Blow ups of complex solutions of the 3D Navier–Stokes system and renormalization group method

Abstract

We consider complex-valued solutions of the three-dimensional Navier-Stokes system without external forcing on R 3 . We show that there exists an open set in the space of 10-parameter families of initial conditions such that for each family from this set there are values of parameters for which the solution develops blow up in finite time. * Since k ∈ R 3 , v(k, t) ∈ C 3 , the order in the inner product is important.assumptions concerning the initial condition v(k, 0) will be discussed later (see §7). In all cases v(k, 0) will be bounded functions whose support is a neighborhood of some point (0, 0, k (0) ). The incompressibility condition implies that the components v 1 (k, 0),v 2 (k, 0) of v(k, 0) are arbitrary functions of k while v 3 (k, 0) can be found from the incompressibility condition v, k = 0.Various methods (see, for example, [K], [C], [S1]) allow to prove in such cases the existence and uniqueness of classical solutions of (1) on finite intervals of time. For these solutionsPresumably, v(k, t) has an asymptotics of this type but this requires more work. According to a conventional wisdom, possible blow ups are connected with the violation of (2).In this paper we fix t and consider one-parameter families of initial conditions v A (k, t) = Av(k, 0). We show that for some special v(k, 0) one can find critical values A cr = A cr (t) such that the solution v Acr (k, s) blows up at t so that for t ′ < t both the energy and the enstrophy are finite while at t ′ = t they both become infinite. Even more, for t ′ < t the solution decays exponentially outside some region depending on t. As t ′ ↑ t this region expands to an unbounded domain in R 3 .Our main approach is based on the renormalization group method which is so useful in probability theory, statistical physics and the theory of dynamical systems. It is rather difficult to give the exact formulation of our result in the introduction because it uses some notions, parameters, etc., which will appear in the later sections. Loosely speaking, we show that in ℓ-parameter families of initial conditions, for ℓ = 10, one can find values of parameters for which the solutions develop blow ups of the type we already described. The meaning of ℓ is explained in §4, §5, §6.We thank C. Fefferman, W.E, K. Khanin and V. Yakhot for many useful discussions. A big part of the text was prepared during the visit of the second author of the Mathematics Department of California Institute of Technology and we thank the Department for its very warm hospitality. We also thank G. Pecht for her excellent typing of the manuscript. The financial support from NSF Grant DMS 0600996 given to the second author is highly acknowledged.

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

“…In the paper [8] it was proved, with techniques inspired by the Renormalization Group Method, that there is a class of real solutions of the integral Eq. (3) which blow up at a finite time.…”

“…They were first introduced by Li and Sinai in 2008 [8], as a contribution to the long-lasting problem on the existence of singular Our review is based both on theoretical work and computer simulations. We will also discuss the behavior of related real solutions, as revealed by computer simulations, and discuss the perspectives of applying to them the techniques introduced in [8].…”

“…Following Li and Sinai [8] the problem (1) is reformulated as a convolution integral equation by introducing the modified Fourier transform…”

The new analytical solution of the 3D Navier-Stokes equation for compressible medium clarifies the sixth Millennium Prize problem

“…Therefore, in 2000 the problem of the existence of a smooth nonstationary vortex solution of the 3D NS equation on an unbounded interval of time was included by Clay Mathematics Institute into the list of the seven fundamental Millennium Prize problems under number six [8,9,11,12]. However, at the same time, in [8] it is proposed to consider this problem solution not for the full NS equation [4], but only for the equation, derived from it in assumption that the divergence of incompressible medium velocity field is equal to zero.…”

“…As a result, for almost two hundred years (since 1827-1845), open remains the issue on the existence of smooth nonstationary divergent and divergent-free solutions of the 3D NS equation in an unbounded (or with periodic boundary condition) space and on an unbounded interval of time [8][9][10][11][12]. And the significance of this problem is determined not only by mathematical, but also by practical interest, owing to both the fundamental and applied problem of predictability in hydrometeorology and other related fields that might be the case with the applications of the methods utilized for the NS equation computational solution [9,10].…”