ИСТИНА

The talk will be devoted to the normal parabolic equation (NPE) connected with 3D Helmholtz system whose nonlinear term B(v) is orthogonal projection of nonlinear term for Helmholtz system on the ray generated by vector v. Interest to NPE arised in connection with attempts to find approaches to solve problem on non local existence of smooth solution for 3D Navier-Stokes equations. As it became clear now the studies of NPE has been opened the way to construct the method of nonlocal stabilization by feedback control for 3D Helmholtz as well as for 3D Navier-Stokes equations. First we describe the structure of dynamical ow corresponding to this NPE [1]. After, the non local stabilization problem for NPE by starting control supported on arbitrary fixed subdomain will be formulated. The main steps of solution to this problem will be discussed [2]. At last how to apply this result for solution of nonlocal stabilization problem with impulse control for 3D Helmholtz system will be explained. References [1] Fursikov A. V. On the Normal-type Parabolic System Corresponding to the three- dimensional Helmholtz System, Advances in Mathematical Analysis of PDEs. AMS Transl.Series 2, 232, 99-118, (2014). [2] Fursikov A. B., Shatina L. S. Nonlocal stabilization of the normal equation con- nected with Helmholtz system by starting control, ArXiv: 1609.08679v2[math.OC] 26 Feb. 2017, 1-55, (2017)