Abstract
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.
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Notes
- 1.
Of course, many other equations including the Helmholtz system possess this property.
- 2.
Note that because of periodic boundary conditions the Stokes system should not contain the pressure term \(\nabla p\).
- 3.
- 4.
Here and below we use for brevity notation \(S(t;y_0)\) instead of \(S(t,\cdot ;y_0)\)
References
Fursikov, A.V.: The simplest semilinear parabolic equation of normal type. Math. Control Relat. Field 2(2), 141–170 (2012)
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)
Fursikov, A.V.: Stabilization of the simplest normal parabolic equation by starting control. Commun. Pur. Appl. Anal. 13(5), 1815–1854 (2014)
Fursikov, A.V.: Stabilization for the 3D Navier-Stokes system by feedback boundary control. Dis. Cont. Dyn. Syst. 10(1&2), 289–314 (2004)
Fursikov, A.V., Gorshkov, A.V.: Certain questions of feedback stabilization for Navier-Stokes equations. Evol. Equat. Contr. Theor. 1(1), 109–140 (2012)
Fursikov, A.V., Shatina, L.S.: On an estimate connected with the stabilization on a normal parabolic equation by start control. J. Math. Sci. 217(6), 803–826 (2016)
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)
Acknowledgments
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.
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Fursikov, A.V. (2018). Normal Equation Generated from Helmholtz System: Nonlocal Stabilization by Starting Control and Properties of Stabilized Solutions. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_11
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DOI: https://doi.org/10.1007/978-3-030-04807-5_11
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