
- Previous Article
- EECT Home
- This Issue
-
Next Article
Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions
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 |
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.
References:
[1] |
A. A. Agrachev, Some open problems, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser. 5, Springer, Cham, 2014.
doi: 10.1007/978-3-319-02132-4_1. |
[2] |
A. A. Agrachev, S. Kuksin, A. V. Sarychev and A. Shirikyan, On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations, Ann. Inst. H. Poincaré Probab. Statist., 43 2007,399–415.
doi: 10.1016/j.anihpb.2006.06.001. |
[3] |
A. A. Agrachev and A. V. Sarychev,
Navier–Stokes equations: Controllability by means of low modes forcing, J. Math. Fluid Mech., 7 (2005), 108-152.
doi: 10.1007/s00021-004-0110-1. |
[4] |
A. A. Agrachev and A. V. Sarychev,
Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys., 265 (2006), 673-697.
doi: 10.1007/s00220-006-0002-8. |
[5] |
A. A. Agrachev and A. V. Sarychev, Solid controllability in fluid dynamics, In Instability in Models Connected with Fluid Flows Ⅰ, Springer, New York, 2008, 1–35.
doi: 10.1007/978-0-387-75217-4_1. |
[6] |
N. V. Chemetov, F. Cipriano and S. Gavrilyuk,
Shallow water model for lakes with friction and penetration, Math. Methods Appl. Sci., 33 (2010), 687-703.
doi: 10.1002/mma.1185. |
[7] |
W. E and J. C. Mattingly,
Ergodicity for the Navier–Stokes equation with degenerate random forcing: Finite dimensional approximation, Comm. Pure Appl. Math., 54 (2001), 1386-1402.
doi: 10.1002/cpa.10007. |
[8] |
E. Fernández-Cara and S. Guerrero,
Null controllability of the Burgers system with distributed controls, Systems Control Lett., 56 (2007), 366-372.
doi: 10.1016/j.sysconle.2006.10.022. |
[9] |
M. Hairer and J. C. Mattingly,
Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. of Math., 164 (2006), 993-1032.
doi: 10.4007/annals.2006.164.993. |
[10] |
A. A. Ilyin and E. S. Titi,
Sharp estimates for the number of degrees of freedom for the damped-driven 2-D Navier–Stokes equations, J. Nonlinear Sci., 16 (2006), 233-253.
doi: 10.1007/s00332-005-0720-7. |
[11] |
J. P. Kelliher,
Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal., 38 (2006), 210-232.
doi: 10.1137/040612336. |
[12] |
V. Nersesyan,
Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force, Nonlinearity, 28 (2015), 825-848.
doi: 10.1088/0951-7715/28/3/825. |
[13] |
H. Nersisyan,
Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var., 16 (2010), 677-694.
doi: 10.1051/cocv/2009017. |
[14] |
H. Nersisyan,
Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564.
doi: 10.1080/03605302.2011.596605. |
[15] |
D. Phan and S. S. Rodrigues, Approximate controllability for equations of fluid mechanics with a few body controls, In Proceedings of the 2015 European Control Conference (ECC), Linz, Austria, 2015, 2682–2687.
doi: 10.1109/ECC.2015.7330943. |
[16] |
D. Phan and S. S. Rodrigues,
Gevrey regularity for Navier–Stokes equations under Lions boundary conditions, J. Funct. Anal., 272 (2017), 2865-2898.
doi: 10.1016/j.jfa.2017.01.014. |
[17] |
D. Phan and S. S. Rodrigues,
Approximate controllability for Navier–Stokes Equations in 3D rectangles under Lions boundary conditions, J. Dyn. Control Syst., 25 (2019), 351-376.
doi: 10.1007/s10883-018-9412-0. |
[18] |
S. S. Rodrigues, Controllability issues for the Navier–Stokes equation on a rectangle, In Proceedings 44th IEEE CDC-ECC'05, Seville, Spain, 2005, 2083–2085.
doi: 10.1109/CDC.2005.1582468. |
[19] |
S. S. Rodrigues,
Navier–Stokes equation on the Rectangle: Controllability by means of low mode forcing, J. Dyn. Control Syst., 12 (2006), 517-562.
doi: 10.1007/s10883-006-0004-z. |
[20] |
S. S. Rodrigues, Controllability of nonlinear pdes on compact Riemannian manifolds, In Proceedings WMCTF'07, Lisbon, Portugal, 2007,462–493. http://people.ricam.oeaw.ac.at/s.rodrigues/. |
[21] |
S. S. Rodrigues, Methods of Geometric Control Theory in Problems of Mathematical Physics, PhD Thesis, Universidade de Aveiro, Portugal, 2008. http://hdl.handle.net/10773/2931. |
[22] |
M. Romito,
Ergodicity of the finite dimensional approximation of the 3D Navier–Stokes equations forced by a degenerate noise, J. Statist. Phys., 114 (2004), 155-177.
