• Previous Article
    Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems
  • MCRF Home
  • This Issue
  • Next Article
    Computation of open-loop inputs for uniformly ensemble controllable systems
doi: 10.3934/mcrf.2021038
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Local null controllability of the penalized Boussinesq system with a reduced number of controls

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

2. 

Chair of Computational Mathematics, Fundación Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country, Spain

3. 

Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33400 Bordeaux, France

* Corresponding author: Jon Asier Bárcena-Petisco

Received  August 2020 Revised  June 2021 Early access July 2021

In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain $ \Omega\subset\mathbb R^N $ for $ N = 2 $ and $ N = 3 $. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter $ \varepsilon > 0 $. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set $ \omega $ contained in $ \Omega $. We also show that the control cost is bounded uniformly with respect to $ \varepsilon \rightarrow 0 $. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.

Citation: Jon Asier Bárcena-Petisco, Kévin Le Balc'h. Local null controllability of the penalized Boussinesq system with a reduced number of controls. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021038
References:
[1]

F. Ammar-KhodjaA. BenabdallahM. González Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[2]

M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations, Math. Control Relat. Fields, 1 (2011), 149-175.  doi: 10.3934/mcrf.2011.1.149.  Google Scholar

[3]

J. A. Bárcena-Petisco, Null controllability of a penalized Stokes problem in dimension two with one scalar control, Asymptot. Anal., 117 (2020), 161-198.  doi: 10.3233/ASY-191550.  Google Scholar

[4]

J. A. Bárcena-Petisco, Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit, SIAM J. Control Optim., 58 (2020), 1597-1625.  doi: 10.1137/19M1252004.  Google Scholar

[5]

K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations, J. Math. Pures Appl. (9), 136 (2020), 22-91.  doi: 10.1016/j.matpur.2020.02.001.  Google Scholar

[6]

M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér., 12 (1978), 211-236.  doi: 10.1051/m2an/1978120302111.  Google Scholar

[7]

N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), 361-382.  doi: 10.3934/mcrf.2012.2.361.  Google Scholar

[8]

N. Carreño and S. Guerrero, Local null controllability of the $N$-dimensional Navier–Stokes system with $N- 1$ scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.  Google Scholar

[9]

N. CarreñoS. Guerrero and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var., 21 (2015), 73-100.  doi: 10.1051/cocv/2014020.  Google Scholar

[10]

F. W. Chaves-Silva, E. Fernández-Cara, K. Le Balc'h, J. L. F. Machado and D. A. Souza, Small-time global exact controllability to the trajectories for the viscous Boussinesq system, preprint, arXiv: 2006.01682. Google Scholar

[11]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar

[12]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar

[13]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.  Google Scholar

[14]

J.-M. CoronF. Marbach and F. Sueur, Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), 22 (2020), 1625-1673.  doi: 10.4171/JEMS/952.  Google Scholar

[15]

M. Duprez and P. Lissy, Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control, preprint, arXiv: 1909.02831. Google Scholar

[16]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[17]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.  Google Scholar

[18]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier–Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.  Google Scholar

[19]

E. Fernádez-CaraJ. Limaco and S. B. de Menezes, Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control, ESAIM Control Optim. Calc. Var., 22 (2016), 1017-1039.  doi: 10.1051/cocv/2016031.  Google Scholar

[20]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[21]

B. Geshkovski and E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59 (2021), 1830-1850.  doi: 10.1137/19M1285354.  Google Scholar

[22]

S. Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029-1054.  doi: 10.1016/j.anihpc.2006.11.001.  Google Scholar

[23]

S. Guerrero and C. Montoya, Local null controllability of the $N$-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and $N-1$ scalar controls, J. Math. Pures Appl. (9), 113 (2018), 37-69.  doi: 10.1016/j.matpur.2018.03.004.  Google Scholar

[24]

V. Hernández-Santamaría and K. Le Balc'h, Local null-controllability of a nonlocal semilinear heat equation, Appl. Math. Optim., (2020), 1–49. Google Scholar

