Long time dynamics for functional three-dimensional Navier-Stokes-Voigt equations

T. Caraballo; A. M. Márquez-Durán; T. Caraballo; A. M. Márquez-Durán

doi:10.3934/math.2020351

AIMS Mathematics

2020, Volume 5, Issue 6: 5470-5494. doi: 10.3934/math.2020351

Previous Article Next Article

Research article

Long time dynamics for functional three-dimensional Navier-Stokes-Voigt equations

T. Caraballo ^{1
,
,},
A. M. Márquez-Durán ²

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, SPAIN
2.
Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1, 41013-Sevilla, SPAIN

Received: 24 April 2020 Accepted: 19 June 2020 Published: 24 June 2020
MSC : 35B40, 35B41, 35Q30, 35Q35, 76F20

In this paper we consider a non-autonomous Navier-Stokes-Voigt model including a variety of delay terms in a unified formulation. Firstly, we prove the existence and uniqueness of solutions by using a Galerkin scheme. Next, we prove the existence and eventual uniqueness of stationary solutions, as well as their exponential stability by using three methods: first, a Lyapunov function which requires differentiability for the delays; next we exploit the Razumikhin technique to weaken the differentiability assumption to just continuity; finally, we use a Gronwall-like type of argument to provide sufficient conditions for the exponential stability in a general case which, in particular, for a situation of variable delay, it only requires measurability of the variable delay function.
- Navier-Stokes-Voigt model,
- delay,
- unified formulation,
- stationary solutions,
- exponential stability,
- Razumikhin
Citation: T. Caraballo, A. M. Márquez-Durán. Long time dynamics for functional three-dimensional Navier-Stokes-Voigt equations[J]. AIMS Mathematics, 2020, 5(6): 5470-5494. doi: 10.3934/math.2020351

Related Papers:

Abstract

In this paper we consider a non-autonomous Navier-Stokes-Voigt model including a variety of delay terms in a unified formulation. Firstly, we prove the existence and uniqueness of solutions by using a Galerkin scheme. Next, we prove the existence and eventual uniqueness of stationary solutions, as well as their exponential stability by using three methods: first, a Lyapunov function which requires differentiability for the delays; next we exploit the Razumikhin technique to weaken the differentiability assumption to just continuity; finally, we use a Gronwall-like type of argument to provide sufficient conditions for the exponential stability in a general case which, in particular, for a situation of variable delay, it only requires measurability of the variable delay function.

