Formation of blow-up singularities for the Navier–Stokes equations ut+(u)u=p+Δu and divu=0 in R3×(0,T) is studied. In cylindrical polar coordinates {r,φ,z} in R3, their restriction to the linear subspace W2=Span{1,z} is shown to be consistent. Using links with blow-up theory for nonlinear reaction-diffusion partial differential equations, the following questions are under scrutiny: (ii) introducing a self-similar blow-up “swirl mechanism” with the angular swirl divergences φ(t)=σln(Tt) and φ̇(t)=(σ/(Tt)) as tT, where σR is a parameter; (ii) existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional/global blow-up of the z component of the velocity; (iii) as an intrinsic part of the construction, convergence of the above rescaled patterns as tT to smooth blow-up self-similar solutions of the corresponding three-dimensional Euler equations ut+(u)u=p and divu=0 in R3×(0,T), which (iv) are also shown to admit single point blow-up in the similarity variables.

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