Complex Singular Solutions of the 3-d Navier–Stokes Equations and Related Real Solutions

Abstract

By applying methods of statistical physics Li and Sinai (J Eur Math Soc 10:267-313, 2008) proved that there are complex solutions of the Navier-Stokes equations in the whole space R 3 which blow up at a finite time. We present a review of the results obtained so far, by theoretical work and computer simulations, for the singular complex solutions, and compare with the behavior of related real solutions. We also discuss the possible application of the techniques introduced in (J Eur Math Soc 10: 2008) to the study of the real ones.

On certain models in the PDE theory of fluid flows

“…One can also consider the usual Navier-Stokes equations and allow the velocity and pressure fields to be complex, see [8]. However, in contract with (3.26), this direct complexification will not have the energy inequality (or, in the inviscid case, energy conservation).…”

Real and complex Li-Sinai solutions of the 3D incompressible Navier-Stokes equations

“…We also show results of simulations for real flows with initial support in Fourier space concentrated in relative small regions around some points ±k 0 , as described above, but otherwise unrelated to the Li-Sinai fixed points. For a particular such solution, which is axial symmetric with nonzero swirl, we found that the initial enstrophy increases by a factor about 20, 10 times more than for the case in [4], [5]. We also report other interesting features of the simulations.…”

“…If now the initial data are chosen as stated in the above theorem, then the asymptotics (2.20) holds for H = H 0 and if we set A = ±(Λ(τ )) −1 , for τ ∈ S, in the expansion (2.8), there is a blow-up at the time t = τ . As shown in the paper [5], if there are cancellations between terms of the series (2.8) with neighboring p, which for the fixed point H 0 holds for positive A, the total enstrophy diverges at the blow-up time as const(τ − t) − 5 2 , and for negative A as const(τ − t) −3 . Another easy result shown in [3], [5], is that the solution v(k, t) tends pointwise to a finite limit as t ↑ τ .…”

“…In the past few years we also performed simulations for the real case obtained by antisymmetrizing the initial data of the case in [13]. The results [4], [5] show that in some range of the parameters the solutions behave initially as the complex ones, showing a rapid increase of the total enstrophy and of the maximal velocity, with concentration of the energy in a small region. Then, after some time there is a relaxation with decrease of the total energy.…”

An antisymmetric solution of the 3D incompressible Navier-Stokes equations

“…(Assuming the standard gaussian is not restrictive, as it can always be obtained by rescaling. ) The fixed point equation for H [13,5,3] has infinitely many solutions that can be explicitly written down [13]. (H is in fact a plane vector, as its component along k (0) vanishes by incompressibility.)…”