September  2015, 20(7): 2051-2067. doi: 10.3934/dcdsb.2015.20.2051

Dynamics of stochastic fractional Boussinesq equations

1. 

College of Science, National University of Defense Technology, Changsha, 410073, China, China

2. 

School of Hydropower and Information Engineer, HuaZhong University of Science and Technology, Wuhan, 430074, China

Received  June 2014 Revised  May 2015 Published  July 2015

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.
Citation: Jianhua Huang, Tianlong Shen, Yuhong Li. Dynamics of stochastic fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2051-2067. doi: 10.3934/dcdsb.2015.20.2051
References:
[1]

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

[2]

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

[3]

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.  doi: 10.1007/BF00280598.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[13]

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

[14]

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

[15]

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.  doi: 10.1090/S0002-9939-09-10197-1.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

E. Stein, Singular Integrals and Differentiablity Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1998).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[21]

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

show all references

References:
[1]

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

[2]

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

[3]

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.  doi: 10.1007/BF00280598.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[13]

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

[14]

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

[15]

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.  doi: 10.1090/S0002-9939-09-10197-1.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

E. Stein, Singular Integrals and Differentiablity Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1998).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[21]

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

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