标题： 跨音速激波对于二维稳态欧拉流在几乎平坦的喷管中具有大重力的多变气体 显示英文标题

标题： Transonic Shocks for 2-D Steady Euler Flows with Large Gravity in an Almost Flat Nozzle for Polytropic Gases

摘要： 本文中，我们关注在水平喷管中垂直重力作用下的二维（2-d）定常欧拉流的可压缩气体的跨音速激波解的存在性。 重力加速度g被假定为一个一般值。 并在喷管出口处施加一个压力条件以确定激波前缘的位置，这是R. Courant和K.O. Friedrichs在其专著《超音速流与激波》中提出的。 首先证明了如果来流马赫数满足某些条件，则可以建立仅依赖于重力方向变量的特殊跨音速激波解的存在性。 然而，特殊解中的激波位置可以在喷管中移动。 然后我们证明，当边界数据是其中一个特殊激波解的小扰动并且满足某些充分条件时，激波位置可以被确定，并且可以建立跨音速激波解的存在性。 数学上，扰动问题可以表述为非线性混合型系统的自由边界问题，分析中的主要困难主要是由垂直重力引起的。 本文发展了方法和技术来处理这些主要困难。 最终结果表明，垂直重力在决定可接受激波位置的机制中起着主导作用。 显示英文摘要

摘要： In this paper, we are concerned with the existence of transonic shock solutions for two-dimensional (2-d) steady Euler flows of polytropic gases under vertical gravity in a horizontal nozzle. The acceleration g of the gravity is assumed to take a generic value. And a pressure condition is imposed at the exit of the nozzle to determine the position of the shock front, as proposed by R. Courant and K.O. Friedrichs in their monograp Supersonic Flow and Shock Waves. It is first shown that the existence of special transonic shock solutions with the flow states depending only on the variable in the gravity direction can be established if and only if the Mach number of the incoming flow satisfies certain conditions. However, the shock position of the special solutions admits shifting in the nozzle. Then we show that the shock position can be determined and the existence of transonic shock solution can be established as the boundary data are small perturbation of one of the special shock solutions and satisfy certain sufficient conditions. Mathematically, the perturbation problem can be formulated as a free boundary problem of a nonlinear mixed-type system, and the key difficulties in the analysis are mainly brought by the vertical gravity. Methods and techniques are developed in this paper to deal with these key difficulties. It finally turns out that the vertical gravity plays a dominant role in the mechanism determining the admissible shock position.