In the context of the Millenium problem for the Navier-Stokes equations, there is a long history of results along the lines of "if global existence fails, then such and such norm has to blow up." There are numerous results on minimal improvements in the specification of "such and such." Of course, no one knows at this point whether any norm really blows up or not.

In the context of the Millenium problem for the Navier-Stokes equations, there is a long history of results along the lines of "if global regularity fails, then such and such norm has to blow up." There are numerous results on minimal improvements in the specification of "such and such." Of course, no one knows at this point whether any norm really blows up or not.