On Blow-up “twistors” for the Navier–stokes Equations in R: a View from Reaction-diffusion Theory

Formation of blow-up singularities for the Navier–Stokes equations (NSEs) ut + (u · ∇)u = −∇p+∆u, divu = 0 in R × R+, with bounded data u0 is discussed. Using natural links with blow-up theory for nonlinear reaction-diffusion PDEs, some possibilities to construct special self-similar and other related solutions that are characterized by blow-up swirl with the angular speed near the blow-up time (this represents simplest ω-limits of rescaled orbits as periodic ones) φ(t) ∼ −σ ln(T − t) =⇒ φ̇(t) ∼ σ T−t → ∞ as t→ T (σ 6= 0). This is done in cylindrical polar coordinates {r, φ, z} in R, using the restriction of the NSEs to the linear subspace W2 = Span{1, z}. Similarly, blow-up twistors with axis precessions in the spherical geometry {r, θ, φ} are introduced. It is shown that other blow-up patterns (a “screwing in tornado”) may correspond to a slow “centre-stable manifold-like drift” about Slezkin–Landau singular or other equilibria of the NSEs. Some approaches to blow-up singularities can be applied to 3D Euler’s equations and to well-posed Burnett equations in 7D (i.e., the NSEs with ∆ 7→ −∆2). Though most of blow-up scenarios were not justified even at a qualitative level, the author hopes that the proposed approaches to families of blow-up and other patterns, including those with blow-up swirl, will give some extra insight into the micro-scale “turbulent” structure of the NSEs. The discussion of possible types of blow-up patterns for the NSEs is going in conjunction with some other classic nonlinear PDEs of mathematical physics.