A note on the development of singularities on solutions to the Navier-Stokes equations under super critical forcing terms

Recently Qi S. Zhang provides examples of solutions to the Navier-Stokes equations which, under suitable hypothesis, blow up in finite time. He considers axially symmetric solutions in a cylinder $D\,$ under appropriate boundary conditions and under the effect of super critical external forces $f\,.$ The loss of boundedness for the velocity field, as $t\rightarrow T\,,$ is the basic case of blow up. However a more general situation is considered below, as explained in the preamble.\par In his main result Zhang exhibits, for each $q<\infty\,,$ a blow up solution with an external force $f\in L^q(0,T;L^1(D))\,.$ Following Zhang, we construct blow up solutions with forcing terms in $L^q(0,T;L^p(D))\,,$ for suitable pairs $(q,p)\,.$ In particular our results contain Zhang's result. A significant particular case is the existence of external forces $f \in L^1(0,T;L^p(D))\,,$ for every $p<2$, for which the velocity blows up in a finite time. The significant case $p=2$ remains open.