# American Institute of Mathematical Sciences

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
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##### 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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