高维超音速流的初边值问题

This project is to study the initial-boundary value problem for multi-dimensional supersonic flow.The well-posedness of its solutions is always a hot topic and challenges in the studying of the partial differential equations.Although some progress for the problems had been made, there are still a lot of difficulties need to be dealt with.In this project,we will plan to study the problems from the supersonic airflow and the stability of the transonic contact discontinuity in nozzles.Based on these,we will further explore the methods for analyzing the initial-boundary value problems for multi-dimensional nonlinear conservation laws as well as their new results.The details are the following:.(1)L^1-stability and uniqueness for the two-dimensional steady supersonic exothermically reacting Euler flow past Lispchitz bending walls;.(2)The global existence and asymptotic behavior of the solutions for steady three-dimensional supersonic full axi-symmetric Euler flow past a Lispchtiz curved cone with vertex angle less than a critical value which is a small perturbation of background state in the sense of BV norm;.(3)The stability of contact discontinuity in the two-dimensional finite and semi-infinity long nozzles which is a small perturbation of the background state in the smooth sense..The possible mathematical tools for discussion are the modified Glimm scheme with quadrangular mesh, wave-front track algorithm as well as the mixed-type equations techniques.

本项目将研究高维超音速流的初边值问题，其解的适定性问题一直是偏微分方程研究的热点和有挑战性的问题。这些问题近年来取得了不少的进展，但仍然面临许多有待解决的诸多困难。本项目计划从超音速绕流问题和管道中的跨音速接触间断的稳定性问题来展开研究，并在此基础上进一步探索高维非线性守恒律初边值问题的分析方法和新结果。具体如下：.(1)证明二维定常超音速放热反应欧拉流绕Lipschitz弯壁流动问题弱解的 L^1稳定性及其唯一性；.(2)研究三维定常轴对称超音速完全欧拉流绕具有任意顶角且不超过某一临界值的Lipschitz圆锥体流动问题整体解关于背景解在BV小扰动意义下的存在性及其在无穷远处的渐近行为；.(3)证明二维有限长和半无限长管道中含有跨音速接触间断在背景意义下光滑小扰动的稳定性。.拟采用改进的平行四边形网格的Glimm格式，波前跟踪格式等方法及混合型型方程的工具。

本项目研究的高维超音速流的初边值问题源于许多实际的工程应用中，比如各种风洞试验、航空发动机设计等等。.本项目主要研究了如下高维超音速流的初边值问题：1)、二维定常等熵无旋高超音速流绕楔形流动问题的相似律；2)、三维定常等熵无旋超音速流绕锥体流动问题含强锥形激波的稳定性；3)、二维有限长管道流中超音速接触间断波和跨音速接触间断波的稳定性。.本项目的主要结果：1)、我们在BV框架下，严格证明了二维定常等熵无旋高超音速流绕薄楔形体流动问题的大变差熵解的整体存在性并且当楔形体的厚度趋于零时， 该熵解趋于一维活塞问题的熵解；2)、我们在BV框架下，证明了三维定常等熵无旋等温超音速流绕锥体流动问题含强锥形激波的稳定性及其沿流动方向在无穷远处的渐近状态；3)、 我们证明了二维有限长管道流中超音速接触间断波和跨音速接触间断波在分片光滑解意义下的稳定性。.本项目的研究成果丰富和发展了高维超音速绕流问题弱解的适定性理论，促进了以管道流中接触间断为自由边界的非线性混合型方程边值问题定解理论的发展，具有重要科学意义和应用价值。..