[25]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.  Google Scholar

[26]

O. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[27]

K. Le Balc'h, Local controllability of reaction-diffusion systems around nonnegative stationary states, ESAIM Control Optim. Calc. Var., 26 (2020), 32pp. doi: 10.1051/cocv/2019033.  Google Scholar

[28]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988.  Google Scholar

[29]

J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, 1996,221–235.  Google Scholar

[30]

Y. LiuT. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2013), 20-42.  doi: 10.1051/cocv/2011196.  Google Scholar

[31]

S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differential Equations, 264 (2018), 3664-3703.  doi: 10.1016/j.jde.2017.11.029.  Google Scholar

[32]

J. T. Oden and O.-P. Jacquotte, Stability of some mixed finite element methods for Stokesian flows, Comput. Methods Appl. Mech. Engrg., 43 (1984), 231-247.  doi: 10.1016/0045-7825(84)90007-0.  Google Scholar

[33]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[34]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations, in Proceedings of the 1994 Beijing Symposium on Nonlinear Evolution Equations and Infinite Dynamical Systems, 1997, 68–78. Available from: https://www.math.purdue.edu/shen7/pub/Pseudo-c.pdf. Google Scholar

[35]

T. Takahashi, Boundary local null-controllability of the Kuramoto–Sivashinsky equation, Math. Control Signals Systems, 29 (2017), 21pp. doi: 10.1007/s00498-016-0182-5.  Google Scholar

[36]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[37]

R. Temam, Une méthode d'approximation de la solution des équations des Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152.  doi: 10.24033/bsmf.1662.  Google Scholar

[38]

E. Zuazua, A uniqueness result for the linear system of elasticity and its control theoretical consequences, SIAM J. Control Optim., 34 (1996), 1473-1495.  doi: 10.1137/S0363012993260070.  Google Scholar

show all references

References:
[1]

F. Ammar-KhodjaA. BenabdallahM. González Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[2]

M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations, Math. Control Relat. Fields, 1 (2011), 149-175.  doi: 10.3934/mcrf.2011.1.149.  Google Scholar

[3]

J. A. Bárcena-Petisco, Null controllability of a penalized Stokes problem in dimension two with one scalar control, Asymptot. Anal., 117 (2020), 161-198.  doi: 10.3233/ASY-191550.  Google Scholar

[4]

J. A. Bárcena-Petisco, Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit, SIAM J. Control Optim., 58 (2020), 1597-1625.  doi: 10.1137/19M1252004.  Google Scholar

[5]

K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations, J. Math. Pures Appl. (9), 136 (2020), 22-91.  doi: 10.1016/j.matpur.2020.02.001.  Google Scholar

[6]

M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér., 12 (1978), 211-236.  doi: 10.1051/m2an/1978120302111.  Google Scholar

[7]

N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), 361-382.  doi: 10.3934/mcrf.2012.2.361.  Google Scholar

[8]

N. Carreño and S. Guerrero, Local null controllability of the $N$-dimensional Navier–Stokes system with $N- 1$ scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.  Google Scholar

[9]

N. CarreñoS. Guerrero and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var., 21 (2015), 73-100.  doi: 10.1051/cocv/2014020.  Google Scholar

[10]

F. W. Chaves-Silva, E. Fernández-Cara, K. Le Balc'h, J. L. F. Machado and D. A. Souza, Small-time global exact controllability to the trajectories for the viscous Boussinesq system, preprint, arXiv: 2006.01682. Google Scholar

[11]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar

[12]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar

[13]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.  Google Scholar

[14]

J.-M. CoronF. Marbach and F. Sueur, Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), 22 (2020), 1625-1673.  doi: 10.4171/JEMS/952.  Google Scholar

[15]

M. Duprez and P. Lissy, Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control, preprint, arXiv: 1909.02831. Google Scholar

[16]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[17]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.  Google Scholar

[18]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier–Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.  Google Scholar

[19]

