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 and 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), 1077-1099. doi: 10.1239/aap/1013540349.

[2]

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.

[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), 237-256. doi: 10.1007/BF00280598.

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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.

[11]

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.

[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), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.

[13]

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

[14]

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.

[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), 1479-1489. doi: 10.1090/S0002-9939-09-10197-1.

[16]

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

[17]

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.

[18]

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.

[19]

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

[20]

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.

[21]

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.

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), 1077-1099. doi: 10.1239/aap/1013540349.

[2]

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.

[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), 237-256. doi: 10.1007/BF00280598.

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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.

[11]

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.

[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), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.

[13]

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

[14]

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.

[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), 1479-1489. doi: 10.1090/S0002-9939-09-10197-1.

[16]

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

[17]

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.

[18]

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.

[19]

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

[20]

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.

[21]

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.

[1]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[2]

Yuncheng You. Random attractor for stochastic reversible Schnackenberg equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1347-1362. doi: 10.3934/dcdss.2014.7.1347

[3]

Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060

[4]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[5]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[6]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122

[7]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326

[8]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745

[9]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[10]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829

[11]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639

[12]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[13]

Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029

[14]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1

[15]

Dingshi Li, Xiaohu Wang, Junyilang Zhao. Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2751-2776. doi: 10.3934/cpaa.2020120

[16]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727

[17]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[18]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635

[19]

Yuri Kifer. Computations in dynamical systems via random perturbations. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 457-476. doi: 10.3934/dcds.1997.3.457

[20]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (163)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]