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.
