MICROPOLAR FLUID FLOWS WITH DELAY ON 2D UNBOUNDED DOMAINS

Wenlong Sun

doi:10.11948/2018.356

2018 Volume 8 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(1): 356-378

Next Article Previous Article

Wenlong Sun. MICROPOLAR FLUID FLOWS WITH DELAY ON 2D UNBOUNDED DOMAINS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 356-378. doi: 10.11948/2018.356

Citation:

Wenlong Sun. MICROPOLAR FLUID FLOWS WITH DELAY ON 2D UNBOUNDED DOMAINS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 356-378. doi: 10.11948/2018.356

MICROPOLAR FLUID FLOWS WITH DELAY ON 2D UNBOUNDED DOMAINS

Wenlong Sun

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Fund Project:

Abstract

Abstract

In this paper, we investigate the incompressible micropolar fluid flows on 2D unbounded domains with external force containing some hereditary characteristics. Since Sobolev embeddings are not compact on unbounded domains, first, we investigate the existence and uniqueness of the stationary solution, and further verify its exponential stability under appropriate conditions { essentially the viscosity δ₁:=min{ν,c_a + c_d} is asked to be large enough. Then, we establish the global well-posedness of the weak solutions via the Galerkin method combined with the technique of truncation functions and decomposition of spatial domain.
- Micropolar fluid flow /
- truncation function /
- well-posedness /
- exponential stability
MSC: 35B40;35Q35;35A01;35A02

FullText(HTML)

References

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar
[2]	J. L. Boldrini, M. Duran and M. A. Rojas-Medar, Existence and uniqueness of strong solution for the incompressible micropolar fluid equations in domains of R³, Annali Delluniversità Di Ferrara Sezione Scienze Matematiche, 2010, 56, 37-51. Google Scholar
[3]	T. Caraballo and J. Real, Navier-Stokes equations with delays, Proc. R. Soc. Lond., 2001, 457, 2441-2453. Google Scholar
[4]	T. Caraballo and J. Real, Asymptotic behaviour of 2D-Navier-Stokes equations with delays, Proc. R. Soc. Lond., 2003, 459, 3181-3194. Google Scholar
[5]	T. Caraballo and J. Real, Attrators for 2D-Navier-Stokes models with delays, J. Differential Equations, 2004, 205, 271-297. Google Scholar
[6]	T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and nonautonomous attractors for differential equations with delays, J. Differential Equations, 2005, 208, 9-41. Google Scholar
[7]	J. Chen, Z. Chen and B. Dong, Existence of H2-global attractors of twodimensional micropolar fluid flows, J. Math. Anal. Appl., 2006, 322, 512-522. Google Scholar
[8]	J. Chen, B. Dong and Z. Chen, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 2007, 20, 1619-1635. Google Scholar
[9]	G. Chen, Pullback attractor for non-homogeneous micropolar fluid flows in nonsmooth domains, Nonlinear Anal., 2009, 10, 3018-3027. Google Scholar
[10]	B. Dong and Z. Chen, Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comp., 2006, 182, 610-620. Google Scholar
[11]	B. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 2010, 249, 200-213. Google Scholar
[12]	A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 1966, 16, 1-18. Google Scholar
[13]	G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, International Journal of Engineering Science, 1977, 15, 105-108. Google Scholar
[14]	M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Analysis, 2006, 64, 1100-1118. Google Scholar
[15]	G. Lukaszewicz, Micropolar fluids:Theory and Applications, Birkhäuser, Boston, 1999. Google Scholar
[16]	G. Lukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Modelling, 2001, 34, 487-509. Google Scholar
[17]	G. Lukaszewicz and A. Tarasińska, On H¹-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Anal., 2009, 71, 782-788. Google Scholar
[18]	A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay:analytical design and numerical simulation, IEEE Trans. Automat Control, 1984, 29, 1058-1068. Google Scholar
[19]	P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 2007, 67, 2784-2799. Google Scholar
[20]	P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a twodimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 2011, 74, 2012-2030. Google Scholar
[21]	B. Nowakowski, Long-time behavior of micropolar fluid equations in cylindrical domains, Nonlinear Anal.-RWA, 2013, 14, 2166-2179. Google Scholar
[22]	R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, Society for Industrial and Applied Mathematics, 1995. Google Scholar
[23]	T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst.-A, 2005, 12, 997-1018. Google Scholar
[24]	Y. Wang and P. E. Kloeden, Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays, Nonlinear Anal., 2013, 90, 86-95. Google Scholar
[25]	N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Mathematical Methods in the Applied Sciences, 2005, 28, 1507-1526. Google Scholar
[26]	C. Zhao, S. Zhou and X. Lian, H¹-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains, Nonlinear Anal.-RWA, 2008, 9, 608-627. Google Scholar
[27]	C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous nonNewtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 2012, 53, 122702-1-23. Google Scholar
[28]	Z. Zhang, Global Regularity for the 2D Micropolar Fluid Flows with Mixed Partial Dissipation and Angular Viscosity, Abstract and Applied Analysis, 2014, 2014, 1-6. Google Scholar
[29]	C. Zhao, W. Sun and C. Hsu, Pullback dynamical behaviors of the nonautonomous micropolar fluid flows, Dynamics of PDE, 2015, 12, 265-288. Google Scholar