入一组新变量将该问题转化为一个具有显式奇异正则结构的线性系统,利用迭代法建立新系统光滑解的存在唯一性,从而证明了原系统解的存在性。

In this paper, we study the solution of transonic flow problems under generalized pressure near the sonic curve. Given the conditions of the sonic curve and the positive characteristic line, the su-personic solution of the two-dimensional isentropic Euler equation is constructed. Since this system is degenerate, it loses hyperbolicity and produces singularities on the sonic curve. Therefore, by introducing a new set of variables, the problem is transformed into a linear system with explicit singular regular structure. The existence and uniqueness of the smooth solution of the new system are established by iterative method, and the existence of the solution of the original system is proved.