Normal Equation Generated from Helmholtz System: Nonlocal Stabilization by Starting Control and Properties of Stabilized Solutions
We consider the semilinear normal parabolic equation (NPE) corresponding to the 3D Helmholtz system with periodic boundary conditions. First, we recall the main definitions and results associated with the NPE including a result on stabilization to zero of the solution for NPE with arbitrary initial condition by starting control. The main content of the paper is to study properties of stabilized solution of NPE.
KeywordsSemilinear normal parabolic equation 3D Helmholtz system Stabilisation theory Navier-Stokes equations
The work has been fulfilled by RAS program “Theoretical problems of modern mathematics", project “Optimization of numerical algorithms of Mathematical Physics problems". The author was supported in part by RFBI Grants 15-01-03576 and 15-01-08023.
- 2.Fursikov, A.V.: On parabolic system of normal type corresponding to 3D Helmholtz system. Advances in Mathematical Analysis of PDEs. In: Proceedings of the St. Petersburg Mathematical Society volume XV, vol. 232, pp. 99–118 (2014) (AMS Transl. Ser. 2)Google Scholar
- 7.Fursikov, A.V., Shatina, L.S.: Nonlocal stabilization of the normal equation connected with Helmholtz system by starting control. Dis. Cont. Dyn. Syst.-A. 38(3), 1187–1242 (2018)Google Scholar