Global solutions of the Navier–Stokes equations in a uniformly rotating space

We consider the Cauchy problem for the Navier–Stokes system of equations in a three-dimensional space rotating uniformly about the vertical axis with the periodicity condition with respect to the spatial variables. Studying this problem is based on expanding given and sought vector functions in Fourier series in terms of the eigenfunctions of the curl and Stokes operators. Using the Galerkin method, we reduce the problem to the Cauchy problem for the system of ordinary differential equations, which has a simple explicit form in the basis under consideration. Its linear part is diagonal, which allows writing explicit solutions of the linear Stokes–Sobolev system, to which fluid flows with a nonzero vorticity correspond. Based on the study of the nonlinear interaction of vortical flows, we find an approach that we can use to obtain families of explicit global solutions of the nonlinear problem.