Finite time blow-up for the hypodissipative Navier Stokes equations with a uniform force in L¹ₜ Cₓ1,ε∩ L²

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 |∇|α for any α∈ [0, α₀) (α₀ = 22-8√7/9 > 0). We construct solutions in R³× [0,T] with a finite $T>0$ and with an external forcing which is in L¹ₜ([0, T]) Cₓ1,ε∩ L², such that on the time interval 0 ≤ t < T, the velocity u is in the space C^∞∩ L² and such that as the time t approaches the blow-up moment T, the integral ∫₀ᵗ |∇ u| ds tends to infinity.