Title: On the continuous properties for the 3D incompressible rotating Euler equations Show Chinese title

Title: 关于三维不可压缩旋转欧拉方程的连续性质

Abstract: In this paper, we consider the Cauchy problem for the 3D Euler equations with the Coriolis force in the whole space. We first establish the local-in-time existence and uniqueness of solution to this system in $B^s_{p,r}(\R^3)$. Then we prove that the Cauchy problem is ill-posed in two different sense: (1) the solution of this system is not uniformly continuous dependence on the initial data in the same Besov spaces, which extends the recent work of Himonas-Misio{\l}ek \cite[Comm. Math. Phys., 296, 2010]{HM1} to the general Besov spaces framework; (2) the solution of this system cannot be H\"{o}lder continuous in time variable in the same Besov spaces. In particular, the solution of the system is discontinuous in the weaker Besov spaces at time zero. To the best of our knowledge, our work is the first one addressing the issue on the failure of H\"{o}lder continuous in time of solution to the classical Euler equations with(out) the Coriolis force. Show Chinese abstract

Abstract: 本文研究了整个空间上带有科里奥利力的三维欧拉方程的柯西问题。 首先，我们在空间$B^s_{p,r}(\R^3)$中建立了该系统的局部存在唯一性。 随后我们证明了该柯西问题在两个不同的意义上是不适定的：（1）系统解在初始数据上的依赖性在相同的 Besov 空间中不是一致连续的，这推广了 Himonas-Misio{\l }ek \cite[Comm. Math. Phys., 296, 2010]{HM1}的近期工作到一般的 Besov 空间框架；（2）系统解在相同 Besov 空间中时间变量上不能是 Hölder 连续的。特别地，在较弱的 Besov 空间中，系统解在时间零点处是不连续的。 据我们所知，我们的工作首次探讨了经典欧拉方程（带或不带科里奥利力）的时间 Hölder 连续性失效的问题。