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Article

The existence of a random attractor for the three dimensional damped Navier–Stokes equations with additive noise

Pages 691-700 | Received 16 Jan 2017, Accepted 23 Mar 2017, Published online: 02 May 2017

References

  • Arnold, L. 1998. Random Dynamical Systems. Berlin: Springer-Verlag.
  • Arnol'd, V. I. 1983. Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer-Verlag.
  • Ball, J. M. 1997. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Journal of Nonlinear Science 7:475–502.
  • Bresch, D., and Desjardins, B. 2003. Existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasigeostrophic model. Communications in Mathematical Physics 238:211–223.
  • Bresch, D., Desjardins, B., and Lin, C. 2003. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Communications in Partial Differential Equations 28:843–868.
  • Cai, X. J., and Jiu, Q. S. 2008. Weak and strong solutions for the incompressible navier-stokes equations with damping. Journal of Mathematical Analysis and Applications 343:799–809.
  • Caraballo, T., and Kloeden, P. E. 2009. Non-autonomous attractor for integro-differential evolution equations. Discrete and Continuous Dynamical Systems 2(Suppl.):17–36.
  • Caraballo, T., Langa, J. A., and Robinson, J. C. 1998.Upper semicontinuity of attractors for small random perturbations of dynamical systems. Communications in Partial Differential Equations 23:1557–1581.
  • Caraballo, T., Łukasiewicz. G., and Real, J. 2006. Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Analysis 64:484–498.
  • Carvalho, A. N., Langa, J. A., and Robinson, J. C. 2012. Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. New York: Springer.
  • Cheban, D. N. 2004. Global Attractors of Nonautonomous Dissipative Dynamical Systems. Hackensack, NJ: World Scientific.
  • Cheskidov, A., and Foias, C. 2006. On global attractors of the 3d Navier-Stokes equations. Journal of Differential Equations 231:714–754.
  • Cheskidov, A., and Lu, S. S. 2014. Uniform global attractors for the nonautonomous 3d Navier-Stokes equations. Advances in Mathematics 267:277–306.
  • Crauel, H., Debussche, A., and Flandoli, F. 1997. Random attractors. Journal of Dynamics and Differential Equations 9:307–341.
  • Crauel, H., and Flandoli, F. 1994. Attractors for random dynamical systems. Probability Theory and Related Fields 100:365–393.
  • Cutland, N. J., and Keisler, H. J. 2004. Global attractors for 3-dimensional stochastic Navier-Stokes equations. Journal of Dynamics and Differential Equations 16:205–266.
  • Cutland, N. J., and Keisler, H. J. 2005. Attractors and neo-attractors for 3d stochastic Navier-Stokes equations. Stochastics and Dynamics 5:487–533.
  • Dong, B. Q., and Jia, Y. 2016. Stability behaviors of Leray weak solutions to the three-dimensional Navier-Stokes equations. Nonlinear Analysis: Real World Applications 30:41–58.
  • Eckmann, J. P., and Ruelle, D. 19895. Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57:617–656.
  • Flandoli, F., and Schmalfuß, B. 1996. Random attractors for the 3d stochastic navier-stokes equation with multiplicative white noise. Stochastics and Stochastic Reports 59:21–45.
  • Flandoli, F., and Schmalfuß, B. 1999. Weak solutions and attractors for three dimensional Navier-Stokes equations with nonregular force. Journal of Dynamics and Differential Equations 11:355–398.
  • Hsiao, L. 1997. Quasilinear Hyperbolic Systems and Dissipative Mechanisms. Hackensack, NJ: World Scientific.
  • Huang, F., and Pan, R. 2003. Convergence rate for compressible Euler equations with damping and vacuum. Archive for Rational Mechanics and Analysis 166:359–376.
  • Jia, Y., Zhang, X. W., and Dong, B. Q. 2011. The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping. Nonlinear Analysis: Real World Applications 12:1736–1747.
  • Jiang, Z. H., and Zhu, M. X. 2012. The large time behavior of solutions to 3d Navier-Stokes equations with nonlinear damping. Mathematical Methods in the Applied Sciences 35:97–102.
  • Kapustyan, A. V., and Valero, J. 2007. Weak and strong attractors for the 3d Navier-Stokes system. Journal of Differential Equations 240:249–278.
  • Kloeden, P. E., and Langa, J. A. 2007. Flattening, squeezing and the existence of random attractors. Proceedings of the Royal Society A 463:163–181.
  • Kloeden, P. E., and Schmalfuß, B. 1997. Non-autonomous systems, cocycle attractors and variable time-step discretization. Numerical Algorithms 14:141–152.
  • Kloeden, P. E., and Stonier, D. J. 1998. Cocycle attractors in nonautonomously perturbed differential equations. Dynamics of Continuous Discrete and Impulsive Systems 4:211–226.
  • Łukaszewicz, G. 2008. Pullback attractors and statistical solutions for 2-d navier-stokes equations. Discrete and Continuous Dynamical Systems 9:643–659.
  • Łukaszewicz, G., and Robinson, J. C. 2014. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems 34:1–12.
  • Marín-Rubio, P., and Robinson, J. C. 2003. Attractors for the stochastic 3d Navier-Stokes equations. Stochastics and Dynamics 3:279–297.
  • Qian, C. Y. 2016. A remark on the global regularity for the 3d navier-stokes equations. Applied Mathematics Letters 57:126–131.
  • Schmalfuß, B. 1997. The random attractor of the stochastic Lorenz system. Zeitschrift fur angewandte Mathematik und Physik 48:951–975.
  • Sell, G. R. 1996. Global attractors for the three-dimensional Navier-Stokes equations. Journal of Dynamics and Differential Equations 8:1–33.
  • Song, X. L., and Hou, Y. R. 2011. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems 31:239–252.
  • Song, X. L., and Hou, Y. R. 2015. Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping. Journal of Mathematical Analysis and Applications 422:337–351.
  • Temam, R. 1997. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer-Verlag.
  • Zhang, Z. J., Wu, X. L., and Lu, M. 2011. On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping. Journal of Mathematical Analysis and Applications 377:414–419.

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