A blow up solution of the Navier-Stokes equations with a super critical forcing term

A forced solution $v$ of the axially symmetric Navier-Stokes equation in a finite cylinder $D$ with suitable boundary condition is constructed. The forcing term is in the super critical space $L_t^q{L}_x^1$ for all $q>1$. The velocity is in the energy space at the final moment when it blows up.