April  2020, 19(4): 1931-1948. doi: 10.3934/cpaa.2020085

Longtime behavior for 3D Navier-Stokes equations with constant delays

1. 

Department of Mathematics and Statistics, University of Wyoming, Laramie 82071 USA

2. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Sevilla, SPAIN, 41012

Dedicated to Tomás Caraballo on his 60th birthday

Received  June 2019 Revised  October 2019 Published  January 2020

Fund Project: The first author was partially supported by Simons Foundation grant 582264. The second author was partially supported by grant PGC2018-096540-I00.

This paper investigates the longtime behavior of delayed 3D Navier-Stokes equations in terms of attractors. The study will strongly rely on the investigation of the linearized Navier-Stokes system, and the relationship between the discrete dynamical flow for the linearized system and the continuous flow associated to the original system. Assuming the viscosity to be sufficiently large, there exists a unique attractor for the delayed 3D Navier-Stokes equations. Moreover, the attractor reduces to a singleton set.

Citation: Hakima Bessaih, María J. Garrido-Atienza. Longtime behavior for 3D Navier-Stokes equations with constant delays. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1931-1948. doi: 10.3934/cpaa.2020085
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

H. BessaihM. J. Garrido-Atienza and B. Schmalfuß, On 3D Navier-Stokes equations: regularization and uniqueness by delays, Physica D: Nonlinear Phenomena, 376/377 (2018), 228-237.  doi: 10.1016/j.physd.2018.03.004.

[3]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Translations of Mathematical Monographs, 187. American Mathematical Society, Providence, RI, 2000.

[4]

J. García-LuengoP. 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.

[5]

S. M. Guzzo and G. Planas, On a class of three dimensional Navier–Stokes equations with bounded delay, Discrete Cont. Dyn. Syst. Series B, 16 (2011), 225-238.  doi: 10.3934/dcdsb.2011.16.225.

[6]

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.

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[8]

G. Planas and E. Hernández, Asymptotic behavior of two–dimensional time–delayed Navier–Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258.  doi: 10.3934/dcds.2008.21.1245.

[9]

R. Temam, Navier-Stokes equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York, 1977.

[10]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[11]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[12]

W. Varnhorn, The Navier–Stokes Equations with Time Delay, Applied Mathematical Sciences, 2 (2008), 947–960.

[13]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-1423-0.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

H. BessaihM. J. Garrido-Atienza and B. Schmalfuß, On 3D Navier-Stokes equations: regularization and uniqueness by delays, Physica D: Nonlinear Phenomena, 376/377 (2018), 228-237.  doi: 10.1016/j.physd.2018.03.004.

[3]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Translations of Mathematical Monographs, 187. American Mathematical Society, Providence, RI, 2000.

[4]

J. García-LuengoP. 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.

[5]

S. M. Guzzo and G. Planas, On a class of three dimensional Navier–Stokes equations with bounded delay, Discrete Cont. Dyn. Syst. Series B, 16 (2011), 225-238.  doi: 10.3934/dcdsb.2011.16.225.

[6]

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.

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[8]

G. Planas and E. Hernández, Asymptotic behavior of two–dimensional time–delayed Navier–Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258.  doi: 10.3934/dcds.2008.21.1245.

[9]

R. Temam, Navier-Stokes equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York, 1977.

[10]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[11]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[12]

W. Varnhorn, The Navier–Stokes Equations with Time Delay, Applied Mathematical Sciences, 2 (2008), 947–960.

[13]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-1423-0.

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