Global solutions for the Navier-Stokes equations in the rotational framework

Abstract

The existence of global unique solutions to the Navier-Stokes equations with the Coriolis force is established in the homogeneous Sobolev spaces \(\dot{H}^s (\mathbb R ^3)^3\) for \(1/2 < s < 3/4\) if the speed of rotation is sufficiently large. This phenomenon is so-called the global regularity. The relationship between the size of initial datum and the speed of rotation is also derived. The proof is based on the space time estimates of the Strichartz type for the semigroup associated with the linearized equations. In the scaling critical space \(\dot{H}^{\frac{1}{2}} (\mathbb R ^3)^3\), the global regularity is also shown.