Ladyzhenskaya流体力学方程组的确定模与确定结点个数估计

Abstract

Abstract: This article estimates the finite number of determining modes and determining nodes for the fluid flow of Ladyzhenskaya model on two-dimensional bounded smooth domains. The finite number of determining modes implies that the solutions of the addressed fluids are determined completely by their first finite number of Fourier modes. The determining nodes reveals that whenever two different solutions of the fluid have the same asymptotic behavior at finite number of points in the physical space, then they also possess the same asymptotic behavior at almost everywhere points of the physical space.

Key words: Ladyzhenskaya model, Determining modes, Determining nodes, Asymptotic behavior

CLC Number:

O212.62

Zhang Mingshu, Zhu Zheqi, Zhao Caidi. Determining Modes and Determining Nodes to the Fluid Flow of Ladyzhenskaya Model[J].Acta mathematica scientia,Series A, 2018, 38(1): 71-82.

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References

[1] Adams R A. Sobolev Spaces. New York:Academic Press, 1975
[2] Bae H. Existence regularity and decay rate of solutions of non-Newtonian flow. J Math Anal Appl, 1999, 231:467-491
[3] Bae H, Choe H. Existence and regularity of solutions of non-Newtonian flow. Quart Appl Math, 2000, 58:101-110
[4] Bloom F, Hao W. Regularization of a non-Newtonian system in an unbounded channel:Existence and uniqueness of solutions. Nonlinear Anal, 2001, 44:281-309
[5] Bellout H, Bloom F, Nečas J. Young measure value solutions for non-Newtonian incompressible fluids. Comm Partial Differential Equations, 1994, 19:1768-1803
[6] Bohme G. Non-Newtonian Fluid Mechanics. Amsterdam:North-Holland, 1987
[7] Boyer F, Fabrie P. Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. New York:Springer, 2013
[8] Cao C, Rammaha M A, Titi E S. The Navier-Stokes equations on the rotating 2-D sphere:Gevrey regularity and asymptotic degrees of freedom. Z Angew Math Phys, 1999, 50:341-360
[9] Cockburn B, Jones D A, Titi E S. Eestimating the number of asymptotic degrees of freedom for nonlinear dissipative systems. Math Comp, 1997, 219:1073-1087
[10] Constantin P, Foias C, Manley O, Temam R. Determining modes and fractal dimension of turbulent flows. J Fluid Mech, 1985, 150:427-440
[11] Constantin P, Foias C, Manley O, Temam R. On the dimension of the attractor in two-dimensional turbulence. Phys D, 1988, 30:427-40
[12] Catania D. Global attractor and determining modes for a hyperbolic MHD turbulence model. J Turbulence, 2011, 40:1-20
[13] Dong B, Li Y. Large time be havior to system of incompressible. J Math Anal Appl, 2004, 209:667-676
[14] Foias C, Temam R. Determination of the solutions of the Navier-Stokes equations by a set of nodal values. Math Comp, 1984, 43:117-133
[15] Foias C, Manley O, Rosa R, Temam R. Navier-Stokes Equations and Turbulence. Cambridge:Cambridge University Press, 2004
[16] Foias C, Jolly M, Kravchenko R, Titi E S. A determining form for the two-dimensional Navier-Stokes equations:The Fourier modes case. J Math Phys, 2012, 53:1-30
[17] Flandolf F, Langa J A, Determining modes for dissipative random dynamical systems. Stochastics:An Inter J Prob Stoch Processes, 1999, 66:1-25
[18] Foias C, Manley O, Temam R, Tréve Y M. Asymptotic analysis of the of the Navier-Stokes equations. Phys D, 1983, 9:157-188
[19] Foias C, Kukavica I. Determining nodes for the Kuramoto-Sivashinsky equation. Dynamics Differential Equations, 1995, 7:365-373
[20] Galdi G P. Determining modes, nodes and volume elements for stationary solutions of the Navier-Stokes problem past a three-dimensional body. Arch Rational Mech Anal, 2006, 180:97-126
[21] Hale J K, Raugel G. Regularity, determining modes and Galerkin methods. J Math Pures Appl, 2003, 82:1075-1136
[22] Hoff D, Ziane M. The global attractor and finite determining nodes for the Navier-Stokes equations of compressible flow with singular initial data. Indiana University Math J, 200349:843-890
[23] Jones D A, Titi E S. On the number of determining nodes for the 2D Navier-Stokes equations. J Math Anal Appl, 1992, 168:72-88
[24] Jones D A, Titi E S. Determining finite volume elements for the 2D Navier-Stokes equations. Physica D, 1992, 60:165-174
[25] Jones D A, Titi E S. Upper bounds on the number of determining modes, and volum elements for the Navier-Stokes equations. Indiana Univ Math J, 1993, 42:875-887
[26] Jollya M S, Sadigov T, Titi E S. A determining form for the damped driven nonlinear Schrödinger equation-Fourier modes case. J Differential Equations, 2015, 258:2711-2744
[27] Ju N. Estimates of asymptotic degrees of freedom for solutions to the Navier-Stokes equations. Nonlinearity, 2000, 13:777-789
[28] Kalantarov V K, Titi E S. Global attractors and determining modes for the 3D Navier-Stokes-Voight equations. Chinese Annals Math-B, 2009, 30:697-714
[29] 郭柏灵, 林国广, 尚亚东. 非牛顿流动力系统. 北京:国防工业出版社, 2006Guo B L, Lin G G, Shang Y D. Non-Newtonian Fluid Dynamic System. Beijing:National Defence Industry Press, 2006
[30] Ladyzhenskaya O. The Mathematical Theory of Viscous Incompressible Fluids. New York:Gordon and Breach, 1969
[31] Ladyzhenskaya O. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problem for them. Trudy Steklov's Math Inst, 1967, 102:85-104
[32] Nečasová Š, Penel P. L² decay for weak solution to equation of non-Newtonian incompressible fluids in the whole space. Nonlinear Anal, 2001, 47:4181-4191
[33] Olson E, Titi E S. Determining modes for continuous data assimilation in 2D turbulence. J Stat Phys, 2003, 113:799-840
[34] Olson E, Titi E S. Determining modes and Grashof number in 2D turbulence:a numerical case study. Theoretical Comp Fluid Dyna, 2008, 22:327-339
[35] Paicu M, Raugel G, Rekalo A. Regularity and finite-dimensional behaviour of the global attractor of the second grade fluids equations. J Differential Equations, 2012, 252:3695-3751
[36] Pokorný M. Cauchy problem for non-Newtonian incompressible fluids. Appl Math, 1996, 41:169-201
[37] Szopa P. Determining modes for 2-D micropolar fluid flows. Math Comp Modelling, 2005, 42:1079-1088
[38] Zhao C, Zhou S. Pullback attractors for nonautonomous incompressible non-Newtonian fuild. J Differential Equations, 2007, 238:394-425
[39] Zhao C, Li Y, Zhou S. Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fuild. J Differential Equations, 2009, 247:2331-2363
[40] Zhao C, Jia X, Yang X. Uniform attractor non-autonomous incompressible non-Newtonian fluid with a new class of external forces. Acta Math Sci, 2011, 31:1803-1812

Determining Modes and Determining Nodes to the Fluid Flow of Ladyzhenskaya Model

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