Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains

Abstract

This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.

Keywords

Navier–Stokes equations multi-delays Lipschitz-like domain robustness

Get full access to this article

View all access options for this article.

References

J.M.

Ball, Global attractors for damped semiliear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), 31–52. doi:10.3934/dcds.2004.10.31.

Barbu and

S.S.

Sritharan, Navier–Stokes equations with hereditary viscosity, Z. Angew Math. Phys. 54 (2003), 449–461. doi:10.1007/s00033-003-1087-y.

R.M.

Brown,

P.A.

Perry and

Shen, On the dimension of the attractor of the non-homogeneous Navier–Stokes equations in non-smooth domains, Inidian University Math. J. 49(1) (2000), 1–34.

Caraballo and

J.A.

Langa, On the upper semi-continuity of cocycle attractors for non-autonomous and random dynamical systems, Discrete Contin. Dyn. Syst. 10 (2003), 491–513.

Caraballo,

J.A.

Langa and

J.C.

Robinson, Upper semi-continuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations 23 (1998), 1557–1581. doi:10.1080/03605309808821394.

Caraballo,

Łukaszewicz and

Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006), 484–498. doi:10.1016/j.na.2005.03.111.

Caraballo and

Real, Navier–Stokes equations with delays, R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 457 (2001), 2441–2453. doi:10.1098/rspa.2001.0807.

Caraballo and

Real, Asymptotic behavior for two-dimensional Navier–Stokes equations with delays, R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 459 (2003), 3181–3194. doi:10.1098/rspa.2003.1166.

Caraballo and

Real, Attractors for 2D Navier–Stokes models with delays, J. Differential Equations 205 (2004), 271–297. doi:10.1016/j.jde.2004.04.012.

10.

A.N.

Carvalho,

J.A.

Langa and

J.C.

Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Springer, New York–Heidelberg–Dordrecht–London, 2013.

11.

V.V.

Chepyzhov and

M.I.

Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence Rhode Island, 2002.

12.

B.E.

Dahlberg and

C.E.

Kenig, Harmonic Analysis and Partial Differential Equations, 1985/1996.

13.

E.B.

Fabes,

C.E.

Kenig and

G.C.

Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), 769–793. doi:10.1215/S0012-7094-88-05734-1.

14.

Foias,

Manley,

Rosa and

Temam, Navier–Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.

15.

García-Luengo and

Marín-Rubio, Attractors for a double time-delayed 2D Navier–Stokes model, Discrete Contin. Dyn. Syst. 252 (2012), 4333–4356.

16.

García-Luengo and

Marín-Rubio, Attractors for a doubled time-delayed 2D Navier–Stokes model, Discrete Contin. Dyn. Syst. 34(10) (2014), 4085–4105. doi:10.3934/dcds.2014.34.4085.

17.

García-Luengo,

Marín-Rubio and

Real, Pullback attractors for 2D Navier–Stokes equations with delays and their regularity, Adv. Nonlinear Study 13 (2013), 331–357. doi:10.1515/ans-2013-0205.

18.

Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, 2013.

19.

O.A.

Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

20.

Li,

Liu and

Ju, Uniform decay estimates for solutions of a class of retarded integral inequalities, J. Differential Equations 271 (2021), 1–38. doi:10.1016/j.jde.2020.08.017.

21.

Li,

Yang,

Li,

Yan and

Lu, Dynamics and stability of the 3D Brinkman–Forchheimer equation with variable delay (I), Asymptotic Anal. 113 (2019), 167–194. doi:10.3233/ASY-181512.

22.

J.L.

Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.

23.

Liu,

Caraballo and

Marín-Rubio, Stability results for 2D Navier–Stokes equations with unbounded delay, J. Differential Equations 265(11) (2018), 5685–5708. doi:10.1016/j.jde.2018.07.008.

24.

Liu and

Fu, Existence and upper semi-continuity of pullback attractors of a p-Laplacian equation with delay, J. Math. Phys. 58 (2017), 082702. doi:10.1063/1.5000076.

25.

Marín-Rubio,

A.M.

Márquez-Durán and

Real, Pullback attractors for globally modified Navier–Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. 31 (2011), 779–796. doi:10.3934/dcds.2011.31.779.

26.

Marín-Rubio and

Real, Pullback attractors for 2D-Navier–Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst. 26 (2010), 989–1006. doi:10.3934/dcds.2010.26.989.

27.

Miranville and

Wang, Upper bounded on the dimension of the attractor for nonhomogeneous Navier–Stokes equations, Discrete Contin. Dyn. Syst. 2 (1996), 95–110. doi:10.3934/dcds.1996.2.95.

28.

Miranville and

Wang, Attractors for non-autonomous nonhomogenerous Navier–Stokes equations, Nonlinearity 10(5) (1997), 1047–1061. doi:10.1088/0951-7715/10/5/003.

29.

Moise,

Rosa and

Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity 11 (1998), 1369–1393. doi:10.1088/0951-7715/11/5/012.

30.

Planas and

Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier–Stokes equations, Discrete Contin. Dyn. Syst. 21 (2008), 1245–1258. doi:10.3934/dcds.2008.21.1245.

31.

J.C.

Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.

32.

Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc. 123 (1995), 801–811. doi:10.1090/S0002-9939-1995-1223521-9.

33.

Taniguchi, The exponential behavior of Navier–Stokes equations with time delay external force, Discrete Contin. Dyn. Syst. 12(5) (2005), 997–1018. doi:10.3934/dcds.2005.12.997.

34.

Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn, Springer, Berlin, 1997.

35.

Temam, Navier–Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition.

36.

Wang and

Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst. 33(7) (2013), 3189–3209. doi:10.3934/dcds.2013.33.3189.

37.

X.-G.

Yang,

Li,

Yan and

Ding, The structure and stability of pullback attractors for 3D Brinkman–Forchheimer equation with delay, Electronic Research Archive 28(4) (2020), 1395–1418. doi:10.3934/era.2020074.

38.

X.-G.

Yang,

Qin,

Lu and

T.F.

Ma, Dynamics of 2D incompressible non-autonomous Navier–Stokes equations on Lipschitz-like domains, Appl. Math. Optim. 83 (2021), 2129–2183. doi:10.1007/s00245-019-09622-w.

39.

X.-G.

Yang,

Wang,

Yan and

Miranville, Dynamics of the 2D Navier–Stokes equations with sublinear operators in Lipschitz-like domains, Discrete Contin. Dyn. Syst. 41(7) (2021), 3343–3366. doi:10.3934/dcds.2020408.

40.

M.C.

Zelati and

C.G.

Gal, Singular limits of Voigt models in fluid dynamics, J. Math. Fluid Mech. 17 (2015), 233–259. doi:10.1007/s00021-015-0201-1.