Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L_t^1{C}_x^{1,\epsilon }\cap L_t^{\mathrm{\infty }}{L}_x^2$

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 $\mathrm{\mid }\mathrm{\nabla }{\mathrm{\mid }}^{\alpha }$ 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 $L_t^1([0,T])C_x^{1,\epsilon }\cap L_t^{\mathrm{\infty }}{L}_x^2$, such that on the time interval $0\le t<T$, the velocity $u$ is in the space $C^{\mathrm{\infty }}\cap L^2$ and such that as the time $t$ approaches the blow-up moment $T$, the integral ${\int }_0^t\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }ds$ tends to infinity.