jerf correctly points out that the mathematicians are investigating the behavior of the mathematical model, not the physical phenomenon which inspired it. I want to add a second point, which is that while it wouldn't be that shocking if Navier-Stokes had blow-up or other non-physical solutions, it's not at all obvious that it has such solutions. People have tried and failed for more than a century to construct them. We know, in fact, that Navier-Stokes is very close, in a technical sense, to not admitting such solutions, as there are very mild deformations of the Navier-Stokes model which do not admit such blowups. Likewise, there are mild deformations which do. The beauty of Navier-Stokes is that it's so perfectly balanced between the two cases.