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March
2021, 10(1): 199-227.
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 |
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.
Mathematics Subject Classification:
Primary: 93B05, 35Q30; Secondary: 93C20.
Citation:
Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations and Control Theory,
2021, 10
(1)
: 199-227.
doi: 10.3934/eect.2020062
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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. |
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