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November  2014, 34(11): 4647-4669. doi: 10.3934/dcds.2014.34.4647

Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity

J. Huang 1, and  Marius Paicu 2,

1. 

Academy of Mathematics & Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2. 

Université Bordeaux 1, Institut de Mathématiques de Bordeaux, F-33405 Talence Cedex, France

Received  September 2013 Revised  December 2013 Published  May 2014

In this paper, we investigate the time decay behavior to weak solution of 2D incompressible inhomogeneous Navier-Stokes equations. Granted the decay estimates, we gain a global well-posed result of these solutions.
Keywords: decay estimates., Inhomogeneous Navier-Stokes equations.
Mathematics Subject Classification: 35Q30, 76D0.
Citation: J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
References:
[1]

H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3-D inhomogeneous Navier-Stokes equations, Comm. Pure. Appl. Math., 64 (2011), 832-881. doi: 10.1002/cpa.20351.  Google Scholar

[2]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

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B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rat. Mech. Anal., 137 (1997), 135-158. doi: 10.1007/s002050050025.  Google Scholar

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G. Gui and P. Zhang, Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity, Chin. Ann. Math., 30 (2009), 607-630.\vspace*{2pt} doi: 10.1007/s11401-009-0027-3.  Google Scholar

[8]

J. Huang, Decay estimate for global solutions of 2-D inhomogeneous Navier-Stokes equations,, submit., ().   Google Scholar

[9]

J. Huang, M. Paicu and P. Zhang, Global solutions to 2-D incompressible inhomogeneous Navier-Stokes system with general velocity, J. Math. Pures Appl., 100 (2013), 806-831. doi: 10.1016/j.matpur.2013.03.003.  Google Scholar

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O. A. Ladyženskaja and V. A. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. (Russian) Boundary value problems of mathematical physics, and related questions of the theory of functions, 8, Zap. Naužn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 52 (1975), 52-109, 218-219.  Google Scholar

[11]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol.1 of Oxford Lecture Series in Mathematics and its Applications 3. New York, Oxford University Press, 1996.  Google Scholar

[12]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763. doi: 10.1080/03605308608820443.  Google Scholar

[13]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Rat. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128.  Google Scholar

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M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $\mathbbR^n$, J. London Math. Soc., 35 (1987), 303-313. doi: 10.1112/jlms/s2-35.2.303.  Google Scholar

show all references

References:
[1]

H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3-D inhomogeneous Navier-Stokes equations, Comm. Pure. Appl. Math., 64 (2011), 832-881. doi: 10.1002/cpa.20351.  Google Scholar

[2]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[3]

J. Bergh and J. L$\ddoto$fstr$\ddoto$m, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften 223, Springer-Verlag Berlin New York, 1976.  Google Scholar

[4]

R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated-Compactness and Hardy spaces, J. Math. Pure Appl., 72 (1993), 247-286.  Google Scholar

[5]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[6]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rat. Mech. Anal., 137 (1997), 135-158. doi: 10.1007/s002050050025.  Google Scholar

[7]

G. Gui and P. Zhang, Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity, Chin. Ann. Math., 30 (2009), 607-630.\vspace*{2pt} doi: 10.1007/s11401-009-0027-3.  Google Scholar

[8]

J. Huang, Decay estimate for global solutions of 2-D inhomogeneous Navier-Stokes equations,, submit., ().   Google Scholar

[9]

J. Huang, M. Paicu and P. Zhang, Global solutions to 2-D incompressible inhomogeneous Navier-Stokes system with general velocity, J. Math. Pures Appl., 100 (2013), 806-831. doi: 10.1016/j.matpur.2013.03.003.  Google Scholar

[10]

O. A. Ladyženskaja and V. A. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. (Russian) Boundary value problems of mathematical physics, and related questions of the theory of functions, 8, Zap. Naužn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 52 (1975), 52-109, 218-219.  Google Scholar

[11]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol.1 of Oxford Lecture Series in Mathematics and its Applications 3. New York, Oxford University Press, 1996.  Google Scholar

[12]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763. doi: 10.1080/03605308608820443.  Google Scholar

[13]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Rat. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128.  Google Scholar

[14]

M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $\mathbbR^n$, J. London Math. Soc., 35 (1987), 303-313. doi: 10.1112/jlms/s2-35.2.303.  Google Scholar

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