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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

Linfang Liu 1, , Tomás Caraballo 2,, and  Xianlong Fu 1,

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  December 2018 Early access  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.

Keywords: Exponential stability, stationary solution, non-Newtonian fluids, delay.
Mathematics Subject Classification: Primary: 34D20, 34K20, 34K21, 37L15, 76A05.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

B. Guo, G. Lin and Y. Shang, Non-Newtonian Fluids Dynamical Systems, National Defense Industry Press, in Chinese, 2006. Google Scholar

[12]

B. Guo, C. Guo and J. Zhang, Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23 (2010), 303-326.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011.  Google Scholar

[21]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013.  Google Scholar

[22]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

B. Guo, G. Lin and Y. Shang, Non-Newtonian Fluids Dynamical Systems, National Defense Industry Press, in Chinese, 2006. Google Scholar

[12]

B. Guo, C. Guo and J. Zhang, Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23 (2010), 303-326.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, 2011.  Google Scholar

[21]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013.  Google Scholar

[22]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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