可压缩流体中疏散波及其相关问题的研究

The compressible Euler equations describe the movement of the ideal fluid flow with no viscous effects, which is one of the hottest frontiers in today's non-linear partial differential equations. The difficuties are the sigularity caused by the vacuum as well as the fine structure of the solution in high-dimension. It is valuable both in mathematical theory and applications.. This project is to study a class of special solutions to the compressible Euler equations: the rarefaction wave as the starting point. One-dimensional case, we plan to adopt lifting the vacuum state and new scale transformation to study the optimal rate of convergence in the maximum norm of zero dissipation limit; zero dissipation limit with vacuum state existing between the two rarefaction waves; hydrodynamic limit and so on. High-dimensional case, we plan to adopt decomposition method to study the connection between the "semi-hyperbolic patch" which we have got and the weak shock wave..The expected results will give a systematic method to solve zero dissipation limit containing the vacuum state as well as enrich and develop study of the high-dimensional nonlinear conservation laws. These results would provide mathmatical support in order to study the supersonic patch around the airplane's wing in practical application of the compressible fluid.

可压缩Euler方程描述了没有粘性影响的理想流体的运动，是当今非线性偏微分方程中最热门的前沿领域之一，其研究瓶颈为真空如何引起奇异性以及高维时解的精细结构。对它的研究具有重要的数学理论及应用价值。.本项目以研究可压缩Euler方程正则性较好的一类特解：疏散波为切入点。一维情形时，拟采用对真空态的提升和新的尺度变换来研究带真空的零耗散极限在最大模意义下最优的收敛速率；真空态存在于两个疏散波之间的零耗散极限；相关的流体动力学极限等。高维情形时，拟采用特征分解的方法研究我们得到的"半双曲斑片解"如何与弱冲击波相连接。.预期成果将给出解决包含真空态的零耗散极限问题研究的一些较系统的方法，丰富和发展对高维非线性守恒律的认知研究，为解决可压缩流体的实际应用中出现的机翼扰流问题提供数学支持。

可压缩Euler方程描述了没有粘性影响的理想流体的运动，是当今非线性偏微分方程中热门的前沿领域之一，对它的研究具有重要的数学理论及应用价值。.本项目以研究可压缩 Euler 方程正则性较好的一类特解:疏散波为切入点。 采用对真空态的提升和新的尺度变换来研究带真空的零耗散极限在最大模意义下的收敛速率。证明了一维可压缩 Navier-Stokes 方程大始值的柯西问题，含真空光滑解的存在性。对于两项Camassa-Holm型方程的守恒解，证明了唯一性，得到了守恒解在爆破点处渐近行为的精细刻画等。.本项目的研究成果给出了解决包含真空态的零耗散极限问题研究的一些较系统的方法，为解决可压缩流体实际应用中出现的问题提供数学支持。