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

Recently, Qi S. Zhang provided examples of solutions to the Navier-Stokes equations which, under suitable hypotheses, blow-up in finite time. He considers axially symmetric solutions in a cylinder \(D\) under appropriate boundary conditions and under the effect of supercritical external forces \(f\). In its main result Zhang exhibits, for each \(q\lt\infty\), a blow-up solution with suitable \(f\in L^q(0,T;L^1(D))\). Following Zhang, we construct blow-up solutions with forcing terms in the space \(L^q(0,T;L^p(D))\), for suitable pairs \((q,p)\). In particular our results contain Zhang's result and provide blow-up examples along the supercritical curve \[\frac{2}{q}+\frac{3}{p}=\frac{7}{2}, \quad 1\leq p\lt 2, \] thereby approaching the classical endpoint \(f\in L^1(0,T;L^2(D))\). A particularly significant case is the existence of external forces \(f \in L^q(0,T;L^p(D))\), for every \(p\lt 2\) and some well determined \(q(p)\gt 1\), for which singularities occur in a finite time. The significant case \(f \in L^1(0,T;L^2(D))\), which corresponds to the classical definition of weak solution, remains open. A particularly significant feature of our approach is that external forces, and solutions, are smooth in \([0,T)\). Blow-up occurs only as \(t\rightarrow T\).