I was looking at the official statement of the Navier-Stokes smoothness and existence problem, and trying to make sense of the hellish maths jargon thsat used. Theres a couple of things I'd like to ask.

The statement has four settings. Two of them are in , which I understand as normal 3D space and two of them are set in . What exactly is this space? I heard that it was a 3-dimensional torus or something. And what is the significance of it?

Also, how come there are two different problems for each space? When you say prove that solutions always exist, and that they are smooth, is it talking about functions that are solutions to the differential equation?.. so that you could find whether a function exists or not by using the definition of a function (using cartesian product?), and then check whether they are differentiable, by looking at the limit at different points?

I don't expect to understand it fully with this post, but I just want to find out more about it.