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A new solution method for nonlinear fractional integro-differential equations
A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions
1. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla |
2. | 221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849 |
References:
[1] |
M. Anguiano, T. Caraballo, J. Real and J. Valero, Pullback attractors for nonautonomous dynamical systems,, Differential and Difference Eqns. with Apps., 47 (2013), 217.
doi: 10.1007/978-1-4614-7333-6_15. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Amsterdam, (1992).
|
[3] |
V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity,, Z. angew. Math. Phys., 54 (2003), 449.
doi: 10.1007/s00033-003-1087-y. |
[4] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Vol. I, (1992). Google Scholar |
[5] |
T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays,, Dyn. Partial Differ. Equ., 11 (2014), 345.
doi: 10.4310/DPDE.2014.v11.n4.a3. |
[6] |
T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays,, J. Math. Anal. Appl., 260 (2001), 421.
doi: 10.1006/jmaa.2000.7464. |
[7] |
T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1775.
doi: 10.1098/rspa.2000.0586. |
[8] |
T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays,, J. Differential Equations, 208 (2005), 9.
doi: 10.1016/j.jde.2003.09.008. |
[9] |
T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems,, Discrete Contin. Dyn. Syst., 34 (2014), 51.
doi: 10.3934/dcds.2014.34.51. |
[10] |
T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity,, J. Difference Equ. Appl., 17 (2011), 161.
doi: 10.1080/10236198.2010.549010. |
[11] |
T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441.
doi: 10.1098/rspa.2001.0807. |
[12] |
T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181.
doi: 10.1098/rspa.2003.1166. |
[13] |
T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.
doi: 10.1016/j.jde.2004.04.012. |
[14] |
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.
doi: 10.1016/j.jmaa.2007.01.038. |
[15] |
D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2 (2002), 125.
|
[16] |
H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 283.
doi: 10.1007/s12044-012-0071-x. |
[17] |
V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).
|
[18] |
P. Constantin and C. Foias, Navier Stokes Equations,, The University of Chicago Press, (1988).
|
[19] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100 (1994), 365.
doi: 10.1007/BF01193705. |
[20] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eq., 9 (1995), 307.
doi: 10.1007/BF02219225. |
[21] |
J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model,, Discret Cont. Dyn. Syst., 34 (2014), 4085.
doi: 10.3934/dcds.2014.34.4085. |
[22] |
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 with infinite delay,, Discret Cont. Dyn. Syst., 34 (2014), 181.
doi: 10.3934/dcds.2014.34.181. |
[23] |
J. García-Luengo, P. Marín-Rubio, J. Real and J. Robinson, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing,, Discret Cont. Dyn. Syst., 34 (2014), 203.
doi: 10.3934/dcds.2014.34.203. |
[24] |
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.
|
[25] |
S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay,, Discret Cont. Dyn. Syst. Series B, 16 (2011), 225.
doi: 10.3934/dcdsb.2011.16.225. |
[26] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988).
|
[27] |
X. Han, Exponential attractors for lattice dynamical systems in weighted spaces,, Discrete Contin. Dyn. Syst., 31 (2011), 445.
doi: 10.3934/dcds.2011.31.445. |
[28] |
X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbbZ^k$ in weighted spaces,, J. Math. Anal. Appl., 397 (2013), 242.
doi: 10.1016/j.jmaa.2012.07.015. |
[29] |
P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems,, American Mathematical Society, (2011).
doi: 10.1090/surv/176. |
[30] |
P. E. Kloeden, Pullback attractors in nonautonomous difference equations,, J. Difference Eqns. Appl., 6 (2000), 33.
doi: 10.1080/10236190008808212. |
[31] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211.
|
[32] |
P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141.
doi: 10.1023/A:1019156812251. |
[33] |
V. B. Kolmanovskii and L. E. Shaikhet, General method of Lyapunov functionals construction for stability investigations of stochastic difference equations,, in Dynamical Systems and Applications, 4 (1995), 397.
doi: 10.1142/9789812796417_0026. |
[34] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge, (1991).
doi: 10.1017/CBO9780511569418. |
[35] |
J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires,, Paris; Dunod, (1969).
|
[36] |
J. Málek and D. Pražák, Large time behavior via the Method of l-trajectories,, J. Diferential Equations, 181 (2002), 243.
doi: 10.1006/jdeq.2001.4087. |
[37] |
P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Analysis, 74 (2011), 2012.
doi: 10.1016/j.na.2010.11.008. |
[38] |
B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay,, Automat. i Telemeh., 21 (1960), 740.
|
[39] |
B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, (1992), 185. Google Scholar |
[40] |
G. Sell, Non-autonomous differential equations and topological dynamics I,, Trans. Amer. Math. Soc., 127 (1967), 241.
|
[41] |
T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.
doi: 10.3934/dcds.2005.12.997. |
[42] |
R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).
|
[43] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4684-0313-8. |
[44] |
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, 2nd Ed., (1995).
doi: 10.1137/1.9781611970050. |
[45] |
L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier-Stokes equations with finite memory,, Journal of Mathematical Physics, 52 (2011).
doi: 10.1063/1.3574630. |
[46] |
S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces,, Nonlinear Anal., 78 (2013), 141.
