Finite Time Blow-Up for the Hypodissipative Navier Stokes Equations with a Force in L¹ₜ Cₓ1,ε∩ L∞ₜ L²ₓ

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 | ∇ | α for any α ∈ [0, α ₀) α ∈ [ 0 , α 0 ) ( α ₀ = 22-8√79 > 0 α 0 = 22 - 8 7 9 > 0 ). We construct solutions in R³× [0,T] R 3 × [ 0 , T ] with a finite $$T>0$$ T > 0 and with an external forcing which is in L¹ₜ([0, T]) Cₓ1,ε∩ L∞ₜ L²ₓ L t 1 ( [ 0 , T ] ) C x 1 , ϵ ∩ L t ∞ L x 2 , such that for each time t ∈ [0, T) t ∈ [ 0 , T ) , the velocity u is in the space C^∞ ∩ L² C ∞ ∩ L 2 and such that as the time t approaches the blow-up moment T , the integral ∫ ₀ᵗ |∇ u| ds ∫ 0 t | ∇ u |