标题： 跨音速激波对于二维定常欧拉流在多变气体的喷管中大重力情况下的研究 显示英文标题

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

摘要： 在本文中，我们关注二维（2-d）定常欧拉流中多聚气体在水平喷管中的跨音速激波解的存在性，该喷管存在垂直重力，并在喷管出口施加压力条件。 假设重力加速度g取一个通用值。 我们首先证明，当且仅当入流的马赫数满足某些条件时，可以建立仅依赖于重力方向变量的特殊跨音速激波解。 然而，特殊解中的激波位置在喷管中是任意的。 我们在一定条件下，当边界数据是特殊激波解的小扰动时，确定激波位置并建立跨音速激波解的存在性。 数学上，扰动问题可以表述为一个非线性双曲-椭圆混合型和复合型的自由边界问题。 分析中的主要困难来自于垂直重力。 本文开发了方法和技术来处理这些主要困难。 最后，结果表明，垂直重力在决定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 with the vertical gravity in a horizontal nozzle under a pressure condition imposed at the exit of the nozzle. The acceleration of the gravity g is assumed to take a generic value. We first show 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 is arbitrary in the nozzle. We determine the shock position and establish the existence of transonic shock solution when the boundary data are small perturbations of the special shock solutions under certain conditions. Mathematically, the perturbation problem can be formulated as a free boundary problem of a nonlinear system of hyperbolic-elliptic mixed type and composite. Key difficulties in the analysis mainly comes from the vertical gravity. Methods and techniques are developed in this paper to deal with these key difficulties. Finally, it turns out that the vertical gravity plays a dominant role in the mechanism determining the shock position.