Uniform local existence for inhomogeneous rotating fluid equations

We investigate the equations of anisotropic incompressible viscous fluids in $\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniform local existence result with respect to the Rossby number in the same functional spaces under the additional assumption that $B = B(t, x_1)$ or $B = B(t, x_2)$. We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law