doi: 10.1016/j.na.2012.10.001. |
show all references
References:
[1] |
M. Anguiano, T. Caraballo, J. Real and J. Valero, Pullback attractors for nonautonomous dynamical systems,, Differential and Difference Eqns. with Apps., 47 (2013), 217.
doi: 10.1007/978-1-4614-7333-6_15. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Amsterdam, (1992).
|
[3] |
V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity,, Z. angew. Math. Phys., 54 (2003), 449.
doi: 10.1007/s00033-003-1087-y. |
[4] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Vol. I, (1992). Google Scholar |
[5] |
T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays,, Dyn. Partial Differ. Equ., 11 (2014), 345.
doi: 10.4310/DPDE.2014.v11.n4.a3. |
[6] |
T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays,, J. Math. Anal. Appl., 260 (2001), 421.
doi: 10.1006/jmaa.2000.7464. |
[7] |
T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1775.
doi: 10.1098/rspa.2000.0586. |
[8] |
T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays,, J. Differential Equations, 208 (2005), 9.
doi: 10.1016/j.jde.2003.09.008. |
[9] |
T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems,, Discrete Contin. Dyn. Syst., 34 (2014), 51.
doi: 10.3934/dcds.2014.34.51. |
[10] |
T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity,, J. Difference Equ. Appl., 17 (2011), 161.
doi: 10.1080/10236198.2010.549010. |
[11] |
T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441.
doi: 10.1098/rspa.2001.0807. |
[12] |
T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181.
doi: 10.1098/rspa.2003.1166. |
[13] |
T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.
doi: 10.1016/j.jde.2004.04.012. |
[14] |
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.
doi: 10.1016/j.jmaa.2007.01.038. |
[15] |
D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2 (2002), 125.
|
[16] |
H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 283.
doi: 10.1007/s12044-012-0071-x. |
[17] |
V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).
|
[18] |
P. Constantin and C. Foias, Navier Stokes Equations,, The University of Chicago Press, (1988).
|
[19] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100 (1994), 365.
doi: 10.1007/BF01193705. |
[20] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eq., 9 (1995), 307.
doi: 10.1007/BF02219225. |
[21] |
J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model,, Discret Cont. Dyn. Syst., 34 (2014), 4085.
doi: 10.3934/dcds.2014.34.4085. |
[22] |
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 with infinite delay,, Discret Cont. Dyn. Syst., 34 (2014), 181.
doi: 10.3934/dcds.2014.34.181. |
[23] |
J. García-Luengo, P. Marín-Rubio, J. Real and J. Robinson, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing,, Discret Cont. Dyn. Syst., 34 (2014), 203.
doi: 10.3934/dcds.2014.34.203. |
[24] |
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.
|
[25] |
S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay,, Discret Cont. Dyn. Syst. Series B, 16 (2011), 225.
doi: 10.3934/dcdsb.2011.16.225. |
[26] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988).
|
[27] |
X. Han, Exponential attractors for lattice dynamical systems in weighted spaces,, Discrete Contin. Dyn. Syst., 31 (2011), 445.
doi: 10.3934/dcds.2011.31.445. |
[28] |
X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbbZ^k$ in weighted spaces,, J. Math. Anal. Appl., 397 (2013), 242.
doi: 10.1016/j.jmaa.2012.07.015. |
[29] |
P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems,, American Mathematical Society, (2011).
doi: 10.1090/surv/176. |
[30] |
P. E. Kloeden, Pullback attractors in nonautonomous difference equations,, J. Difference Eqns. Appl., 6 (2000), 33.
doi: 10.1080/10236190008808212. |
[31] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211.
|
[32] |
P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141.
doi: 10.1023/A:1019156812251. |
[33] |
V. B. Kolmanovskii and L. E. Shaikhet, General method of Lyapunov functionals construction for stability investigations of stochastic difference equations,, in Dynamical Systems and Applications, 4 (1995), 397.
doi: 10.1142/9789812796417_0026. |
[34] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge, (1991).
doi: 10.1017/CBO9780511569418. |
[35] |
J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires,, Paris; Dunod, (1969).
|
[36] |
J. Málek and D. Pražák, Large time behavior via the Method of l-trajectories,, J. Diferential Equations, 181 (2002), 243.
doi: 10.1006/jdeq.2001.4087. |
[37] |
P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Analysis, 74 (2011), 2012.
doi: 10.1016/j.na.2010.11.008. |
[38] |
B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay,, Automat. i Telemeh., 21 (1960), 740.
|
[39] |
B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, (1992), 185. Google Scholar |
[40] |
G. Sell, Non-autonomous differential equations and topological dynamics I,, Trans. Amer. Math. Soc., 127 (1967), 241.
|
[41] |
T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.
doi: 10.3934/dcds.2005.12.997. |
[42] |
R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).
|
[43] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4684-0313-8. |
[44] |
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, 2nd Ed., (1995).
doi: 10.1137/1.9781611970050. |
[45] |
L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier-Stokes equations with finite memory,, Journal of Mathematical Physics, 52 (2011).
doi: 10.1063/1.3574630. |
[46] |
S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces,, Nonlinear Anal., 78 (2013), 141.
doi: 10.1016/j.na.2012.10.001. |
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