Will the existence and smoothness properties of the Navier-Stokes equations in three dimensions turn out to depend on the compactness of the universe over which they are defined?

The Navier-Stokes existence and smoothness conjecture is an important open problem in fluid dynamics and the theory of partial differential equations. It's been designated as one of the Clay Institute's Millennium Prize Problems in 2000 and there is a 1 million dollar bounty available for either proving or disproving the conjecture. In the official introduction to the problem here, the Clay Institute splits the problem into four statements A, B, C and D; and the problem is considered to have been settled if any one of them is proven.

Importantly for this question, the statements A and C are about the Navier-Stokes equations defined on the noncompact space ( \mathbb R^3 ), while B and D are about the equations defined on the compact torus ( (\mathbb R/\mathbb Z)^3 ).