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## On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, , Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012 - Sevilla, Spain 2 Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam 3 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 4 Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam

* Corresponding author: Tran Ngoc Thach (tranngocthach@tdtu.edu.vn)

Received  May 2020 Revised  June 2020 Published  October 2020

In this paper, we study two stochastic problems for time-fractional Rayleigh-Stokes equation including the initial value problem and the terminal value problem. Here, two problems are perturbed by Wiener process, the fractional derivative are taken in the sense of Riemann-Liouville, the source function and the time-spatial noise are nonlinear and satisfy the globally Lipschitz conditions. We attempt to give some existence results and regularity properties for the mild solution of each problem.

Citation: Tomás Caraballo, Tran Bao Ngoc, Tran Ngoc Thach, Nguyen Huy Tuan. On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020289
##### References:
 [1] E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.  Google Scholar [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar [3] L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains., J. Math. Fluid Mech., 18 (2016), 25-69.  doi: 10.1007/s00021-015-0234-5.  Google Scholar [4] M. Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications., Numer. Methods Partial Differential Equations, 22 (2006), 220-257.  doi: 10.1002/num.20071.  Google Scholar [5] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fract., 32 (2007), 661-675.  doi: 10.1016/j.chaos.2005.11.010.  Google Scholar [6] M. Dehghan and M. Abbaszadeh, A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Eng Comput., 33 (2017), 587-605.   Google Scholar [7] C. Fetecau, M. Jamil, C. Fetecau and D. Vieru, The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys., 60 (2009), 921-933.  doi: 10.1007/s00033-008-8055-5.  Google Scholar [8] G. Hu, Y. Lou and P. D. Christofides, Dynamic output feedback covariance control of stochastic dissipative partial differential equations, Chem. Eng. Sci., 63 (2008), 4531-4542.   Google Scholar [9] Y. Jiang, T. Wei and X. Zhou, Stochastic generalized Burgers equations driven by fractional noises, J. Differ. Equ., 252 (2012), 1934-1961.  doi: 10.1016/j.jde.2011.07.032.  Google Scholar [10] M. Khan, The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl., 10 (2009), 3190-3195.  doi: 10.1016/j.nonrwa.2008.10.002.  Google Scholar [11] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4.  Google Scholar [12] M. Lakestani and M. Dehghan, The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math., 235 (2010), 669-678.  doi: 10.1016/j.cam.2010.06.020.  Google Scholar [13] P. D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.  Google Scholar [14] F. Li, Y. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar [15] F. Li, Y. Li and R. Wang, Limiting dynamics for stochastic reaction-diffusion equations on the Sobolev space with thin domains, Comput. Math. Appl., 79 (2020), 457-475.  doi: 10.1016/j.camwa.2019.07.009.  Google Scholar [16] Y. Li and Y. Wang., The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay., J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.  Google Scholar [17] J. Liang, X. Qian, T. Shen and S. Song, Analysis of time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by multiplicative white noise, J. Math. Anal. Appl., 466 (2018), 1525-1544.  doi: 10.1016/j.jmaa.2018.06.066.  Google Scholar [18] T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Z. Yong, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Meth. Appl. Sci., (2020), 1–27. Google Scholar [19] H. L. Nguyen, H. T. Nguyen, K. Mokhtar and X. T. Duong Dang, Identifying initial condition of the Rayleigh-Stokes problem with random noise, Math. Meth. Appl. Sci., 42 (2019), 1561-1571.  doi: 10.1002/mma.5455.  Google Scholar [20] H. L. Nguyen, H. T. Nguyen and Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci., 42 (2019), 3481-3495.  doi: 10.1002/mma.5593.  Google Scholar [21] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002, 23 pp. doi: 10.1088/1361-6420/ab532c.  Google Scholar [22] J.-C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.  Google Scholar [23] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar [24] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.  Google Scholar [25] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar [26] F. Shen, W. Tan, Y. Zhao and T. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model., Nonlinear Anal. Real World Appl., 7 (2006), 1072-1080.  doi: 10.1016/j.nonrwa.2005.09.007.  Google Scholar [27] X. Su and M. Li, The regularity of fractional stochastic evolution equations in Hilbert space, Stoch. Anal. Appl., 36 (2018), 639-653.  doi: 10.1080/07362994.2018.1436973.  Google Scholar [28] N. H. Tuan, Y. Zhou, T. N. Thach and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data., Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104873, 18 pp. doi: 10.1016/j.cnsns.2019.104873.  Google Scholar [29] R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.  Google Scholar [30] R. Wang, L. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\Bbb R^N$, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.  Google Scholar [31] C. Xue and J. Nie, Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space, App. Math. Model, 33 (2009), 524-531.  doi: 10.1016/j.apm.2007.11.015.  Google Scholar [32] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar [33] M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 75 (2018), 2243-2258.  doi: 10.1016/j.camwa.2017.12.004.  Google Scholar [34] C. Zhao and C. Yang, Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels, Appl. Math. Comput., 211 (2009), 502-509.  doi: 10.1016/j.amc.2009.01.068.  Google Scholar [35] G. Zou, G. Lv and J.-L. Wu, Stochastic Navier–Stokes equations with Caputo derivative driven by fractional noises, J. Math. Anal. Appl., 461 (2018), 595-609.  doi: 10.1016/j.jmaa.2018.01.027.  Google Scholar [36] G. Zou and B. Wang, Stochastic Burgers' equation with fractional derivative driven by multiplicative noise, Comput. Math. Appl., 74 (2017), 3195-3208.  doi: 10.1016/j.camwa.2017.08.023.  Google Scholar

