# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022079
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## New well-posedness results for stochastic delay Rayleigh-Stokes equations

 1 Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Viet Nam, Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam 2 Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Viet Nam 3 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

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

In memorian to María J. Garrido-Atienza

Received  August 2021 Revised  March 2022 Early access April 2022

In this work, the following stochastic Rayleigh-Stokes equations are considered
 \begin{align*} \partial_t \big[ x(t)+f(t,x_\rho(t)) \big] = \big( A +\vartheta &\partial_t^\beta A \big) \big[ x(t)+f(t,x_\rho(t)) \big] \\ &+ g(t,x_\tau(t)) + B(t,x_\xi(t)) \dot{W}(t), \end{align*}
which involve the Riemann-Liouville fractional derivative in time, delays and standard Brownian motion. Under two different conditions for the non-linear external forcing terms, two existence and uniqueness results for the mild solution are established respectively, in the continuous space
 $\mathcal{C}([-h,T];L^p(\Omega,V_q))$
,
 $p \ge 2$
,
 $q \ge 0$
. Our study was motivated and inspired by a series of papers by T. Caraballo and his colleagues on stochastic differential equations containing delays.
Citation: Nguyen Huy Tuan, Nguyen Duc Phuong, Tran Ngoc Thach. New well-posedness results for stochastic delay Rayleigh-Stokes equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022079
##### References:
 [1] W. Arendt, A. F. Ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differential Equations, 43 (2018), 1-24.  doi: 10.1080/03605302.2017.1363229. [2] 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. [3] A. Boudaoui, T. Caraballo and A. Ouahab, Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Appl. Anal., 95 (2016), 2039-2062.  doi: 10.1080/00036811.2015.1086756. [4] B. Boufoussi and S. Hajji, Transportation inequalities for neutral stochastic differential equations driven by fractional brownian motion with hurst parameter lesser than 1/2, Mediterr. J. Math., 14 (2017), Paper No. 192, 16 pp. doi: 10.1007/s00009-017-0992-9. [5] T. Caraballo and R. Colucci, A qualitative description of microstructure formation and coarsening phenomena for an evolution equation, Nonlinear Differential Equations Appl., 24 (2017), Paper No. 14, 24 pp. doi: 10.1007/s00030-017-0437-y. [6] T. Caraballo and M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.  doi: 10.1007/s11464-013-0300-3. [7] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047. [8] T. Caraballo, T. B. Ngoc, T. N. Thach and N. H. Tuan, On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4299–4323. doi: 10.3934/dcdsb.2020289. [9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513. [10] 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. [11] 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. [12] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fractals, 32 (2007), 661-675.  doi: 10.1016/j.chaos.2005.11.010. [13] 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, Engineering with Computers, 33 (2017), 587-605. [14] K. T. Dinh, L. Do and T. P. Thanh, On stability for semilinear generalized Rayleigh-Stokes equation involving delays, arXiv preprint, arXiv: 2011.00545. [15] 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. [16] T. Kato, Perturbation Theory for Linear Operators, "Mir", Moscow, 1972. [17] 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. [18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006). [19] L. Liu, T. Caraballo and P. Marín-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008. [20] 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. Methods Appl. Sci., 42 (2019), 1561-1571.  doi: 10.1002/mma.5455. [21] 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. [22] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999. [23] 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. [24] N. H. Tuan, V. V. Tri, J. Singh and T. N. Thach, On a fractional Rayleigh-Stokes equation driven by fractional Brownian motion, Mathematical Methods in the Applied Sciences. [25] 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. [26] J. Wang, C. Zhao and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105459, 14 pp. doi: 10.1016/j.cnsns.2020.105459. [27] J. Xu and T. Caraballo, Long time behavior of fractional impulsive stochastic differential equations with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2719-2743.  doi: 10.3934/dcdsb.2018272. [28] J. Xu, Z. Zhang and T. Caraballo, Mild solutions to time fractional stochastic 2D-Stokes equations with bounded and unbounded delay, J. Dynam. Differential Equations, 34 (2022), 583-603.  doi: 10.1007/s10884-019-09809-3. [29] J. Xu, Z. Zhang and T. Caraballo, Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay, Commun. Nonlinear Sci. Numer. Simul., 75 (2019), 121-139.  doi: 10.1016/j.cnsns.2019.03.002. [30] J. Xu, Z. Zhang and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505-546.  doi: 10.1016/j.jde.2020.07.037. [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, Appl. Math. Model., 33 (2009), 524-531.  doi: 10.1016/j.apm.2007.11.015. [32] 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. [33] 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. [34] Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, 2$^{nd}$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