doi: 10.1023/B:JOSS.0000003108.92097.5c. |
[23] |
A. Sarychev,
Controllability of the cubic Schroedinger equation via a low-dimensional source term, Math. Control Relat. Fields, 2 (2012), 247-270.
doi: 10.3934/mcrf.2012.2.247. |
[24] |
A. Shirikyan,
Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys., 266 (2006), 123-151.
doi: 10.1007/s00220-006-0007-3. |
[25] |
A. Shirikyan, Controllability of nonlinear PDEs: Agrachev–Sarychev approach, Journées Équations aux Dérivées Partielles. Évian, 4 juin–8 juin. Exposé no. IV, 2007, 1–11. https://eudml.org/doc/10631. |
[26] |
A. Shirikyan,
Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537.
doi: 10.1016/j.anihpc.2006.04.002. |
[27] |
A. Shirikyan,
Euler equations are not exactly controllable by a finite-dimensional external force, Phys. D, 237 (2008), 1317-1323.
doi: 10.1016/j.physd.2008.03.021. |
[28] |
A. Shirikyan, Global exponential stabilisation for the burgers equation with localised control, J. Éc. polytech. Math., 4: 613–632, 2017.
doi: 10.5802/jep.53. |
[29] |
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, 2nd edition, Philadelphia, PA, 1995. |
[30] |
Y. Xiao and Z. Xin,
On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.
doi: 10.1002/cpa.20187. |
[31] |
Y. Xiao and Z. Xin,
On the inviscid limit of the 3D Navier–Stokes equations with generalized Navier-slip boundary conditions, Commun. Math. Stat., 1 (2013), 259-279.
doi: 10.1007/s40304-013-0014-6. |
show all references
References:
[1] |
A. A. Agrachev, Some open problems, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser. 5, Springer, Cham, 2014.
doi: 10.1007/978-3-319-02132-4_1. |
[2] |
A. A. Agrachev, S. Kuksin, A. V. Sarychev and A. Shirikyan, On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations, Ann. Inst. H. Poincaré Probab. Statist., 43 2007,399–415.
doi: 10.1016/j.anihpb.2006.06.001. |
[3] |
A. A. Agrachev and A. V. Sarychev,
Navier–Stokes equations: Controllability by means of low modes forcing, J. Math. Fluid Mech., 7 (2005), 108-152.
doi: 10.1007/s00021-004-0110-1. |
[4] |
A. A. Agrachev and A. V. Sarychev,
Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys., 265 (2006), 673-697.
doi: 10.1007/s00220-006-0002-8. |
[5] |
A. A. Agrachev and A. V. Sarychev, Solid controllability in fluid dynamics, In Instability in Models Connected with Fluid Flows Ⅰ, Springer, New York, 2008, 1–35.
doi: 10.1007/978-0-387-75217-4_1. |
[6] |
N. V. Chemetov, F. Cipriano and S. Gavrilyuk,
Shallow water model for lakes with friction and penetration, Math. Methods Appl. Sci., 33 (2010), 687-703.
doi: 10.1002/mma.1185. |
[7] |
W. E and J. C. Mattingly,
Ergodicity for the Navier–Stokes equation with degenerate random forcing: Finite dimensional approximation, Comm. Pure Appl. Math., 54 (2001), 1386-1402.
doi: 10.1002/cpa.10007. |
[8] |
E. Fernández-Cara and S. Guerrero,
Null controllability of the Burgers system with distributed controls, Systems Control Lett., 56 (2007), 366-372.
doi: 10.1016/j.sysconle.2006.10.022. |
[9] |
M. Hairer and J. C. Mattingly,
Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. of Math., 164 (2006), 993-1032.
doi: 10.4007/annals.2006.164.993. |
[10] |
A. A. Ilyin and E. S. Titi,
Sharp estimates for the number of degrees of freedom for the damped-driven 2-D Navier–Stokes equations, J. Nonlinear Sci., 16 (2006), 233-253.
doi: 10.1007/s00332-005-0720-7. |
[11] |
J. P. Kelliher,
Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal., 38 (2006), 210-232.
doi: 10.1137/040612336. |
[12] |
V. Nersesyan,
Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force, Nonlinearity, 28 (2015), 825-848.
