关于Courant-Friedrichs的跨音速激波问题

On the Courant-Friedrichs′ transonic shock wave in a nozzle

高维定常可压缩流体的系统数学理论一直是人们长期非常关注且悬而未决的偏微分方程中的核心问题之一.由于会出现退化,变形,自由边界,激波等重要现象和困难,人们的注意力主要集中于空气动力学中有重要应用意义且有许多实验和数值模拟结果的典型波形,比如,绕流和管道流的研究.而Courant-Friedrichs的关于有限弯曲管道中的跨音速激波问题就是这样一个重要问题.该问题涉及混合型非线性偏微分方程的带有非平凡边界条件的自由边值问题,对其研究有着许多挑战.本文主要介绍该问题的物理背景,严格数学描述,已经取得的重要进展和方法,特别是在二维De Laval管道时的适定性结果.最后会指出三维时的主要困难和问题.

A systematic mathematical theory for the high-dimensional steady compressible flows has been one ofthe key problems in the field of partial differential equations.Due to important phenomena and difficulties suchas degeneracy,change type,free boundary,and shock wave,etc.,researches have focused mainly on the typicalwaves with important applications in aerodynamics and many experimental and numerical simulation results,such as the studies of flows past a solid body and flows in a nozzle with variable sections.Courant-friedrichs′transonic shock wave in finite curved nozzle is such an important problem.This problem involves a free boundaryvalue problem with non-trivial boundary conditions for mixed type nonlinear partial differential equations,whosesolution leads to many challenges.In this paper,we present the physical background,rigorous mathematicaldescription of the problem,and the important progress and methods that have been made,especially the wellposednessresults in the two-dimensional De Laval nozzle.Finally,the main difficulties and problems in thethree-dimensional De Laval nozzle are discussed.