-
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] |
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 |
[2] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
[3] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[4] |
Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227 |
[5] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
[6] |
Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[7] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348 |
[8] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174 |
[9] |
Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380 |
[10] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[11] |
Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 |
[12] |
Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020050 |
[13] |
Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 |
[14] |
Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020385 |
[15] |
Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 |
[16] |
Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 |
[17] |
Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003 |
[18] |
P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 |
[19] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021015 |
[20] |
Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021015 |
2019 Impact Factor: 0.953
Tools
Article outline
[Back to Top]