A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Previous Article
A new solution method for nonlinear fractional integro-differential equations
DCDS-S Home
This Issue
Next Article
Research on operating mechanism for creative products supply chain based on game theory

December 2015, 8(6): 1079-1101. doi: 10.3934/dcdss.2015.8.1079

A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Tomás Caraballo ^1, and Xiaoying Han ^2,

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

Received June 2015 Revised September 2015 Published December 2015

Abstract
Related Papers

In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research.

Keywords: Galerkin, variable delay, stability, measurable delay., Navier-Stokes equations, distributed delay.

Mathematics Subject Classification: Primary: 35R10, 35B40; Secondary: 47H20, 58F3.

Citation: Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079

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. Google Scholar
[2]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Amsterdam, (1992). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar*
[17]	V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar
[18]	P. Constantin and C. Foias, Navier Stokes Equations,, The University of Chicago Press, (1988). Google Scholar
[19]	H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100* (1994), 365. doi: 10.1007/BF01193705. Google Scholar*
[20]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eq., 9 (1995), 307. doi: 10.1007/BF02219225. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988). Google Scholar
[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. Google Scholar
[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. Google Scholar*
[29]	P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems,, American Mathematical Society, (2011). doi: 10.1090/surv/176. Google Scholar
[30]	P. E. Kloeden, Pullback attractors in nonautonomous difference equations,, J. Difference Eqns. Appl., 6 (2000), 33. doi: 10.1080/10236190008808212. Google Scholar
[31]	P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge, (1991). doi: 10.1017/CBO9780511569418. Google Scholar
[35]	J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires,, Paris; Dunod, (1969). Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[42]	R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979). Google Scholar
[43]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar
[44]	R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, 2nd Ed., (1995). doi: 10.1137/1.9781611970050. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Amsterdam, (1992). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar*
[17]	V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar
[18]	P. Constantin and C. Foias, Navier Stokes Equations,, The University of Chicago Press, (1988). Google Scholar
[19]	H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100* (1994), 365. doi: 10.1007/BF01193705. Google Scholar*
[20]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eq., 9 (1995), 307. doi: 10.1007/BF02219225. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988). Google Scholar
[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. Google Scholar
[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. Google Scholar*
[29]	P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems,, American Mathematical Society, (2011). doi: 10.1090/surv/176. Google Scholar
[30]	P. E. Kloeden, Pullback attractors in nonautonomous difference equations,, J. Difference Eqns. Appl., 6 (2000), 33. doi: 10.1080/10236190008808212. Google Scholar
[31]	P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge, (1991). doi: 10.1017/CBO9780511569418. Google Scholar
[35]	J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires,, Paris; Dunod, (1969). Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[42]	R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979). Google Scholar
[43]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar
[44]	R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, 2nd Ed., (1995). doi: 10.1137/1.9781611970050. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[2]	Sandro M. Guzzo, Gabriela Planas. On a class of three dimensional Navier-Stokes equations with bounded delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 225-238. doi: 10.3934/dcdsb.2011.16.225
[3]	Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[4]	Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[5]	Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312
[6]	Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[7]	Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[8]	Julia García-Luengo, Pedro Marín-Rubio, José Real. Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 181-201. doi: 10.3934/dcds.2014.34.181
[9]	Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[10]	Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
[11]	Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[12]	Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457
[13]	Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369
[14]	Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497
[15]	Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[16]	Mickaël D. Chekroun, Michael Ghil, Honghu Liu, Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133
[17]	Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[18]	Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
[19]	Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128
[20]	Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences