可压缩欧拉方程的最优正则性研究

可压缩欧拉方程的最优正则性研究是偏微分方程领域的关键课题。对于无旋的可压缩欧拉方程以及不可压缩欧拉方程的初值问题，其控制局部解存在的最优正则性问题已经得到解决。然而，对于有旋的可压缩欧拉方程（波动方程与运输方程的耦合组），其最优正则性问题是公开问题。本项目拟采用波包分解方法、几何向量场方法以及构造性方法重点讨论其两维初值问题：(1)当初值的正则性高于临界Sobolev指标时，低正则解的适定性；(2)初值足够光滑时，解的生命跨度估计；(3)当初值的正则性低于临界Sobolev指标时，低正则解的不适定性。这些问题是连接双曲型方程与调和分析、几何、物理的一座桥梁，它们的解决无论在理论上还是在应用方面均具有重要的科学价值。

The research on the sharp regularity of compressible Euler equations is a key problem in the area of partial differential equations. For the initial-value problem of the irrotational compressible Euler equations and the incompressible Euler equations , it’s solved for controlling the existence of solutions under sharp Sobolev regularity. However, for the compressible Euler equations with non-zero vorticity(a coupled system with wave equations and transport equations), the sharp Sobolev regularity problem is open. In this project, we plan to use wave-packet decomposition methods, geometry vector-field approaches and construction methods to study the two-dimensional initial-value problem: (1) the well-posedness of low regularity solutions if the regularity of the initial data is greater than the critical Sobolev regularity; (2) the life-span of the local solutions if the initial data is smooth enough; (3) the ill-posedness of low regularity solutions if the regularity of initial data is less than the critical Sobolev regularity. These problems connect the hyperbolic equations with harmonic analysis, geometry and physics. Moreover, it plays a crucial role to us whenever in theory or in applications for solving these problems.