A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations

In this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” equations of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Pure Appl Math 53:1536–1602, 2000), we establish a priori estimates of Sobolev norms for free boundary problem of inviscid rotating Boussinesq equations under the Taylor-type sign condition on the initial free boundary. Using the same method, we can also obtain a priori estimates for the incompressible inviscid rotating MHD system with damping.