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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

 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  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.

Citation: Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056
##### References:
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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] G. Da. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [7] J. Lions, Sur l'existence de solution des équation de Navier-Stokes, C. R. Acad. Sci. Pairs, 248 (1959), 2847-2849. Google Scholar [8] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperrial College Press, London, 2010. Google Scholar [9] M. Shinbrot, Fractional derivatives of solutions of the Navier-stokes equations, Arch. Ration. Mech. Anal., 40 (1971), 139-154. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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