# American Institute of Mathematical Sciences

August  2019, 24(8): 3929-3946. doi: 10.3934/dcdsb.2018337

## Some regularity results for a double time-delayed 2D-Navier-Stokes model

 1 Departamento de Matemática Aplicada a las TIC, Universidad Politécnica de Madrid, C/ Nikola Tesla s/n, 28031, Madrid, Spain 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012, Sevilla, Spain 3 Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda 651, 13083-859 Campinas, SP, Brazil

Dedicated to Professor Peter E. Kloeden on occasion of his Seventieth Birthday

Received  May 2018 Revised  September 2018 Published  January 2019

In this paper we analyze some regularity properties of a double time-delayed 2D-Navier-Stokes model, that includes not only a delay force but also a delay in the convective term. The interesting feature of the model -from the mathematical point of view- is that being in dimension two, it behaves similarly as a 3D-model without delay, and extra conditions in order to have uniqueness were required for well-posedness. This model was previously studied in several papers, being the existence of attractor in the $L^2$-framework obtained by the authors [Discrete Contin. Dyn. Syst. 34 (2014), 4085-4105]. Here regularization properties of the solutions and existence of (regular) attractors for several associated dynamical systems are established. Moreover, relationships among these objects are also provided.

Citation: Julia García-Luengo, Pedro Marín-Rubio, Gabriela Planas. Some regularity results for a double time-delayed 2D-Navier-Stokes model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3929-3946. doi: 10.3934/dcdsb.2018337
##### References:
 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807. Google Scholar [2] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. 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), 271-297. doi: 10.1016/j.jde.2004.04.012. Google Scholar [4] J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105. doi: 10.3934/dcds.2014.34.4085. Google Scholar [5] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. Google Scholar [6] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. doi: 10.1515/ans-2013-0205. Google Scholar [7] J. García-Luengo, P. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst., 34 (2014), 181-201. doi: 10.3934/dcds.2014.34.181. Google Scholar [8] J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621. doi: 10.3934/cpaa.2015.14.1603. Google Scholar [9] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057. Google Scholar [10] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238. doi: 10.3934/dcdsb.2011.16.225. Google Scholar [11] S. M. Guzzo and G. Planas, Existence of solutions for a class of Navier-Stokes equations with infinite delay, Appl. Anal. 94 (2015), 840-855. doi: 10.1080/00036811.2014.905677. Google Scholar [12] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56. doi: 10.3934/dcdsb.2002.2.47. Google Scholar [13] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799. doi: 10.1016/j.na.2006.09.035. Google Scholar [14] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065. Google Scholar [15] 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), 989-1006. doi: 10.3934/dcds.2010.26.989. Google Scholar [16] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008. Google Scholar [17] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258. doi: 10.3934/dcds.2008.21.1245. Google Scholar [18] T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. Google Scholar [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar [20] W. Varnhorn, The Navier-Stokes equations with time delay, Applied Mathematical Sciences, 2 (2008), 947-960. Google Scholar

show all references

##### References:
 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807. Google Scholar [2] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. 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), 271-297. doi: 10.1016/j.jde.2004.04.012. Google Scholar [4] J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105. doi: 10.3934/dcds.2014.34.4085. Google Scholar [5] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. Google Scholar [6] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. doi: 10.1515/ans-2013-0205. Google Scholar [7] J. García-Luengo, P. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst., 34 (2014), 181-201. doi: 10.3934/dcds.2014.34.181. Google Scholar [8] J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621. doi: 10.3934/cpaa.2015.14.1603. Google Scholar [9] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057. Google Scholar [10] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238. doi: 10.3934/dcdsb.2011.16.225. Google Scholar [11] S. M. Guzzo and G. Planas, Existence of solutions for a class of Navier-Stokes equations with infinite delay, Appl. Anal. 94 (2015), 840-855. doi: 10.1080/00036811.2014.905677. Google Scholar [12] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56. doi: 10.3934/dcdsb.2002.2.47. Google Scholar [13] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799. doi: 10.1016/j.na.2006.09.035. Google Scholar [14] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065. Google Scholar [15] 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), 989-1006. doi: 10.3934/dcds.2010.26.989. Google Scholar [16] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008. Google Scholar [17] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258. doi: 10.3934/dcds.2008.21.1245. Google Scholar [18] T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. Google Scholar [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar [20] W. Varnhorn, The Navier-Stokes equations with time delay, Applied Mathematical Sciences, 2 (2008), 947-960. Google Scholar
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