American Institute of Mathematical Sciences

Advanced
doi: 10.3934/eect.2021002
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.

Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations

Do Lan

Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, 100000, Vietnam

Received  June 2020 Revised  October 2020 Early access January 2021

Fund Project: This research is funded by Thuyloi University Foundation for Science and Technology under grant number TLU.STF.19-04

We study the generalized Rayleigh-Stokes problem involving a fractional derivative and nonlinear perturbation. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and asymptotic stability of solutions. In particular, if the nonlinearity is Lipschitzian then the mild solution of the mentioned problem becomes a classical one and its convergence to equilibrium point is proved.

Keywords: Rayleigh-Stokes problem, stability, nonlocal PDE.
Mathematics Subject Classification: Primary: 35B40, 35R11, 35C15; Secondary: 45D05, 45K05.
Citation: Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2021002
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]

X. Bi, S. Mu, Q. Liu, Q. Liu, B. Liu, P. Zhuang, J. Gao, H. Jiang, X. Li and B. Li, Advanced implicit meshless approaches for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, Int. J. Comput. Methods, 15 (2018), 1850032, 27 pp. doi: 10.1142/S0219876218500329.  Google Scholar

[3]

C. M. Chen, F. Liu, K. Burrage and Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78 (2013), 924-944.  doi: 10.1093/imamat/hxr079.  Google Scholar

[4]

C. M. Chen, F. Liu and V. Anh, Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Appl. Math. Comput., 204 (2008), 340-351.  doi: 10.1016/j.amc.2008.06.052.  Google Scholar

[5]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[6]

L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  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]

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

[9]

N. H. Luc, N. H. Tuan 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

[10]

T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci., (2020), 1–27. doi: 10.1002/mma.6162.  Google Scholar

[11]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[12]

F. Salehi, H. Saeedi and M. M. Moghadam, Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem, Comput. Appl. Math., 37 (2018), 5274-5292.  doi: 10.1007/s40314-018-0631-5.  Google Scholar

[13]

F. Shen, W. Tan, Y. Zhao and Y. 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

[14]

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

[15]

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.  Google Scholar

[16]

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

[17]

J. Zierep, R. Bohning and C. Fetecau, Rayleigh-Stokes problem for non-Newtonian medium with memory, ZAMM Z. Angew. Math. Mech., 87 (2007), 462-467.  doi: 10.1002/zamm.200710328.  Google Scholar

[18]

Y. Zhou and J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Method Appl. Sci., (2019), 1–8. doi: 10.1002/mma.5926.  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]

X. Bi, S. Mu, Q. Liu, Q. Liu, B. Liu, P. Zhuang, J. Gao, H. Jiang, X. Li and B. Li, Advanced implicit meshless approaches for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, Int. J. Comput. Methods, 15 (2018), 1850032, 27 pp. doi: 10.1142/S0219876218500329.  Google Scholar

[3]

C. M. Chen, F. Liu, K. Burrage and Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78 (2013), 924-944.  doi: 10.1093/imamat/hxr079.  Google Scholar

[4]

C. M. Chen, F. Liu and V. Anh, Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Appl. Math. Comput., 204 (2008), 340-351.  doi: 10.1016/j.amc.2008.06.052.  Google Scholar

[5]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[6]

L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  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]

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

[9]

N. H. Luc, N. H. Tuan 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

[10]

T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci., (2020), 1–27. doi: 10.1002/mma.6162.  Google Scholar

[11]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[12]

F. Salehi, H. Saeedi and M. M. Moghadam, Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem, Comput. Appl. Math., 37 (2018), 5274-5292.  doi: 10.1007/s40314-018-0631-5.  Google Scholar

[13]

F. Shen, W. Tan, Y. Zhao and Y. 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

[14]

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

[15]

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.  Google Scholar

[16]

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

[17]

J. Zierep, R. Bohning and C. Fetecau, Rayleigh-Stokes problem for non-Newtonian medium with memory, ZAMM Z. Angew. Math. Mech., 87 (2007), 462-467.  doi: 10.1002/zamm.200710328.  Google Scholar

[18]

Y. Zhou and J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Method Appl. Sci., (2019), 1–8. doi: 10.1002/mma.5926.  Google Scholar

[1]

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
[2]

Hung-Wen Kuo. The initial layer for Rayleigh problem. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 137-170. doi: 10.3934/dcdsb.2011.15.137
[3]

Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925
[4]

Jhean E. Pérez-López, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis. Journal of Computational Dynamics, 2020, 7 (1) : 159-181. doi: 10.3934/jcd.2020006
[5]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[6]

Annalisa Cesaroni, Matteo Novaga. The isoperimetric problem for nonlocal perimeters. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 425-440. doi: 10.3934/dcdss.2018023
[7]

George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207
[8]

Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53
[9]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
[10]

Annamaria Canino, Luigi Montoro, Berardino Sciunzi. The jumping problem for nonlocal singular problems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6747-6760. doi: 10.3934/dcds.2019293
[11]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[12]

Peter E. Kloeden, Stefanie Sonner, Christina Surulescu. A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2233-2254. doi: 10.3934/dcdsb.2016045
[13]

Fatimzehrae Ait Bella, Aissam Hadri, Abdelilah Hakim, Amine Laghrib. A nonlocal Weickert type PDE applied to multi-frame super-resolution. Evolution Equations & Control Theory, 2021, 10 (3) : 633-655. doi: 10.3934/eect.2020084
[14]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[15]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323
[16]

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053
[17]

Bettina Klaus, Frédéric Payot. Paths to stability in the assignment problem. Journal of Dynamics & Games, 2015, 2 (3&4) : 257-287. doi: 10.3934/jdg.2015004
[18]

Hui-Ling Li, Heng-Ling Wang, Xiao-Liu Wang. A quasilinear parabolic problem with a source term and a nonlocal absorption. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1945-1956. doi: 10.3934/cpaa.2018092
[19]

Jong-Shenq Guo, Nikos I. Kavallaris. On a nonlocal parabolic problem arising in electrostatic MEMS control. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1723-1746. doi: 10.3934/dcds.2012.32.1723
[20]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102

2020 Impact Factor: 1.081

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences