Аннотация:PARABOLIC EQUATION OF NORMAL TYPE
CONNECTED WITH 3D HELMHOLTZ SYSTEM
AND ITS NONLOCAL STABILIZATION
A. V. Fursikov
Lomonosov Moscow State University,
Voronezh State University
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 flow corresponding to this
NPE (see [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 (see [2]). At last how to
apply this result for solution of nonlocal stabilization problem with impulse
control for 3D Helmholtz system will be explained.
Literature
[1] A.V.Fursikov. "On the Normal-type Parabolic System Corresponding
to the three-dimensional Helmholtz System".- Advances in Mathematical
Analysis of PDEs. AMS Transl.Series 2, v.232 (2014), 99-118.
[2] A.V.Fursikov, L.S.Shatina. "Nonlocal stabilization of the normal equation
connected with Helmholtz system by starting control.ArXiv: 1609.08679v2[math.OC]
26 Feb. 2017, p.1-55