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Abstrakty
In this article, we consider a non-autonomous nonlinear bipolar with phase transition in a two-dimensional bounded domain. We assume that the external force is singularly oscillating and depends on a small parameter ε. We prove the existence of the uniform global attractor Aε. Furthermore, using the method of [9] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of Aε as e goes to zero.
Wydawca
Czasopismo
Rocznik
Tom
Strony
123--146
Opis fizyczny
Bibliogr. 35 poz.
Twórcy
autor
- Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, USA
Bibliografia
- [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies Math. Appl. 25, North-Holland, Amsterdam, 1992.
- [2] H. Bellout, F. Bloom and J. Nečas, Weak and measure-valued solutions for non-Newtonian fluids, C. R. Acad. Sci. Paris 317 (1993), 795-800.
- [3] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible viscous fluids, Comm. Partial Differential Equations 19 (1994), 1763-1803.
- [4] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Phys. D32 (1999), 1119-1123.
- [5] F. Bloom, Attractors of non-Newtonian fluids, J. Dynam. Differential Equations 7 (1995), no. 1, 109-140.
- [6] F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal. 44 (2001), 281-309.
- [7] F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence of a maximal compact attractor, Nonlinear Anal. 43 (2001), 743-766.
- [8] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal. 92 (1986), no. 3, 205-245.
- [9] V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity 22 (2009), no. 2, 351-370.
- [10] V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating coefficients, Funct. Differ. Equ. 8 (2001), no. 1-2, 123-140.
- [11] V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, 5b. Math 192 (2001), no. 1-2, 11-47.
- [12] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ. 49, American Mathematical Society, Providence, 2002.
- [13] V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dynam. Differential Equations 19 (2007), no. 3, 655-684.
- [14] V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous Sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst. 12 (2005), no. 1, 27-38.
- [15] S. C. Cowin, The theory of polar fluids, in: Advances in Applied Mechanics. Vol. 14, Academic Press, New York (1974), 279-347.
- [16] Q. Du and M. Gunzberger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl. 155 (1991), 21-45.
- [17] E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci. 20 (2010), no. 7,1129-1160.
- [18] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincare Anal. Non Linéaire, 27 (2010), no. 1, 401-436.
- [19] C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst. 28 (2010), no. 1, 1-39.
- [20] C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B 31 (2010), no. 5, 655-678.
- [21] A. E. Green and R. S. Rivlin, Multipolar continunnm mechanics, Arch. Ration. Mech. Anal. 17 (1964), 113-147.
- [22] A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Arch. Ration. Mech. Anal. 16 (1964), 325-353.
- [23] A. Haraux, Systemes Dynamiques Dissipatifs et Applications, Rech. Math. Appl. 17, Mason, Paris, 1991.
- [24] S. Kaniel, On the initial-value problem for an incompressible fluid with nonlinear viscosity, J. Math. Mech. 19(1970), 681-706.
- [25] O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1965.
- [26] O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in: Bounday Value Problems of Mathematical Physics V, American Mathematical Society, Providence (1970).
- [27] S. Lu, H. Wu, and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 701-719.
- [28] J. Nečas and M. Silhavy, Multipolar Viscous Fluids, Quart. Appl. Math. 49 (1991), 247-265.
- [29] Y. R. Ou and M. Sritharan, Analysis of regularized Navier-Stokes equation. I, Quart. Appl. Math. 49 (1991), 651-685.
- [30] Y. R. Ou and M. Sritharan, Analysis of regularized Navier-Stokes equation. II, Quart. Appl. Math. 49 (1991), 687-728.
- [31] T. Tachim Medjo, Longtime behavior of a nonlinear bipolar fluid model with phase transition, preprint (2013).
- [32] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci. 68, Springer, New York, 1988.
- [33] C. Zhao and Y. Li, h2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal. 56 (2004), 1091-1103.
- [34] C. Zhao and S. Zhou, Pullback trajectory attractors for evolution equations and application to 3D incompressible non-Newtonian fluid, Nonlinearity 21 (2008), 1691-1717.
- [35] C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous ncompressible non-Newtonian fluid, Appl. Math. Comput. 201 (2008), 688-700.
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Bibliografia
Identyfikator YADDA
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