×
Luo, Jianxin

Upper semicontinuity of attractors for a non-Newtonian fluid under small random perturbations. (English) Zbl 1474.35536

Abstr. Appl. Anal. 2014, Article ID 546898, 7 p. (2014).
Summary: This paper investigates the limiting behavior of attractors for a two-dimensional incompressible non-Newtonian fluid under small random perturbations. Under certain conditions, the upper semicontinuity of the attractors for diminishing perturbations is shown.

MSC:

35Q35 PDEs in connection with fluid mechanics
35R60 PDEs with randomness, stochastic partial differential equations
35B41 Attractors
76A05 Non-Newtonian fluids
PDF BibTeX XML Cite
\textit{J. Luo}, Abstr. Appl. Anal. 2014, Article ID 546898, 7 p. (2014; Zbl 1474.35536)
Full Text: DOI

References:

[1] Málek, J.; Nečas, J.; Rokyta, M.; Růžička, M., Weak and Measure-Valued Solutions to Evolutionary PDE, 13 (1996), New York, NY, USA: Chapman & Hall, New York, NY, USA · Zbl 0851.35002
[2] Ladyzhenskaya, O., New equations for the description of the viscous incompressible fluids and solvability in large of the boundary value problems for them, Boundary Value Problems of Mathematical Physics (1970), Providence, RI, USA: American Mathematical Society, Providence, RI, USA
[3] Bellout, H.; Bloom, F.; Nečas, J., Young measure-valued solutions for non-Newtonian incompressible fluids, Communications in Partial Differential Equations, 19, 11-12, 1763-1803 (1994) · Zbl 0840.35079
[4] Bloom, F.; Hao, W., Regularization of a non-Newtonian system in an unbounded channel: existence and uniqueness of solutions, Nonlinear Analysis: Theory, Methods & Applications, 44, 281-309 (2001) · Zbl 0993.76004
[5] Bloom, F.; Hao, W., Regularization of a non-Newtonian system in an unbounded channel: existence of a maximal compact attractor, Nonlinear Analysis: Theory, Methods & Applications, 43, 6, 743-766 (2001) · Zbl 0989.76003
[6] Guo, C.; Guo, B.; Han, Y., Random attractors of stochastic non-Newtonian fluids, Acta Mathematicae Applicatae Sinica, 28, 1, 165-180 (2012) · Zbl 1355.76006
[7] Pokorný, M., Cauchy problem for the non-Newtonian viscous incompressible fluid, Applications of Mathematics, 41, 3, 169-201 (1996) · Zbl 0863.76003
[8] Zhao, C.; Li, Y., \(L^2\)-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Analysis: Theory, Methods & Applications, 56, 7, 1091-1103 (2004) · Zbl 1073.35044
[9] Zhao, C.; Zhou, S., \(L^2\)-compact uniform attractors for a nonautonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space, Journal of Mathematical Physics, 48, 3 (2007) · Zbl 1137.37332
[10] Zhao, C.; Duan, J., Random attractor for the Ladyzhenskaya model with additive noise, Journal of Mathematical Analysis and Applications, 362, 1, 241-251 (2010) · Zbl 1185.35347
[11] Chepyzhov, V. V.; Vishik, M. I., Attractors for Equations of Mathematical Physics. Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49 (2002), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0986.35001
[12] Hale, J. K., Asymptotic behavior of dissipative systems. Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs (1988), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0642.58013
[13] Robinson, J. C., Infinite-dimensional dynamical systems (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1026.37500
[14] Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, 68 (1997), Berlin, Germany: Springer, Berlin, Germany · Zbl 0871.35001
[15] Arnold, L., Random Dynamical Systems (1998), Springer: Berlin, Germany, Springer
[16] da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions. Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications (1992), Cambridge University Press, Cambridge · Zbl 1140.60034
[17] Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probability Theory and Related Fields, 100, 3, 365-393 (1994) · Zbl 0819.58023
[18] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, Journal of Dynamics and Differential Equations, 9, 2, 307-341 (1997) · Zbl 0884.58064
[19] Flandoli, F.; Schmalfuss, B., Random attractors for the \(H^2\) D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics and Stochastics Reports, 59, 1-2, 21-45 (1996) · Zbl 0870.60057
[20] Caraballo, T.; Langa, J. A.; Valero, J., Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise, Journal of Mathematical Analysis and Applications, 260, 2, 602-622 (2001) · Zbl 0987.60074
[21] Caraballo, T.; Langa, J. A.; Melnik, V. S.; Valero, J., Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11, 2, 153-201 (2003) · Zbl 1018.37048
[22] Caraballo, T.; Langa, J. A.; Valero, J., Global attractors for multivalued random dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 48, 6, 805-829 (2002) · Zbl 1004.37035
[23] Caraballo, T.; Lu, K., Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3, 3, 317-335 (2008) · Zbl 1155.60324
[24] Caraballo, T.; Garrido-Atienza, M. J.; Schmalfuss, B.; Valero, J., Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete and Continuous Dynamical Systems A, 21, 2, 415-443 (2008) · Zbl 1155.60025
[25] Caraballo, T.; Langa, J. A.; Robinson, J. C., Upper semicontinuity of attractors for small random perturbations of dynamical systems, Communications in Partial Differential Equations, 23, 9-10, 1557-1581 (1998) · Zbl 0917.35169
[26] Caraballo, T.; Langa, J. A.; Robinson, J. C., Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete and Continuous Dynamical Systems, 6, 4, 875-892 (2000) · Zbl 1011.37031
[27] Caraballo, T.; Kloeden, P. E.; Schmalfuß, B., Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics and Optimization, 50, 3, 183-207 (2004) · Zbl 1066.60058
[28] Robinson, J. C., Stability of random attractors under perturbation and approximation, Journal of Differential Equations, 186, 2, 652-669 (2002) · Zbl 1020.37033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.