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June  2018, 23(4): 1523-1533. doi: 10.3934/dcdsb.2018056

Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise

Tianlong Shen 1, , Jianhua Huang 1,, and  Caibin Zeng 2,

1. 

College of Science, National University of Defense Technology, Changsha, 410073, China
2. 

School of Mathematics, South China University of Technology, Guangzhou, 510640, China

* Corresponding author: Jianhua Huang

Received  April 2017 Revised  August 2017 Published  June 2018 Early access  February 2018

Fund Project: The authors are supported by NSF of China(11371367,11771449).

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.

Keywords: Caputo-type time fractional drivative, fractional Laplacian operator, Mittag-Leffler functions, mild solution.
Mathematics Subject Classification: Primary: 37L55; Secondary: 60H15.
Citation: Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056
References:
[1]

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.

[2]

P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.

[3]

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.

[4]

G. Da. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.

[5]

A. Heibig and L. I. Lalade, Well-posedness of a linearized fractional derivative fluid model, J. Math. Anal. Appl., 380 (2011), 211-255.  doi: 10.1016/j.jmaa.2011.02.047.

[6]

J. Huang, T. Shen and Y. Li, Dynamics of stochastic fractional Boussinesq equations, Discre. Continu. Dynam. Syst.-B, 20 (2015), 2051-2067.  doi: 10.3934/dcdsb.2015.20.2051.

[7]

J. Lions, Sur l'existence de solution des équation de Navier-Stokes, C. R. Acad. Sci. Pairs, 248 (1959), 2847-2849. 

[8]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperrial College Press, London, 2010.

[9]

M. Shinbrot, Fractional derivatives of solutions of the Navier-stokes equations, Arch. Ration. Mech. Anal., 40 (1971), 139-154. 

[10]

Y. Wang, J. Xu and P. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlin. Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[11]

C. Zeng and Q. Yang, Mild solution of time fractional Navier-Stokes equations driven by fractional Brownian motion, Preprint, 2017. doi: 10.1016/j.spa.2017.03.013.

[12]

Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Compu. Math. Appl., 73 (2017), 1016-1027.  doi: 10.1016/j.camwa.2016.07.007.

[13]

Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Compu. Math. Appl., 73 (2017), 874-891.  doi: 10.1016/j.camwa.2016.03.026.

show all references

References:
[1]

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.

[2]

P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.

[3]

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.

[4]

G. Da. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.

[5]

A. Heibig and L. I. Lalade, Well-posedness of a linearized fractional derivative fluid model, J. Math. Anal. Appl., 380 (2011), 211-255.  doi: 10.1016/j.jmaa.2011.02.047.

[6]

J. Huang, T. Shen and Y. Li, Dynamics of stochastic fractional Boussinesq equations, Discre. Continu. Dynam. Syst.-B, 20 (2015), 2051-2067.  doi: 10.3934/dcdsb.2015.20.2051.

[7]

J. Lions, Sur l'existence de solution des équation de Navier-Stokes, C. R. Acad. Sci. Pairs, 248 (1959), 2847-2849. 

[8]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperrial College Press, London, 2010.

[9]

M. Shinbrot, Fractional derivatives of solutions of the Navier-stokes equations, Arch. Ration. Mech. Anal., 40 (1971), 139-154. 

[10]

Y. Wang, J. Xu and P. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlin. Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[11]

C. Zeng and Q. Yang, Mild solution of time fractional Navier-Stokes equations driven by fractional Brownian motion, Preprint, 2017. doi: 10.1016/j.spa.2017.03.013.

[12]

Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Compu. Math. Appl., 73 (2017), 1016-1027.  doi: 10.1016/j.camwa.2016.07.007.

[13]

Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Compu. Math. Appl., 73 (2017), 874-891.  doi: 10.1016/j.camwa.2016.03.026.

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