January  2012, 32(1): 57-79. doi: 10.3934/dcds.2012.32.57

Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation

1. 

School of Mathematical Science, Anhui University, Hefei 230039, China, China

Received  July 2010 Revised  January 2011 Published  September 2011

This paper is concerned with the Cauchy problem of three-dimensional modified Navier-Stokes equations with fractional dissipation $ \nu (-\Delta)^{\alpha} u$. The results are three-fold. We first prove the global existence of weak solutions for $0<\alpha\leq1 $ and global smooth solution for $\frac{3}{4}<\alpha\leq1.$ Second, we obtain the optimal decay rates of both weak solutions and the higher-order derivative of the smooth solution. Finally, we investigate the asymptotic stability of the large solution to the system under large initial and external forcing perturbation.
Citation: Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
References:
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T. Caraballo, J. Real and P. E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations,, Advanced Nonlinear Studies, 6 (2006), 411.   Google Scholar

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T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete and Continuous Dynamical Systems Series B, 10 (2008), 761.   Google Scholar

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B.-Q. Dong and Z.-M. Chen, Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation,, Nonlinearity, 19 (2006), 2919.  doi: 10.1088/0951-7715/19/12/011.  Google Scholar

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B.-Q. Dong and Z.-M. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows,, Discrete and Continuous Dynamical Systems, 23 (2009), 765.   Google Scholar

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B.-Q. Dong and W. Jiang, On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows,, Science in China Series A: Mathematics, 51 (2008), 925.  doi: 10.1007/s11425-007-0196-z.  Google Scholar

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T. Kawanago, Stability estimate for strong solutions of the Navier-Stokes system and its applications,, Electron. J. Differential Equations, 15 (1998), 1.   Google Scholar

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P. E. Kloeden, J. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 6 (2007), 937.  doi: 10.3934/cpaa.2007.6.937.  Google Scholar

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H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations,, Comm. Math. Phys., 214 (2000), 191.  doi: 10.1007/s002200000267.  Google Scholar

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H. Kozono, Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations,, J. Funct. Anal., 176 (2000), 153.  doi: 10.1006/jfan.2000.3625.  Google Scholar

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J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta. Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

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F.-H. Lin, P. Zhang and Z. Zhang, On the global existence of smooth solution to the 2-D FENE dumbbell model,, Comm. Math. Phys., 277 (2008), 531.  doi: 10.1007/s00220-007-0385-1.  Google Scholar

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K. Masuda, Weak solutions of the Navier-Stokes equations,, Tohoku Math. J., 36 (1984), 623.  doi: 10.2748/tmj/1178228767.  Google Scholar

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G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3D Navier-Stokes equations,, Comm. Math. Phys., 159 (1994), 329.  doi: 10.1007/BF02102642.  Google Scholar

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M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch Rational Mech. Anal., 88 (1985), 209.  doi: 10.1007/BF00752111.  Google Scholar

[19]

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

[20]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations in $H^m$ spaces,, Comm. Partial Differential Equations, 20 (1995), 103.   Google Scholar

[21]

M. E. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows,, Discrete and Continuous Dynamical Systems, 13 (2005), 1277.  doi: 10.3934/dcds.2005.13.1277.  Google Scholar

[22]

R. Teman, "The Navier-Stokes Equations,", Studies in Mathematics and its Applications, 2 (1977).   Google Scholar

[23]

Y. Zhou, Asymptotic stability to the 3D Navier-Stokes equations,, Comm. Partial Differential Equations, 30 (2005), 323.   Google Scholar

[24]

L. Zhang, Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations,, Comm. Partial Differential Equations, 20 (1995), 119.   Google Scholar

[25]

L. Zhang, New results of general $n$-dimensional incompressible Navier-Stokes equations,, J. Differential Equations, 245 (2008), 3470.   Google Scholar

show all references

References:
[1]

T. Caraballo, J. Real and P. E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations,, Advanced Nonlinear Studies, 6 (2006), 411.   Google Scholar

[2]

T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete and Continuous Dynamical Systems Series B, 10 (2008), 761.   Google Scholar

[3]

P. Constantin, "Near Identity Transformations for the Navier-Stokes Equations,", Handbook of Mathematical Fluid Dynamics, II (2003), 117.   Google Scholar

[4]

J.-Y. Chemin, "Perfect Incompressible Fluids,", New York: Oxford University Press 1998., (1998).   Google Scholar

[5]

B.-Q. Dong and Z.-M. Chen, Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation,, Nonlinearity, 19 (2006), 2919.  doi: 10.1088/0951-7715/19/12/011.  Google Scholar

[6]

B.-Q. Dong and Z.-M. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows,, Discrete and Continuous Dynamical Systems, 23 (2009), 765.   Google Scholar

[7]

B.-Q. Dong and W. Jiang, On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows,, Science in China Series A: Mathematics, 51 (2008), 925.  doi: 10.1007/s11425-007-0196-z.  Google Scholar

[8]

B.-Q. Dong and Y. Li, Large time behavior to the system of incompressible non-Newtonian fluids in $\R^2$,, J. Math. Anal. Appl., 298 (2004), 667.  doi: 10.1016/j.jmaa.2004.05.032.  Google Scholar

[9]

N. H. Katz and N. Pavlovic, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation,, Geom. Funct. Anal., 12 (2002), 355.  doi: 10.1007/s00039-002-8250-z.  Google Scholar

[10]

T. Kawanago, Stability estimate for strong solutions of the Navier-Stokes system and its applications,, Electron. J. Differential Equations, 15 (1998), 1.   Google Scholar

[11]

P. E. Kloeden, J. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 6 (2007), 937.  doi: 10.3934/cpaa.2007.6.937.  Google Scholar

[12]

H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations,, Comm. Math. Phys., 214 (2000), 191.  doi: 10.1007/s002200000267.  Google Scholar

[13]

H. Kozono, Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations,, J. Funct. Anal., 176 (2000), 153.  doi: 10.1006/jfan.2000.3625.  Google Scholar

[14]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta. Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[15]

F.-H. Lin, P. Zhang and Z. Zhang, On the global existence of smooth solution to the 2-D FENE dumbbell model,, Comm. Math. Phys., 277 (2008), 531.  doi: 10.1007/s00220-007-0385-1.  Google Scholar

[16]

K. Masuda, Weak solutions of the Navier-Stokes equations,, Tohoku Math. J., 36 (1984), 623.  doi: 10.2748/tmj/1178228767.  Google Scholar

[17]

G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3D Navier-Stokes equations,, Comm. Math. Phys., 159 (1994), 329.  doi: 10.1007/BF02102642.  Google Scholar

[18]

M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch Rational Mech. Anal., 88 (1985), 209.  doi: 10.1007/BF00752111.  Google Scholar

[19]

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

[20]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations in $H^m$ spaces,, Comm. Partial Differential Equations, 20 (1995), 103.   Google Scholar

[21]

M. E. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows,, Discrete and Continuous Dynamical Systems, 13 (2005), 1277.  doi: 10.3934/dcds.2005.13.1277.  Google Scholar

[22]

R. Teman, "The Navier-Stokes Equations,", Studies in Mathematics and its Applications, 2 (1977).   Google Scholar

[23]

Y. Zhou, Asymptotic stability to the 3D Navier-Stokes equations,, Comm. Partial Differential Equations, 30 (2005), 323.   Google Scholar

[24]

L. Zhang, Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations,, Comm. Partial Differential Equations, 20 (1995), 119.   Google Scholar

[25]

L. Zhang, New results of general $n$-dimensional incompressible Navier-Stokes equations,, J. Differential Equations, 245 (2008), 3470.   Google Scholar

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