show all references

##### References:
 [1] W. Arendt, A. F. Ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differential Equations, 43 (2018), 1-24.  doi: 10.1080/03605302.2017.1363229. [2] 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. [3] A. Boudaoui, T. Caraballo and A. Ouahab, Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Appl. Anal., 95 (2016), 2039-2062.  doi: 10.1080/00036811.2015.1086756. [4] B. Boufoussi and S. Hajji, Transportation inequalities for neutral stochastic differential equations driven by fractional brownian motion with hurst parameter lesser than 1/2, Mediterr. J. Math., 14 (2017), Paper No. 192, 16 pp. doi: 10.1007/s00009-017-0992-9. [5] T. Caraballo and R. Colucci, A qualitative description of microstructure formation and coarsening phenomena for an evolution equation, Nonlinear Differential Equations Appl., 24 (2017), Paper No. 14, 24 pp. doi: 10.1007/s00030-017-0437-y. [6] T. Caraballo and M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.  doi: 10.1007/s11464-013-0300-3. [7] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047. [8] T. Caraballo, T. B. Ngoc, T. N. Thach and N. H. Tuan, On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4299–4323. doi: 10.3934/dcdsb.2020289. [9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513. [10] 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. [11] 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. [12] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fractals, 32 (2007), 661-675.  doi: 10.1016/j.chaos.2005.11.010. [13] 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, Engineering with Computers, 33 (2017), 587-605. [14] K. T. Dinh, L. Do and T. P. Thanh, On stability for semilinear generalized Rayleigh-Stokes equation involving delays, arXiv preprint, arXiv: 2011.00545. [15] 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. [16] T. Kato, Perturbation Theory for Linear Operators, "Mir", Moscow, 1972. [17] 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. [18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006). [19] L. Liu, T. Caraballo and P. Marín-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008. [20] 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. Methods Appl. Sci., 42 (2019), 1561-1571.  doi: 10.1002/mma.5455. [21] 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. [22] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999. [23] 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. [24] N. H. Tuan, V. V. Tri, J. Singh and T. N. Thach, On a fractional Rayleigh-Stokes equation driven by fractional Brownian motion, Mathematical Methods in the Applied Sciences. [25] 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. [26] J. Wang, C. Zhao and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105459, 14 pp. doi: 10.1016/j.cnsns.2020.105459. [27] J. Xu and T. Caraballo, Long time behavior of fractional impulsive stochastic differential equations with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2719-2743.  doi: 10.3934/dcdsb.2018272. [28] J. Xu, Z. Zhang and T. Caraballo, Mild solutions to time fractional stochastic 2D-Stokes equations with bounded and unbounded delay, J. Dynam. Differential Equations, 34 (2022), 583-603.  doi: 10.1007/s10884-019-09809-3. [29] J. Xu, Z. Zhang and T. Caraballo, Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay, Commun. Nonlinear Sci. Numer. Simul., 75 (2019), 121-139.  doi: 10.1016/j.cnsns.2019.03.002. [30] J. Xu, Z. Zhang and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505-546.  doi: 10.1016/j.jde.2020.07.037. [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, Appl. Math. Model., 33 (2009), 524-531.  doi: 10.1016/j.apm.2007.11.015. [32] 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. [33] 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. [34] Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, 2$^{nd}$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
 [1] Tran Bao Ngoc, Nguyen Huy Tuan, R. Sakthivel, Donal O'Regan. Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative. Evolution Equations and Control Theory, 2022, 11 (2) : 439-455. doi: 10.3934/eect.2021007 [2] Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025 [3] María Guadalupe Morales, Zuzana Došlá, Francisco J. Mendoza. Riemann-Liouville derivative over the space of integrable distributions. Electronic Research Archive, 2020, 28 (2) : 567-587. doi: 10.3934/era.2020030 [4] Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations and Control Theory, 2022, 11 (1) : 259-282. doi: 10.3934/eect.2021002 [5] Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control and Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016 [6] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure and Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [7] Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837 [8] Paul Eloe, Jaganmohan Jonnalagadda. Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2719-2734. doi: 10.3934/dcdss.2020220 [9] Paulo Mendes de Carvalho Neto, Renato Fehlberg Júnior. The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022118 [10] 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 [11] Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031 [12] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [13] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [14] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [15] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [16] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [17] Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084 [18] Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 [19] Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719 [20] Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic and Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001

2021 Impact Factor: 1.497