doi: 10.3934/dcdsb.2022079
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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. ArendtA. 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. 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.

[3]

A. BoudaouiT. 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. CaraballoM. 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. 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.

[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. LiuT. 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. NguyenH. T. NguyenK. 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. 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.

[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. 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.

[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. XuZ. 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. XuZ. 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. XuZ. 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. ArendtA. 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. 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.

[3]

A. BoudaouiT. 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. CaraballoM. 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. 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.

[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. LiuT. 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. NguyenH. T. NguyenK. 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. 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.

[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. 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.

[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. XuZ. 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. XuZ. 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. XuZ. 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.

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