On the continuous properties for the 3D incompressible rotating Euler equations

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-Misiołek \cite[Comm. Math. Phys., 296, 2010]{HM1} to the more general framework of Besov spaces; (2) the solution of this system cannot be Hölder 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 Hölder continuous in time of solution to the classical Euler equations with(out) the Coriolis force.