Strange aspect of the proposed Navier-Stokes millennium solution

Doesn't it seem odd that the proposed solution for the Millennium problem is posed on the whole space, but they claim it has compact support in space for the whole time interval [0,1)? Seems at odds with parabolic delocalization... 🤔 Actually the delocalization which occurs due to the non local pressure is often much worse, for instance as investigated by Brandolese:

https://www.esaim-cocv.org/articles/cocv/abs/2002/02/brandolese/brandolese.html

Author: Adventurous_Cook_809