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October  2022, 15(10): 2965-2980. doi: 10.3934/dcdss.2022127

The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations

 School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Dedicated to Professor Georg Hetzer's 75th birthday

Received  April 2021 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The author is supported by Initial Scientific Research Fund in Northwest Normal University under Grant No. 202103101206/6014

This paper is concerned with the general stability of the solution to a stochastic functional 2D Navier-Stokes equation driven by a multiplicative white noise when the viscosity coefficient is time varying. First we give some sufficient conditions ensuring the existence and uniqueness of global solutions. Then the general stability of the solution in the sense of p-th ($p\geq2$) moment is established. From this fact we further prove that the null solution is almost surely stable with the general decay rate. The convergence in probability of the solution is also analyzed.

Citation: Tongtong Liang. The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 2965-2980. doi: 10.3934/dcdss.2022127
References:
 [1] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359.  doi: 10.4310/DPDE.2014.v11.n4.a3. [2] T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101.  doi: 10.3934/dcdss.2015.8.1079. [3] T. Caraballo, J. A. Langa and T. Taniguchi, The exponential behaviour and stabilizability of stochastic 2D-Navier-Stokes equations, J. Differential Equations, 179 (2002), 714-737.  doi: 10.1006/jdeq.2001.4037. [4] 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. [5] 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. [6] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci. Math. Sci., 122 (2012), 283-295.  doi: 10.1007/s12044-012-0071-x. [7] H. Chen, P. Shi and C.-C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching, Systems Control Lett., 110 (2017), 38-48.  doi: 10.1016/j.sysconle.2017.10.008. [8] H. Chen and C. Yuan, On the asymptotic behavior for neutral stochastic differential delay equations, IEEE Trans. Automat. Control, 64 (2019), 1671-1678.  doi: 10.1109/TAC.2018.2852607. [9] X. Gao and H. Gao, Existence and uniqueness of weak solutions to stochastic 3D Navier-Stokes equations with delays, Appl. Math. Lett., 95 (2019), 158-164.  doi: 10.1016/j.aml.2019.03.037. [10] J. García-Luengo and P. Marín-Rubio, Pullback attractors for 2D Navier-Stokes equations with delays and the flattening property, Commun. Pure Appl. Anal., 19 (2020), 2127-2146.  doi: 10.3934/cpaa.2020094. [11] 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 (2004), 181-201.  doi: 10.3934/dcds.2014.34.181. [12] 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. [13] J. Li, Y. Wang and X.-G. Yang, Pullback attractors of 2D Navier-Stokes equations with weak damping, distributed delay, and continuous delay, Math. Methods Appl. Sci., 39 (2016), 3186-3203.  doi: 10.1002/mma.3762. [14] T. Liang and Y. Wang, Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4697-4726.  doi: 10.3934/dcdsb.2020309. [15] L. Liu and T. Caraballo, Analysis of a stochastic $2D$-Navier-Stokes model with infinite delay, J. Dynam. Differential Equations, 31 (2019), 2249-2274.  doi: 10.1007/s10884-018-9703-x. [16] L. Liu, T. Caraballo and P. Marín-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008. [17] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779. [18] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Asymptotic behavior of solutions for a three dimensional system of globally modified Navier-Stokes equations with a locally Lipschitz delay term, Nonlinear Anal., 79 (2013), 68-79.  doi: 10.1016/j.na.2012.11.006. [19] 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. [20] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, in: Lecture Notes in Mathematics, Springer, Berlin, 2007. [21] T. Taniguchi, The exponential 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. [22] T. Taniguchi, The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by Lévy processes, J. Math. Anal. Appl., 385 (2012), 634-654.  doi: 10.1016/j.jmaa.2011.06.076. [23] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. [24] M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay, Appl. Math. J. Chin. Univ. Ser. A, 24 (2009), 493-500.

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References:
 [1] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359.  doi: 10.4310/DPDE.2014.v11.n4.a3. [2] T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101.  doi: 10.3934/dcdss.2015.8.1079. [3] T. Caraballo, J. A. Langa and T. Taniguchi, The exponential behaviour and stabilizability of stochastic 2D-Navier-Stokes equations, J. Differential Equations, 179 (2002), 714-737.  doi: 10.1006/jdeq.2001.4037. [4] 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. [5] 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. [6] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci. Math. Sci., 122 (2012), 283-295.  doi: 10.1007/s12044-012-0071-x. [7] H. Chen, P. Shi and C.-C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching, Systems Control Lett., 110 (2017), 38-48.  doi: 10.1016/j.sysconle.2017.10.008. [8] H. Chen and C. Yuan, On the asymptotic behavior for neutral stochastic differential delay equations, IEEE Trans. Automat. Control, 64 (2019), 1671-1678.  doi: 10.1109/TAC.2018.2852607. [9] X. Gao and H. Gao, Existence and uniqueness of weak solutions to stochastic 3D Navier-Stokes equations with delays, Appl. Math. Lett., 95 (2019), 158-164.  doi: 10.1016/j.aml.2019.03.037. [10] J. García-Luengo and P. Marín-Rubio, Pullback attractors for 2D Navier-Stokes equations with delays and the flattening property, Commun. Pure Appl. Anal., 19 (2020), 2127-2146.  doi: 10.3934/cpaa.2020094. [11] 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 (2004), 181-201.  doi: 10.3934/dcds.2014.34.181. [12] 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. [13] J. Li, Y. Wang and X.-G. Yang, Pullback attractors of 2D Navier-Stokes equations with weak damping, distributed delay, and continuous delay, Math. Methods Appl. Sci., 39 (2016), 3186-3203.  doi: 10.1002/mma.3762. [14] T. Liang and Y. Wang, Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4697-4726.  doi: 10.3934/dcdsb.2020309. [15] L. Liu and T. Caraballo, Analysis of a stochastic $2D$-Navier-Stokes model with infinite delay, J. Dynam. Differential Equations, 31 (2019), 2249-2274.  doi: 10.1007/s10884-018-9703-x. [16] L. Liu, T. Caraballo and P. Marín-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008. [17] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779. [18] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Asymptotic behavior of solutions for a three dimensional system of globally modified Navier-Stokes equations with a locally Lipschitz delay term, Nonlinear Anal., 79 (2013), 68-79.  doi: 10.1016/j.na.2012.11.006. [19] 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. [20] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, in: Lecture Notes in Mathematics, Springer, Berlin, 2007. [21] T. Taniguchi, The exponential 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. [22] T. Taniguchi, The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by Lévy processes, J. Math. Anal. Appl., 385 (2012), 634-654.  doi: 10.1016/j.jmaa.2011.06.076. [23] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. [24] M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay, Appl. Math. J. Chin. Univ. Ser. A, 24 (2009), 493-500.
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