The boundedness of pullback attractors for delayed non-Newtonian fluids

Abstract

We investigated the boundedness of pullback attractors for non-Newtonian fluids with delay in two-dimensional bounded domains. Under suitable assumptions on the delay term, we established the boundedness of the pullback attractor, defined in the phase space L², which was bounded in H⁴ by means of refined energy estimates.

Keywords

boundedness / non-Newtonian fluid / pullback attractor

References

[1] H. Bellout, F. Bloom, Incompressible bipolar and non-Newtonian viscous fluid flow, Springer, New York, 2014. https://doi.org/10.1007/978-3-319-00891-2

[2] G. Liu, Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid, J. Math. Fluid Mech., 19 (2017), 623-643. http://doi.org/10.1007/s00021-016-0299-9

[3] C. Zhao, S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. http://doi.org/10.1016/j.jde.2007.04.001

[4] T. Caraballo, J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. https://doi.org/10.1098/rspa.2001.0807

[5] T. Caraballo, J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. http://doi.org/10.1016/j.jde.2004.04.012

[6] C. J. Niche, G. Planas, Existence and decay of solutions to the dissipative quasi-geostrophic equation with delays, Nonlinear Anal., 75 (2012), 3936-3950. http://doi.org/10.1016/j.na.2012.02.017

[7] X. Song, Y. Xiong, Pullback attractors for 2D MHD equations with delays, J. Math. Phys., 62 (2021), 072704. http://doi.org/10.1063/5.0020351

[8] M. T. Tachim, A Cahn-Hilliard-Navier-Stokes model with delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2663-2685. http://doi.org/10.3934/dcdsb.2016067

[9] P. Zhang, L. Huang, Stability for a 3D Ladyzhenskaya fluid model with unbounded variable delay, Electron. Res. Arch., 31 (2023), 7602-7627. http://doi.org/10.3934/era.2023384

[10] C. Zhao, S. Zhou, Y. Li, Existence and regularity of pullback attractors for an incompressible non- Newtonian fluid with delays, Quart. Appl. Math., 67 (2009), 503-540. http://doi.org/10.1090/S0033-569X-09-01146-2

[11] J. Jeong, J. Park, Pullback attractors for a 2D non-automous incompressible non-Newtonian fluid with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2687-2702. http://doi.org/10.3934/dcdsb.2016068

[12] L. Liu, T. Caraballo, X. Fu, Dynamics of a non-automous incompressible non-Newtonian fluid with delay, Dyn. Partial Differ. Equ., 14 (2017), 375-402. http://doi.org/10.4310/DPDE.2017.v14.n4.a4

[13] G. Liu, H. Xu, C. Zhao, Existence and regularity of pullback attractors for a non-Newtonian fluid with delays in 2D unbounded domains, Evol. Equ. Control Theory, 13 (2024), 382-399. http://doi.org/10.3934/eect.2023050

[14] J. C. Robinson, Infinite-dimensional dynamical system, Cambridge University Press, Cambridge, 2001.

[15] J. Garcia-Luengo, P. Marin-Rubio, J. Rose, H²-boundedness of the pullback attractors for non- autonomous 2D Navier-Stokes equations in bounded domains, Nonlinear Anal., 74 (2011), 4882-4887. https://doi.org/10.1016/j.na.2011.04.063

[16] C. Zhao, W. Sun, C. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ., 12 (2015), 265-288. https://doi.org/10.4310/DPDE.2015.v12.n3.a4

[17] G. Liu, C. Zhao, J. Cao, H⁴-boundedness of pullback attractor for a 2D non-Newtonian fluid flow, Front. Math. China, 8 (2013), 1377-1390. http://doi.org/10.1007/s11464-013-0250-9

[18] C. Zhao, L. Yang, G. Liu, C. Hsu, Global well-posenness and pullback attractor for a delayed non- Newtonian fluid on two dimensional unbounded domains, Acta Math. Appl. Sin., 40 (2017), 287-311.

[19] J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and measure-valued solutions to evolutionary PDE, Champman-Hall, New York, 1996. http://doi.org/10.1201/9780367810771