# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020062

## Approximate controllability for Navier–Stokes equations in $\rm3D$ cylinders under Lions boundary conditions by an explicit saturating set

 Institut für Mathematik, Universität Innsbruck, Technikerstraße 13/7, A-6020 Innsbruck, Austria

* Corresponding author: duy.phan-duc@uibk.ac.at

Received  May 2019 Revised  January 2020 Published  June 2020

Fund Project: The author is supported by Universität Innsbruck. The author would like to appreciate Sérgio S. Rodrigues (RICAM Linz Austria) for his fruitful discussions to improve this work and Sy Nguyen-Ky (HAMK Finland) for all helpful figures

An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier–Stokes equations in $\rm 3D$ Cylinders under Lions boundary conditions.

Citation: Duy Phan. Approximate controllability for Navier–Stokes equations in $\rm3D$ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, doi: 10.3934/eect.2020062
##### References:

show all references

##### References:
Induction Step based on the decomposition (19)
Step 3.3.1
Step 3.3.2
Step 3.3.3
 [1] Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165 [2] Oscar P. Manley. Some physical considerations attendant to the approximate inertial manifolds for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 585-593. doi: 10.3934/dcds.1996.2.585 [3] Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719 [4] Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171 [5] Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations & Control Theory, 2020, 9 (3) : 733-751. doi: 10.3934/eect.2020031 [6] Andrei Fursikov. Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 269-289. doi: 10.3934/dcdss.2010.3.269 [7] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [8] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [9] Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953 [10] Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations & Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025 [11] Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020076 [12] Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027 [13] Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 [14] D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251 [15] Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\beta$-plane. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 687-701. doi: 10.3934/dcdsb.2011.16.687 [16] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [17] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [18] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [19] Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189 [20] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

2019 Impact Factor: 0.953

## Metrics

• HTML views (27)
• Cited by (0)

• on AIMS