Advanced Search
Article Contents
Article Contents

Dynamics of stochastic fractional Boussinesq equations

Abstract Related Papers Cited by
  • The current paper is devoted to the asymptotic behavior of the stochastic fractional Boussinesq equations (SFBE). The global well-posedness of SFBE is proved, and the existence of a random attractor for the random dynamical system generalized by the SFBE are also provided.
    Mathematics Subject Classification: Primary: 37L55; Secondary: 60H15.


    \begin{equation} \\ \end{equation}
  • [1]

    J. Angulo, M. Ruiz-Medina, V. Anh and W. Grecksch, Fractional diffusion and fractional heat equation, Adv. Appl. Probab., 32 (2000), 1077-1099.doi: 10.1239/aap/1013540349.


    P. Azerad and M. Mellouk, On a Stochastic partial differential equation with non-local diffusion, Potential. Anal., 27 (2007), 183-197.doi: 10.1007/s11118-007-9052-6.


    C. Bardos, P. Penel, U. Frisch and P. Sulem, Modifed dissipativity for a nonlinear evolution equation arising in turbulence, Arch. Ration. Mech. Anal., 71 (1979), 237-256.doi: 10.1007/BF00280598.


    P. Biler, T. Funaki and W. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46.doi: 10.1006/jdeq.1998.3458.


    L. Bo, K. Shi and Y. Wang, On a nonlocal stochastic Kuramoto-Sivashinsky equation with jumps, Stoch. Dyn., 7 (2007), 439-457.doi: 10.1142/S0219493707002104.


    Z. Brzeźniak, L. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, preprint, arXiv:1106.1918, (2011).


    H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn.Diff. Eqs.,9 (1997), 307-341.doi: 10.1007/BF02219225.


    H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.doi: 10.1007/BF01193705.


    J. Debbi and M. Dozzi, On the solution of nonlinear stochastic fractional partial equations in one spatial dimension, Stoch. Proc. Appl., 115 (2005), 1764-1781.doi: 10.1016/j.spa.2005.06.001.


    J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.doi: 10.1016/j.jmaa.2008.03.061.


    T. Kato and G. Ponce, Commutator estimates and Euler and Navier-Stokes equation, Comm. Pure Appl. Math., 41 (1988), 891-907.doi: 10.1002/cpa.3160410704.


    C. Kening, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-DeVries equation, J.Amer. Math. Soc., 4 (1991), 323-347.doi: 10.1090/S0894-0347-1991-1086966-0.


    B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Numerical Solution (in Chinese), Science Press, Beijing, 2011.


    B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.doi: 10.1016/j.jmaa.2009.09.009.


    M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by sapce-time white noise, Proceeding of the American Mathematical Society, 138 (2010), 1479-1489.doi: 10.1090/S0002-9939-09-10197-1.


    G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.doi: 10.1017/CBO9780511666223.


    X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.doi: 10.1016/j.jmaa.2010.06.035.


    X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, 92 (2013), 318-334.doi: 10.1080/00036811.2011.614601.


    E. Stein, Singular Integrals and Differentiablity Properties of Functions, Princeton University Press, Princeton, NJ, 1970.


    R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1998.doi: 10.1007/978-1-4684-0313-8.


    X. Xu, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Analysis, TMA, 72 (2010), 677-681.doi: 10.1016/j.na.2009.07.008.

  • 加载中

Article Metrics

HTML views() PDF downloads(179) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint