American Institute of Mathematical Sciences

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August  2021, 26(8): 4299-4323. doi: 10.3934/dcdsb.2020289

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

Tomás Caraballo 1, , Tran Bao Ngoc 2, , Tran Ngoc Thach 3,, and  Nguyen Huy Tuan 4,

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  August 2021 Early access  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.

Keywords: Time-fractional Rayleigh-Stokes equation, Wiener process, existence and regularity properties of solution.
Mathematics Subject Classification: 60H15, 60G15, 60G22, 60H50.
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 and Continuous Dynamical Systems - B, 2021, 26 (8) : 4299-4323. 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.

[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.
[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[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.

[11]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4.

[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.

[13]

P. D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[24]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.
[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[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.

[11]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4.

[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.

[13]

P. D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[24]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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