show all references

##### References:
 [1] E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.  Google Scholar [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar [3] L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains., J. Math. Fluid Mech., 18 (2016), 25-69.  doi: 10.1007/s00021-015-0234-5.  Google Scholar [4] M. Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications., Numer. Methods Partial Differential Equations, 22 (2006), 220-257.  doi: 10.1002/num.20071.  Google Scholar [5] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fract., 32 (2007), 661-675.  doi: 10.1016/j.chaos.2005.11.010.  Google Scholar [6] M. Dehghan and M. Abbaszadeh, A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Eng Comput., 33 (2017), 587-605.   Google Scholar [7] C. Fetecau, M. Jamil, C. Fetecau and D. Vieru, The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys., 60 (2009), 921-933.  doi: 10.1007/s00033-008-8055-5.  Google Scholar [8] G. Hu, Y. Lou and P. D. Christofides, Dynamic output feedback covariance control of stochastic dissipative partial differential equations, Chem. Eng. Sci., 63 (2008), 4531-4542.   Google Scholar [9] Y. Jiang, T. Wei and X. Zhou, Stochastic generalized Burgers equations driven by fractional noises, J. Differ. Equ., 252 (2012), 1934-1961.  doi: 10.1016/j.jde.2011.07.032.  Google Scholar [10] M. Khan, The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl., 10 (2009), 3190-3195.  doi: 10.1016/j.nonrwa.2008.10.002.  Google Scholar [11] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4.  Google Scholar [12] M. Lakestani and M. Dehghan, The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math., 235 (2010), 669-678.  doi: 10.1016/j.cam.2010.06.020.  Google Scholar [13] P. D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.  Google Scholar [14] F. Li, Y. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar [15] F. Li, Y. Li and R. Wang, Limiting dynamics for stochastic reaction-diffusion equations on the Sobolev space with thin domains, Comput. Math. Appl., 79 (2020), 457-475.  doi: 10.1016/j.camwa.2019.07.009.  Google Scholar [16] Y. Li and Y. Wang., The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay., J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.  Google Scholar [17] J. Liang, X. Qian, T. Shen and S. Song, Analysis of time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by multiplicative white noise, J. Math. Anal. Appl., 466 (2018), 1525-1544.  doi: 10.1016/j.jmaa.2018.06.066.  Google Scholar [18] T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Z. Yong, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Meth. Appl. Sci., (2020), 1–27. Google Scholar [19] H. L. Nguyen, H. T. Nguyen, K. Mokhtar and X. T. Duong Dang, Identifying initial condition of the Rayleigh-Stokes problem with random noise, Math. Meth. Appl. Sci., 42 (2019), 1561-1571.  doi: 10.1002/mma.5455.  Google Scholar [20] H. L. Nguyen, H. T. Nguyen and Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci., 42 (2019), 3481-3495.  doi: 10.1002/mma.5593.  Google Scholar [21] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002, 23 pp. doi: 10.1088/1361-6420/ab532c.  Google Scholar [22] J.-C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.  Google Scholar [23] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar [24] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.  Google Scholar [25] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar [26] F. Shen, W. Tan, Y. Zhao and T. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model., Nonlinear Anal. Real World Appl., 7 (2006), 1072-1080.  doi: 10.1016/j.nonrwa.2005.09.007.  Google Scholar [27] X. Su and M. Li, The regularity of fractional stochastic evolution equations in Hilbert space, Stoch. Anal. Appl., 36 (2018), 639-653.  doi: 10.1080/07362994.2018.1436973.  Google Scholar [28] N. H. Tuan, Y. Zhou, T. N. Thach and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data., Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104873, 18 pp. doi: 10.1016/j.cnsns.2019.104873.  Google Scholar [29] R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.  Google Scholar [30] R. Wang, L. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\Bbb R^N$, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.  Google Scholar [31] C. Xue and J. Nie, Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space, App. Math. Model, 33 (2009), 524-531.  doi: 10.1016/j.apm.2007.11.015.  Google Scholar [32] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar [33] M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 75 (2018), 2243-2258.  doi: 10.1016/j.camwa.2017.12.004.  Google Scholar [34] C. Zhao and C. Yang, Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels, Appl. Math. Comput., 211 (2009), 502-509.  doi: 10.1016/j.amc.2009.01.068.  Google Scholar [35] G. Zou, G. Lv and J.-L. Wu, Stochastic Navier–Stokes equations with Caputo derivative driven by fractional noises, J. Math. Anal. Appl., 461 (2018), 595-609.  doi: 10.1016/j.jmaa.2018.01.027.  Google Scholar [36] G. Zou and B. Wang, Stochastic Burgers' equation with fractional derivative driven by multiplicative noise, Comput. Math. Appl., 74 (2017), 3195-3208.  doi: 10.1016/j.camwa.2017.08.023.  Google Scholar
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