doi: 10.1088/0951-7715/28/3/825. |
[13] |
H. Nersisyan,
Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var., 16 (2010), 677-694.
doi: 10.1051/cocv/2009017. |
[14] |
H. Nersisyan,
Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564.
doi: 10.1080/03605302.2011.596605. |
[15] |
D. Phan and S. S. Rodrigues, Approximate controllability for equations of fluid mechanics with a few body controls, In Proceedings of the 2015 European Control Conference (ECC), Linz, Austria, 2015, 2682–2687.
doi: 10.1109/ECC.2015.7330943. |
[16] |
D. Phan and S. S. Rodrigues,
Gevrey regularity for Navier–Stokes equations under Lions boundary conditions, J. Funct. Anal., 272 (2017), 2865-2898.
doi: 10.1016/j.jfa.2017.01.014. |
[17] |
D. Phan and S. S. Rodrigues,
Approximate controllability for Navier–Stokes Equations in 3D rectangles under Lions boundary conditions, J. Dyn. Control Syst., 25 (2019), 351-376.
doi: 10.1007/s10883-018-9412-0. |
[18] |
S. S. Rodrigues, Controllability issues for the Navier–Stokes equation on a rectangle, In Proceedings 44th IEEE CDC-ECC'05, Seville, Spain, 2005, 2083–2085.
doi: 10.1109/CDC.2005.1582468. |
[19] |
S. S. Rodrigues,
Navier–Stokes equation on the Rectangle: Controllability by means of low mode forcing, J. Dyn. Control Syst., 12 (2006), 517-562.
doi: 10.1007/s10883-006-0004-z. |
[20] |
S. S. Rodrigues, Controllability of nonlinear pdes on compact Riemannian manifolds, In Proceedings WMCTF'07, Lisbon, Portugal, 2007,462–493. http://people.ricam.oeaw.ac.at/s.rodrigues/. |
[21] |
S. S. Rodrigues, Methods of Geometric Control Theory in Problems of Mathematical Physics, PhD Thesis, Universidade de Aveiro, Portugal, 2008. http://hdl.handle.net/10773/2931. |
[22] |
M. Romito,
Ergodicity of the finite dimensional approximation of the 3D Navier–Stokes equations forced by a degenerate noise, J. Statist. Phys., 114 (2004), 155-177.
doi: 10.1023/B:JOSS.0000003108.92097.5c. |
[23] |
A. Sarychev,
Controllability of the cubic Schroedinger equation via a low-dimensional source term, Math. Control Relat. Fields, 2 (2012), 247-270.
doi: 10.3934/mcrf.2012.2.247. |
[24] |
A. Shirikyan,
Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys., 266 (2006), 123-151.
doi: 10.1007/s00220-006-0007-3. |
[25] |
A. Shirikyan, Controllability of nonlinear PDEs: Agrachev–Sarychev approach, Journées Équations aux Dérivées Partielles. Évian, 4 juin–8 juin. Exposé no. IV, 2007, 1–11. https://eudml.org/doc/10631. |
[26] |
A. Shirikyan,
Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537.
doi: 10.1016/j.anihpc.2006.04.002. |
[27] |
A. Shirikyan,
Euler equations are not exactly controllable by a finite-dimensional external force, Phys. D, 237 (2008), 1317-1323.
doi: 10.1016/j.physd.2008.03.021. |
[28] |
A. Shirikyan, Global exponential stabilisation for the burgers equation with localised control, J. Éc. polytech. Math., 4: 613–632, 2017.
doi: 10.5802/jep.53. |
[29] |
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, 2nd edition, Philadelphia, PA, 1995. |
[30] |
Y. Xiao and Z. Xin,
On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.
doi: 10.1002/cpa.20187. |
[31] |
Y. Xiao and Z. Xin,
On the inviscid limit of the 3D Navier–Stokes equations with generalized Navier-slip boundary conditions, Commun. Math. Stat., 1 (2013), 259-279.
doi: 10.1007/s40304-013-0014-6. |




[1] |
Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and 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 and 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 and Control Theory, 2021, 10 (3) : 471-489. doi: 10.3934/eect.2020076 |
[12] |
Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations and Control Theory, 2021, 10 (4) : 749-766. doi: 10.3934/eect.2020091 |
[13] |
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 and Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027 |
[14] |
Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 |
[15] |
D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251 |
[16] |
Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\beta$-plane. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 687-701. doi: 10.3934/dcdsb.2011.16.687 |
[17] |
Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 |
[18] |
Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 |
[19] |
C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 |
[20] |
Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]