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:
[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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[23]

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

[24]

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

[25]

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

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[23]

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

[24]

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

[25]

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

[1]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[2]

Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269

[3]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[4]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361

[5]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761

[6]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779

[7]

Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245

[8]

Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2018312

[9]

Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033

[10]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101

[11]

Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic & Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009

[12]

Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052

[13]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

[14]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997

[15]

P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785

[16]

P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937

[17]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655

[18]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[19]

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

[20]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]