There are two different settings for the one-million-dollar-prize Navier–Stokes existence and smoothness problem. The original problem is in the whole space R 3 , which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that they are no longer working on the whole space R 3 but in the 3-dimensional torus T 3 = R 3 / Z 3 . Each case will be treated separately.