E. Fernádez-CaraJ. Limaco and S. B. de Menezes, Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control, ESAIM Control Optim. Calc. Var., 22 (2016), 1017-1039.  doi: 10.1051/cocv/2016031.  Google Scholar

[20]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[21]

B. Geshkovski and E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59 (2021), 1830-1850.  doi: 10.1137/19M1285354.  Google Scholar

[22]

S. Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029-1054.  doi: 10.1016/j.anihpc.2006.11.001.  Google Scholar

[23]

S. Guerrero and C. Montoya, Local null controllability of the $N$-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and $N-1$ scalar controls, J. Math. Pures Appl. (9), 113 (2018), 37-69.  doi: 10.1016/j.matpur.2018.03.004.  Google Scholar

[24]

V. Hernández-Santamaría and K. Le Balc'h, Local null-controllability of a nonlocal semilinear heat equation, Appl. Math. Optim., (2020), 1–49. Google Scholar

[25]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.  Google Scholar

[26]

O. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[27]

K. Le Balc'h, Local controllability of reaction-diffusion systems around nonnegative stationary states, ESAIM Control Optim. Calc. Var., 26 (2020), 32pp. doi: 10.1051/cocv/2019033.  Google Scholar

[28]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988.  Google Scholar

[29]

J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, 1996,221–235.  Google Scholar

[30]

Y. LiuT. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2013), 20-42.  doi: 10.1051/cocv/2011196.  Google Scholar

[31]

S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differential Equations, 264 (2018), 3664-3703.  doi: 10.1016/j.jde.2017.11.029.  Google Scholar

[32]

J. T. Oden and O.-P. Jacquotte, Stability of some mixed finite element methods for Stokesian flows, Comput. Methods Appl. Mech. Engrg., 43 (1984), 231-247.  doi: 10.1016/0045-7825(84)90007-0.  Google Scholar

[33]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[34]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations, in Proceedings of the 1994 Beijing Symposium on Nonlinear Evolution Equations and Infinite Dynamical Systems, 1997, 68–78. Available from: https://www.math.purdue.edu/shen7/pub/Pseudo-c.pdf. Google Scholar

[35]

T. Takahashi, Boundary local null-controllability of the Kuramoto–Sivashinsky equation, Math. Control Signals Systems, 29 (2017), 21pp. doi: 10.1007/s00498-016-0182-5.  Google Scholar

[36]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[37]

R. Temam, Une méthode d'approximation de la solution des équations des Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152.  doi: 10.24033/bsmf.1662.  Google Scholar

[38]

E. Zuazua, A uniqueness result for the linear system of elasticity and its control theoretical consequences, SIAM J. Control Optim., 34 (1996), 1473-1495.  doi: 10.1137/S0363012993260070.  Google Scholar

[1]

Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control & Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149

[2]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

[3]

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

[4]

Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737

[5]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5449-5463. doi: 10.3934/dcdsb.2020353

[6]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169

[7]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

[8]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[9]

Claudia Valls. Stability of some waves in the Boussinesq system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 929-939. doi: 10.3934/cpaa.2006.5.929

[10]

A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213

[11]

Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311

[12]

Nicolás Carreño. Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Mathematical Control & Related Fields, 2012, 2 (4) : 361-382. doi: 10.3934/mcrf.2012.2.361

[13]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207

[14]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665

[15]

Min Chen. Numerical investigation of a two-dimensional Boussinesq system. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1169-1190. doi: 10.3934/dcds.2009.23.1169

[16]

Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125

[17]

Debayan Maity. On the null controllability of the Lotka-Mckendrick system. Mathematical Control & Related Fields, 2019, 9 (4) : 719-728. doi: 10.3934/mcrf.2019048

[18]

Karine Adamy. Existence of solutions for a Boussinesq system on the half line and on a finite interval. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 25-49. doi: 10.3934/dcds.2011.29.25

[19]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035

[20]

Jerry L. Bona, Zoran Grujić, Henrik Kalisch. A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1121-1139. doi: 10.3934/dcds.2010.26.1121

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (81)
  • HTML views (159)
  • Cited by (0)

Other articles
by authors

[Back to Top]