-
Previous Article
Decay rates for the moore-gibson-thompson equation with memory
- EECT Home
- This Issue
-
Next Article
Deterministic control of stochastic reaction-diffusion equations
Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations
Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, 100000, Vietnam |
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.
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] |
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. |
[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. |
[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. |
[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. |
[6] |
L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[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] |
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. |
[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. |
[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. |
[11] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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] |
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. |
[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. |
[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. |
[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. |
[6] |
L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[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] |
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. |
[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. |
[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. |
[11] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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 |
[10] |
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 |
[11] |
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 |
[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, 2020 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 |
2019 Impact Factor: 0.953
Tools
Metrics
Other articles
by authors
[Back to Top]