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

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  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.

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, 2021, 26 (8) : 4299-4323. doi: 10.3934/dcdsb.2020289
References:
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C. FetecauM. JamilC. 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

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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

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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

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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

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

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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. ShenW. TanY. 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. WangY. 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. WangL. 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. YeJ. 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. ZouG. 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. BazhlekovaB. JinR. 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. FetecauM. JamilC. 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. HuY. 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. JiangT. 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. LiY. 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. LiY. 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. LiangX. QianT. 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. NguyenH. T. NguyenK. 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. NguyenH. 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. ShenW. TanY. 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. WangY. 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. WangL. 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. YeJ. 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. ZouG. 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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