# American Institute of Mathematical Sciences

July  2011, 16(1): 225-238. doi: 10.3934/dcdsb.2011.16.225

## On a class of three dimensional Navier-Stokes equations with bounded delay

 1 Universidade Estadual do Oeste do Paraná - UNIOESTE, Colegiado do curso de Matemática, Rua Universitária, 2069. Cx.P. 711, 85819-110 Cascavel, PR, Brazil 2 Departamento de Matemática, IMECC - UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859 Campinas, SP, Brazil

Received  June 2010 Revised  September 2010 Published  April 2011

In this paper we consider a three dimensional Navier-Stokes type equations with delay terms. We discuss the existence of weak and strong solutions and we study the asymptotic behavior of the strong solutions. Moreover, under suitable assumptions, we show the exponential stability of stationary solutions.
Citation: Sandro M. Guzzo, Gabriela Planas. On a class of three dimensional Navier-Stokes equations with bounded delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 225-238. doi: 10.3934/dcdsb.2011.16.225
##### References:
 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays,, Proc. R. Soc. Lond. A, 457 (2001), 2441.  doi: 10.1098/rspa.2001.0807.  Google Scholar [2] T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays,, Proc. R. Soc. Lond. A, 459 (2003), 3181.  doi: 10.1098/rspa.2003.1166.  Google Scholar [3] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [4] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains,, Nonlinear Anal., 64 (2006), 1100.  doi: 10.1016/j.na.2005.05.057.  Google Scholar [5] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation,, Discrete Contin. Dyn. Syst. Ser. B., 2 (2002), 47.  doi: 10.3934/dcdsb.2002.2.47.  Google Scholar [6] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,, Nonlinear Anal., 67 (2007), 2784.  doi: 10.1016/j.na.2006.09.035.  Google Scholar [7] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,, Discrete Contin. Dyn. Syst., 26 (2010), 989.  doi: 10.3934/dcds.2010.26.989.  Google Scholar [8] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations,, Discrete Contin. Dyn. Syst., 21 (2008), 1245.  doi: 10.3934/dcds.2008.21.1245.  Google Scholar [9] J. C. Robinson, "Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,", Cambridge texts in applied mathematics, (2001).   Google Scholar [10] Y. Tang and M. Wan, A remark on exponential stability of time-delayed Burgers equation,, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 219.  doi: 10.3934/dcdsb.2009.12.219.  Google Scholar [11] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.  doi: 10.3934/dcds.2005.12.997.  Google Scholar [12] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis,", Studies in Mathematics and its applications. Volume \textbf{2}, 2 (1984).   Google Scholar

show all references

##### References:
 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays,, Proc. R. Soc. Lond. A, 457 (2001), 2441.  doi: 10.1098/rspa.2001.0807.  Google Scholar [2] T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays,, Proc. R. Soc. Lond. A, 459 (2003), 3181.  doi: 10.1098/rspa.2003.1166.  Google Scholar [3] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [4] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains,, Nonlinear Anal., 64 (2006), 1100.  doi: 10.1016/j.na.2005.05.057.  Google Scholar [5] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation,, Discrete Contin. Dyn. Syst. Ser. B., 2 (2002), 47.  doi: 10.3934/dcdsb.2002.2.47.  Google Scholar [6] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,, Nonlinear Anal., 67 (2007), 2784.  doi: 10.1016/j.na.2006.09.035.  Google Scholar [7] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,, Discrete Contin. Dyn. Syst., 26 (2010), 989.  doi: 10.3934/dcds.2010.26.989.  Google Scholar [8] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations,, Discrete Contin. Dyn. Syst., 21 (2008), 1245.  doi: 10.3934/dcds.2008.21.1245.  Google Scholar [9] J. C. Robinson, "Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,", Cambridge texts in applied mathematics, (2001).   Google Scholar [10] Y. Tang and M. Wan, A remark on exponential stability of time-delayed Burgers equation,, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 219.  doi: 10.3934/dcdsb.2009.12.219.  Google Scholar [11] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.  doi: 10.3934/dcds.2005.12.997.  Google Scholar [12] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis,", Studies in Mathematics and its applications. Volume \textbf{2}, 2 (1984).   Google Scholar
 [1] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [2] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [3] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [4] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [5] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [6] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [7] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [8] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [9] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [10] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [11] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [12] Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $-\Delta u = \lambda f(u)$ as $\lambda\to-\infty$. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293 [13] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [14] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [15] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [16] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [17] Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218 [18] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [19] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 [20] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2019 Impact Factor: 1.27