# American Institute of Mathematical Sciences

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

 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.
Citation: J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
##### References:
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##### 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. [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. [3] J. Bergh and J. L$\ddot{O}$fstr$\ddot{O}$m, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften 223, Springer-Verlag Berlin New York, 1976. [4] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated-Compactness and Hardy spaces, J. Math. Pure Appl., 72 (1993), 247-286. [5] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386. [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. [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. [8] J. Huang, Decay estimate for global solutions of 2-D inhomogeneous Navier-Stokes equations, submit. [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. [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. [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. [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. [13] M. Vishik, Hydrodynamics in Besov spaces, Arch. Rat. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128. [14] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $\mathbb{R}^{N}$, J. London Math. Soc., 35 (1987), 303-313. doi: 10.1112/jlms/s2-35.2.303.
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