References

[1]	L. C. Berselli, L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal-Theor., 75 (2012),117-130. doi: 10.1016/j.na.2011.08.011
[2]	L. C. Berselli, T. Y. Kim, L. G. Rebholz, Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization, Discrete Cont. Dyn-B., 21 (2016),1027-1050. doi: 10.3934/dcdsb.2016.21.1027
[3]	L. C. Berselli, S. Spirito, Suitable weak solutions to the 3D Navier-Stokes equations are constructed with the Voigt approximation, J. Differ. Equations, 262 (2017), 3285-3316. doi: 10.1016/j.jde.2016.11.027
[4]	Y. P. Cao, E. M. Lunasin, E. S. Titi, Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848. doi: 10.4310/CMS.2006.v4.n4.a8
[5]	T. Caraballo, X. Han, A survey on Navier-Stokes models with delays: existence, uniqueness and asymptotic behavior of solutions, Discrete Cont. Dyn-S., 8 (2015), 1079-1101.
[6]	T. Caraballo, X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dynam. Part. Differ. Eq., 11 (2014), 345-359. doi: 10.4310/DPDE.2014.v11.n4.a3
[7]	T. Caraballo, A. M. Márquez-Durán, J. Real, Asymptotic behaviour of the three-dimensionalNavier-Stokes model with delays, J. Math. Anal. Appl., 340 (2008), 410-423. doi: 10.1016/j.jmaa.2007.08.011
[8]	T. Caraballo, K. Liu, A. Truman, Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property, Proc. R. Soc. Lond. A., 456 (2000), 1775-1802.
[9]	T. Caraballo, J. Real, Navier-Stokes equations with delay, Proc. R. Soc. Lond. A., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807
[10]	T. Caraballo, J. Real, Asymptotic behaviour of Navier-Stokes equations with delays, Proc. R. Soc. Lond. A., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807
[11]	T. Caraballo, J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differ. Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012
[12]	H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Math. Sci., 122 (2012), 283-295. doi: 10.1007/s12044-012-0071-x
[13]	P. Constantin, C. Foias, Navier Stokes Equations, University of Chicago Press, 1988.
[14]	M. A. Ebrahimi, M. Holst, E. Lunasin, The Navier-Stokes-Voight model for image inpainting, IMA J. App. Math., 78 (2013), 869-894. doi: 10.1093/imamat/hxr069
[15]	C. G. Gal, T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Method. Appl. Sci., 36 (2013), 2507-2523. doi: 10.1002/mma.2771
[16]	S. M. Guzzo, G. Planas, Existence of solutions for a class of Navier-Stokes equations with infinite delay, Appl. Anal., 94 (2015), 840-855. doi: 10.1080/00036811.2014.905677
[17]	L. Haiyan, Q. Yuming, Pullback attractors for three-dimensional Navier-Stokes-Voigt equations with delays, Bound. value probl., 2013 (2013), 1-17. doi: 10.1186/1687-2770-2013-1
[18]	V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. LOMI, 152 (1986), 50-54.
[19]	V. K. Kalantarov, B. Levant, E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-StokeVoight equations, J. Nonlinear Sci., 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7
[20]	V. K. Kalantarov, E. S. Titi, Global attractors and determining modes for the 3D Navier-StokesVoight equations, Chinese Ann. Math. B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3
[21]	P. Kuberry, A. Larios, L. G. Rebholz, aet al. Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662.
[22]	O. A. Ladyzhenskaya, In memory of A. P. Olskolkov, J. Math. Sci., 99 (2000), 799-801. doi: 10.1007/BF02673588
[23]	W. J. Layton, L. G. Rebholz, On relaxation times in the Navier-Stokes-Voigt model, Int. J. Comput. Fluid D., 27 (2013), 184-187. doi: 10.1080/10618562.2013.766328
[24]	B. Levant, F. Ramos, E. S. Titi, On the statistical properties of the 3D incompressible NavierStokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14
[25]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
[26]	J. M. García Luengo, P. Marín-Rubio, J. Real, Pullback attractors for three-dimensional nonautonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930. doi: 10.1088/0951-7715/25/4/905
[27]	J. M. García Luengo, P. Marín Rubio, J. Real Anguas, Regularity of pullback attractors and attraction in H1 in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Cont. Dyn-A., 34 (2014), 181-201. doi: 10.3934/dcds.2014.34.181
[28]	P. Marín Rubio, A. M. Márquez Durán, J. Real Anguas, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Cont. Dyn-B, 14 (2010), 655-673.
[29]	P. Marín Rubio, A. M. Márquez Durán, J. Real Anguas, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Cont. Dyn-A, 31 (2011), 779-796. doi: 10.3934/dcds.2011.31.779
[30]	P. Marín Rubio, J. Real, J. Valero, Pullback attractors for a two-dimensional Navier-Stokes equations in an infinite delay case, Nonlinear Anal-Theor., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008
[31]	C. J. Niche, Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum, J. Differ. Equations, 260 (2016), 4440-4453. doi: 10.1016/j.jde.2015.11.014
[32]	A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, (Russian) Boundary value problems of mathematical physics and related questions in the theory of functions, 7. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 98-116.
[33]	A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, J. Math. Sci., 28 (1985), 751-758. doi: 10.1007/BF02112340
[34]	D. T. Quyet, Pullback attractors for 2D g-Navier-Stokes equations with infinite delays, Commun, Korean Math. Soc., 31 (2016), 519-532. doi: 10.4134/CKMS.c150186
[35]	F. Ramos, E. S. Titi, Invariant measures for the 3D Navier-Stokes-Voigt equations and their NavierStokes limit, Discrete Cont. Dyn-A., 28 (2010), 375-403. doi: 10.3934/dcds.2010.28.375
[36]	B. S. Razumikhin, On stability of systems with retardation, Prikl. Mat. Meh., 20 (1956), 500-512.
[37]	B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay, Avtomat. Telemeh. 21 (1960), 740-749.
[38]	K. Su, M. Zhao, J. Cao, Pullback attractors of 2D Navier-Stokes-Voigt equations with delay on a non-smooth domain, Bound. Value Probl., 2015 (2015), 1-27. doi: 10.1186/s13661-014-0259-3
[39]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Revised edition, With an appendix by F. Thomasset. Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979.
[40]	G. Yue, C. Zhong, Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations, Discrete Contin. Dyn-B, 16 (2011), 985-1002.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)