摘要 : Formation of blow-up singularities for the Navier-Stokes equations u(t)+(u center dot del)u=-del p+Delta u and div u=0 in R(3)x(0,T) is studied. In cylindrical polar coordinates {r,phi,z} in R-3, their restriction to the linear su... 展开 Formation of blow-up singularities for the Navier-Stokes equations u(t)+(u center dot del)u=-del p+Delta u and div u=0 in R(3)x(0,T) is studied. In cylindrical polar coordinates {r,phi,z} in R-3, their restriction to the linear subspace W-2=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 phi(t)=-sigma ln(T-t) and phi(t)=(sigma/(T-t))->infinity as t -> T-, where sigma is an element of R 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 t -> T- to smooth blow-up self-similar solutions of the corresponding three-dimensional Euler equations u(t)+(u center dot del)u=-del p and div u=0 in R(3)x(0,T), which (iv) are also shown to admit single point blow-up in the similarity variables. 收起