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Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping
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 |
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.
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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