ИСТИНА

Let consider 3D Helmholtz system that describes evolution of the velocity vortex of viscous incompressible fluid with periodic boundary conditions for spatial variables (i.e. defined on 3D torus T^3) and with arbitrary smooth initial condition. One has to find impulse control supported in a given subdomain of ω⊂T^3 that will ensure the tendency to zero of the solution’s L_2(T^3) - norm with increasing time. This problem is full of content because the millennium problem is not solved yet, i.e. existence in whole of smooth solutions for 3D Helmholtz system (or, what is equivalent, for 3D Navier-Stokes system) is not proved. The quadratic operator in the Helmholtz system consists of the sum of a normal operator Φ(y)y whose image is collinear to the argument y, and the tangential operator B_τ(y), whose image is orthogonal to yinL_2(T^3). At the first stage, we omit the operator B_τ(y) of the Helmholtz system with the aim to stabilize by impulse control the obtained boundary value problem. As is known, (see [1]) the solution of the boundary value problem obtained has an explicit formula that allows us to solve the stabilization problem (see [2, 3]). At the second stage, after returning the operator B_τ(y), only the first steps were made, more precisely, the stabilization problem solution was obtained, but not for the Helmholtz system. Solution of stabilization problem has been obtained only in the case of a model problem for the differentiated Burgers equation (see [4]).The main content of the report is related to the presentation of the results of the second stage. This research was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). References 1. A.V.Fursikov, On the normal type parabolic system corresponding to 3D Helmholtz System.- Adv. in Math. Anal. of PDEs. Vol.232, Providence, AMS, 2014, p.99-118. 2. A.V.Fursikov, L.S.Shatina, Nonlocal stabilization of the normal equation connected with Helmholtz system by starting control.- Discr. Contin. Dyn. Syst., Vol.38, 2018, p.1187-1242. 3. A.V.Fursikov, L.S.Osipova, On the nonlocal stabilization by starting control of the normal equation generated from the Helmhotz system.- SCIENCE CHINA Math.Vol.61, No11, 2018, p.2017-2032. 4. А.В.Фурсиков, Л.С.Осипова, Об одном методе нелокальной стабилизации уравнения типа Бюргерса посредством импульсного управления.- Дифф. Уравн.,Том 55, №5, 2019, с.702-716.