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

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

Received  May 2019 Revised  January 2020 Published  March 2021 Early access  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 and Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062
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. ChemetovF. 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. ChemetovF. 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.

Figure 1.  Induction Step based on the decomposition (19)
Figure 2.  Step 3.3.1
Figure 3.  Step 3.3.2
Figure 4.  Step 3.3.3
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