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December  2018, 23(10): 4285-4303. doi: 10.3934/dcdsb.2018138

## Exponential stability of an incompressible non-Newtonian fluid with delay

 1 Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China 2 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Fac. Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain

* Corresponding author: caraball@us.es

Received  December 2017 Revised  January 2018 Published  April 2018

Fund Project: This research was partially supported by the projects MTM2015-63723-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía), and by NSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407), respectively.

The existence and uniqueness of stationary solutions to an incompressible non-Newtonian fluid are first established. The exponential stability of steady-state solutions is then analyzed by means of four different approaches. The first is the classical Lyapunov function method, while the second one is based on a Razumikhin type argument. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. Some comments concerning several open research directions about this model are also included.

Citation: Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
##### References:
 [1] H.-O. Bae, Existence, regularity, and decay rate of solutions of non-Newtonian flow, J. Math. Anal. Appl., 231 (1999), 467-491. doi: 10.1006/jmaa.1998.6242. [2] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations, 19 (1994), 1763-1803. doi: 10.1080/03605309408821073. [3] F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3. [4] T. Caraballo and A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281. doi: 10.4310/DPDE.2013.v10.n3.a3. [5] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [6] 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. [7] T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464. [8] T. Caraballo, A. M. Márquez-Durán and F. Rivero, Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540021, 11pp. [9] 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. [10] 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. [11] B. Guo, G. Lin and Y. Shang, Non-Newtonian Fluids Dynamical Systems, National Defense Industry Press, in Chinese, 2006. [12] B. Guo, C. Guo and J. Zhang, Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23 (2010), 303-326. [13] J. U. Jeong and J. Park, Pullback attractors for a 2D-non-autonomous incompressible non-Newtonian fluid with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2687-2702. doi: 10.3934/dcdsb.2016068. [14] V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results, Math. Comput. Modelling, 36 (2002), 691–716. Lyapunov's methods in stability and control. doi: 10.1016/S0895-7177(02)00168-1. [15] V. Kolmanovskii and L. Shaikhet, General method of Lyapunov functionals construction for stability investigation of stochastic difference equations, In Dynamical Systems and Applications, volume 4 of World Sci. Ser. Appl. Anal., pages 397–439. World Sci. Publ., River Edge, NJ, 1995. [16] O. Ladyzhenskaya, New Equations for the Description of the Viscous Incompressible Fluids and Solvability in the Large of the Boundary Value Problems for Them, in: Boundary Value Problem of Mathematical Physics, American Mathematical Society, Providence, 1970. [17] L. Liu and T. Caraballo, Dynamics of a non-autonomous incompressible non-newtonian fluid with delay, Dynamics of PDE, 14 (2017), 375-402. doi: 10.4310/DPDE.2017.v14.n4.a4. [18] J. Málek, J. Nečas, M. Rokyta and M. Ružička, Weak and Measure-Valued Solutions to Evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1996. [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] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011. [21] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. [22] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997. [23] C. Zhao and Y. Li, H2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006. [24] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001. [25] C. Zhao, S. Zhou and Y. Li, Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid, J. Math. Anal. Appl., 325 (2007), 1350-1362. doi: 10.1016/j.jmaa.2006.02.069.

show all references

##### References:
 [1] H.-O. Bae, Existence, regularity, and decay rate of solutions of non-Newtonian flow, J. Math. Anal. Appl., 231 (1999), 467-491. doi: 10.1006/jmaa.1998.6242. [2] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations, 19 (1994), 1763-1803. doi: 10.1080/03605309408821073. [3] F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3. [4] T. Caraballo and A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281. doi: 10.4310/DPDE.2013.v10.n3.a3. [5] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [6] 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. [7] T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464. [8] T. Caraballo, A. M. Márquez-Durán and F. Rivero, Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540021, 11pp. [9] 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. [10] 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. [11] B. Guo, G. Lin and Y. Shang, Non-Newtonian Fluids Dynamical Systems, National Defense Industry Press, in Chinese, 2006. [12] B. Guo, C. Guo and J. Zhang, Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23 (2010), 303-326. [13] J. U. Jeong and J. Park, Pullback attractors for a 2D-non-autonomous incompressible non-Newtonian fluid with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2687-2702. doi: 10.3934/dcdsb.2016068. [14] V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results, Math. Comput. Modelling, 36 (2002), 691–716. Lyapunov's methods in stability and control. doi: 10.1016/S0895-7177(02)00168-1. [15] V. Kolmanovskii and L. Shaikhet, General method of Lyapunov functionals construction for stability investigation of stochastic difference equations, In Dynamical Systems and Applications, volume 4 of World Sci. Ser. Appl. Anal., pages 397–439. World Sci. Publ., River Edge, NJ, 1995. [16] O. Ladyzhenskaya, New Equations for the Description of the Viscous Incompressible Fluids and Solvability in the Large of the Boundary Value Problems for Them, in: Boundary Value Problem of Mathematical Physics, American Mathematical Society, Providence, 1970. [17] L. Liu and T. Caraballo, Dynamics of a non-autonomous incompressible non-newtonian fluid with delay, Dynamics of PDE, 14 (2017), 375-402. doi: 10.4310/DPDE.2017.v14.n4.a4. [18] J. Málek, J. Nečas, M. Rokyta and M. Ružička, Weak and Measure-Valued Solutions to Evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1996. [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] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011. [21] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. [22] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997. [23] C. Zhao and Y. Li, H2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006. [24] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001. [25] C. Zhao, S. Zhou and Y. Li, Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid, J. Math. Anal. Appl., 325 (2007), 1350-1362. doi: 10.1016/j.jmaa.2006.02.069.
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