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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 - A, 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.  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, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

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[5]

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

[6]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids,, Arch. Rat. Mech. Anal., 137 (1997), 135.  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.  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.  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, 52 (1975), 52.   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, (1996).   Google Scholar

[12]

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

[13]

M. Vishik, Hydrodynamics in Besov spaces,, Arch. Rat. Mech. Anal., 145 (1998), 197.  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.  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.  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, (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, (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.   Google Scholar

[5]

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

[6]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids,, Arch. Rat. Mech. Anal., 137 (1997), 135.  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.  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.  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, 52 (1975), 52.   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, (1996).   Google Scholar

[12]

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

[13]

M. Vishik, Hydrodynamics in Besov spaces,, Arch. Rat. Mech. Anal., 145 (1998), 197.  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.  doi: 10.1112/jlms/s2-35.2.303.  Google Scholar

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