Global well-posedness and ill-posedness for the Navier–Stokes equations with the Coriolis force in function spaces of Besov type

We consider the initial value problems for the Navier–Stokes equations in the rotational framework. We introduce function spaces B˙p,qs(R3) of Besov type, and prove the global in time existence and the uniqueness of the mild solution for small initial data in our space B˙1,2−1(R3) near BMO−1(R3)BMO−1(R3). Furthermore, we also discuss the ill-posedness for the Navier–Stokes equations with the Coriolis force, which implies the optimality of our function space B˙1,2−1(R3) for the global well-posedness.