In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by \(|\nabla |^{\alpha }\) for any \(\alpha \in [0, \alpha _0)\) ( \(\alpha _0 = \frac{22-8\sqrt{7}}{9} > 0\) ). We construct solutions in \(\mathbb {R}^3\times [0,T]\) with a finite \(T>0\) and with an external forcing which is in \(L^1_t([0, T]) C_x^{1,\epsilon }\cap L^{\infty }_{t} L^{2}_{x}\) , such that for each time \(t \in [0, T)\) , the velocity u is in the space \(C^\infty \cap L^2\) and such that as the time t approaches the blow-up moment T, the integral \(\int _0^t |\nabla u| \text {d}s\) tends to infinity. Since this result lays in a well-posedness class, the blow-up is generated by the dynamics of the equation and not by the force itself. This is the first blow-up result for hypodissipative Navier–Stokes in a well-posedness class.
