Journal of Applied Mathematics

The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations

Wen-Juan Wang and Yan Jia

Full-text: Open access

Abstract

We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solution u of the Navier-Stokes equations lies in the regular class uLp(0,;Bq,0(3)), (2α/p)+(3/q)=2α, 2<q<, 0<α<1, then every weak solution v(x,t) of the perturbed system converges asymptotically to u(x,t) as vt-utL20, t.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 321427, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808195

Digital Object Identifier
doi:10.1155/2013/321427

Mathematical Reviews number (MathSciNet)
MR3127446

Zentralblatt MATH identifier
06950617

Citation

Wang, Wen-Juan; Jia, Yan. The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations. J. Appl. Math. 2013 (2013), Article ID 321427, 6 pages. doi:10.1155/2013/321427. https://projecteuclid.org/euclid.jam/1394808195


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References

  • R. Teman, The Navier-Stokes Equations, North-Holland, Amsterdam, The Netherlands, 1977.
  • L. Caffarelli, R. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 35, no. 6, pp. 771–831, 1982.
  • B.-Q. Dong and Z.-M. Chen, “Regularity criteria of weak solutions to the three-dimensional micropolar flows,” Journal of Mathematical Physics, vol. 50, no. 10, p. 103525, 13, 2009.
  • B.-Q. Dong and Z. Zhang, “Global regularity of the 2D micropolar fluid flows with zero angular viscosity,” Journal of Differential Equations, vol. 249, no. 1, pp. 200–213, 2010.
  • P. Constantin and J. Wu, “Behavior of solutions of 2D quasi-geostrophic equations,” SIAM Journal on Mathematical Analysis, vol. 30, no. 5, pp. 937–948, 1999.
  • B.-Q. Dong and J. Song, “Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation,” Discrete and Continuous Dynamical Systems, vol. 32, no. 1, pp. 57–79, 2012.
  • Z. Q. Luo, “Numerical solution of potential flow equations with a predictor-corrector finite difference method,” Journal of Zhejiang University: Science C, vol. 13, no. 5, pp. 393–402, 2012.
  • Y. Jia, X. Zhang, and B.-Q. Dong, “The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping,” Nonlinear Analysis, vol. 12, no. 3, pp. 1736–1747, 2011.
  • J. Wu, “Generalized MHD equations,” Journal of Differential Equations, vol. 195, no. 2, pp. 284–312, 2003.
  • Z. Q. Luo and Z. M. Chen, “Numerical simulation of standing wave with 3D Predictor-Corrector finite difference method for potential flow equations,” Applied Mathematics and Mechanic, vol. 35, no. 8, pp. 931–944, 2013.
  • J. Wu, “The generalized incompressible Navier-Stokes equations in Besov spaces,” Dynamics of Partial Differential Equations, vol. 1, no. 4, pp. 381–400, 2004.
  • J. Fan and T. Ozawa, “Asymptotic stability for the Navier-Stokes equations,” Journal of Evolution Equations, vol. 8, no. 2, pp. 379–389, 2008.
  • H. Kozono, “Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations,” Journal of Functional Analysis, vol. 176, no. 2, pp. 153–197, 2000.
  • B.-Q. Dong and Z.-M. Chen, “Asymptotic stability of non-Newtonian flows with large perturbation in ${R}^{2}$,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 243–250, 2006.
  • B.-Q. Dong and Z.-M. Chen, “Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation,” Nonlinearity, vol. 19, no. 12, pp. 2919–2928, 2006.
  • Z.-Q. Luo and Z.-M. Chen, “Sloshing simulation of standing wave with time-independent finite difference method for Euler equations,” Applied Mathematics and Mechanics, vol. 32, no. 11, pp. 1475–1488, 2011.
  • Y. Zhou, “Asymptotic stability for the 3D Navier-Stokes equations,” Communications in Partial Differential Equations, vol. 30, no. 1–3, pp. 323–333, 2005.
  • J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, New York, NY, USA, 1998.
  • J.-M. Bony, “Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires,” Annales Scientifiques de l'École Normale Supérieure, vol. 14, no. 2, pp. 209–246, 1981.
  • B.-Q. Dong and Z.-M. Chen, “On the weak-strong uniqueness of the dissipative surface quasi-geostrophic equation,” Nonlinearity, vol. 25, no. 5, pp. 1513–1524, 2012.
  • M. E. Schonbek, “${L}^{2}$ decay for weak solutions of the Navier-Stokes equations,” Archive for Rational Mechanics and Analysis, vol. 88, no. 3, pp. 209–222, 1985.
  • B. Dong and Y. Li, “Large time behavior to the system of incompressible non-Newtonian fluids in ${\mathbb{R}}^{2}$,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 667–676, 2004.