Implementation for blow up of tornado-type solutions for complex version of 3D Navier-Stokes system

We consider Cauchy problem for Fourier transformation of 3-dimensional Navier-Stokes system with zero external force. Using initial data purposed by Dong Li and Ya.G.Sinai in [1] we implement self-similar regime producing fast growing behavior of the energy of solution while time tends to critical value T cr. Fourier transform of the 3D Navier-Stokes system with purely imaginary initial data can be written in the form v(k, t) = e −t|k| 2 v(k, 0) + t 0 e −(t−s)|k| 2 R 3 k, v(k − l, s)P k (v(l, s))dlds (1) where k ∈ R 3 corresponds to Fourier mode of the solution, t ∈ [0, T cr ] ⊂ R + is time, v(k, t) corresponds to imaginary part of the solution, ·, · denotes Euclidian inner product. P k is a Leray projection on the subspace orthogonal to k and has the form P k = Id − k, · |k| 2 k. Viscosity supposed to be equal 1, external forcing is put to zero. Incompressibility condition means that for all t ∈ [0, T cr ] v(k, t)⊥k for any k ∈ R 3 (2) For given function c 0 (k) consider one-parametric family of initial conditions v A (k, 0) = Ac 0 (k). Accordingly to Ya.G. Sinai [2], [3] for any c 0 with fast decay at infinity, for every A there exists such constant T cr that for t ∈ [0, T cr ] there exists unique solution of the system (1) which can be represented as a power series of the parameter A: v A (k, t) = Ae −t|k| 2 c 0 (k) + t 0 e −(t−s)|k| 2 p>1 A p h p (k, s)ds (3) Substituting (3) into (1) one can express coefficient h p (k, s) trough coefficients with fewer indices and from initial data c 0 with the following recurrent relations: