American Institute of Mathematical Sciences

Advanced
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

Duy Phan ,

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.

Keywords: Navier–Stokes equations, approximate controllability, saturating set.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Nersisyan, Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564.  doi: 10.1080/03605302.2011.596605.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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/. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, 2nd edition, Philadelphia, PA, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Nersisyan, Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564.  doi: 10.1080/03605302.2011.596605.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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/. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, 2nd edition, Philadelphia, PA, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Induction Step based on the decomposition (19)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Step 3.3.1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Step 3.3.2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Step 3.3.3
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[2]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[3]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[4]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201
[5]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119
[6]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[7]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[8]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[9]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027
[10]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[11]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[12]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[13]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[14]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[15]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[16]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[17]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[18]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[19]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[20]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (84)
  • HTML views (283)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences