ISSN 1064-5624, Doklady Mathematics, 2013, Vol. 87, No. 2, pp. 148–152. © Pleiades Publishing, Ltd., 2013. Original Russian Text © M.E. Bogovskii, 2013, published in Doklady Akademii Nauk, 2013, Vol. 449, No. 2, pp. 136–140. 148 INTRODUCTION If the nonlinear nonstationary Navier–Stokes problem has a strong solution, it can be constructed using an iterative method in which each iteration step involves solving a problem with convective terms lin- earized about the solution obtained at the previous iteration. The global convergence of this iterative method was targeted in [1], for which reason the method itself is below referred to for brevity as the method [1]. It was shown in [1] that, for any initial approximation, the global convergence to the sought solution in the norm of the class of Hopf weak solu- tions is faster than a geometric progression rate with a ratio arbitrarily close to zero. However, the proof of the global convergence in the norm of the class of strong solutions for the method [1] faces rather significant though not insurmountable technical difficulties. In this paper, a modification of the method [1] is pro- posed that substantially facilitates the proof of its glo- bal convergence from any initial approximation in the norm of the class of strong solutions assuming that the sought strong solution does exist. More specifically, for iterated convective terms, certain cutoff factors are introduced that formally damp the hypothetical growth of the strong norms of the successively approx- imated solutions with the number of iterations increasing. When the sought strong solution does exist, the iterative method thus modified is almost identical to the original method [1], since the cutoff factors are taking effect only in a number of first iteration steps. After several iterations, whose number is estimated from above in terms of the strong norm of the sought solution, the cutoff factors become identically equal to unity and hence disappear, letting the modified itera- tions take their original unmodified form [1]. It is established below that the existence of a strong solution to the nonlinear Navier–Stokes problem with any initial approximation from the class of strong solu- tions is equivalent to the global convergence of the modified iteration sequence to the sought solution in the norm of the class of strong solutions at a conver- gence rate higher than a geometric progression rate with a ratio arbitrarily close to zero. Interestingly, the sought strong solution is not involved in the construc- tion of the modified iteration sequence and, in fact, the global boundedness of this sequence in the norm of the class of strong solutions becomes a criterion for the global solvability of the nonlinear Navier–Stokes problem in the class of strong solutions (for more detail, see Section 2 below). In a bounded domain Ω ⊂  3 with the time-inde- pendent smooth boundary ∂Ω ∈ C 2 , we consider the nonlinear evolution Navier–Stokes problem, i.e., the initial–boundary value problem for the nonstationary Navier–Stokes equations (1) describing the dynamics of an incompressible fluid with a kinematic viscosity ν > 0, velocity field v: Q T →  3 , and hydrodynamic pressure p: Q T → . To define a strong solution of the nonlinear initial– boundary value problem (1), we need to introduce suitable function spaces. Let L r (Q T ;  3 ) denote the Lebesgue space of vector fields v: Q T →  3 that are integrable to the power r ∈ (1, ∞) over the cylinder Q T . Let (Q T ;  3 ) denote the anisotropic Sobolev space of vector fields v: Q T →  3 , where the super- scripts l and m designate, as usual, the orders of smoothness in x and t, while the corresponding deriv- atives belong to L r (Q T ;  3 ). The symbol H T stands for the (Q T ;  3 )-closed subspace of x-solenoidal v t v ∇ , ( ) v νΔ v – ∇p + + fxt , ( ) , = div v 0 , xt , ( ) Q T ∈ Ω 0 T , ( ) , × = = v t 0 = ax () , div a 0 , x Ω, ∈ = = v ∂Ω 0 , t 0 T , ( ) , a ∂Ω ∈ 0 , = = def W rxt , , lm , W 2 xt , , 21 , Global Convergence in the Strong Norm of an Iterative Method for the Nonstationary Navier–Stokes Problem M. E. Bogovskii Presented by Academician V.P. Dymnikov October 2, 2012 Received October 10, 2012 DOI: 10.1134/S1064562413020075 Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia e-mail: bogovskii@ccas.ru MATHEMATICS