doi: 10.3934/eect.2021002

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

Received  June 2020 Revised  October 2020 Published  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.

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. 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.  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. ChenF. LiuK. 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. ChenF. 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. 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.  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. LucN. 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. SalehiH. 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. ShenW. TanY. 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. ZierepR. 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. 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.  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. ChenF. LiuK. 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. ChenF. 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. 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.  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. LucN. 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. SalehiH. 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. ShenW. TanY. 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. ZierepR. 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]

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

[2]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

[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]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[5]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021

[6]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129

[7]

Alessandro Gondolo, Fernando Guevara Vasquez. Characterization and synthesis of Rayleigh damped elastodynamic networks. Networks & Heterogeneous Media, 2014, 9 (2) : 299-314. doi: 10.3934/nhm.2014.9.299

[8]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003

[9]

Danielle Hilhorst, Pierre Roux. A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. Coli colonies. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021033

[10]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934/dcds.2020385

[11]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003

[12]

Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021018

[13]

José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025

[14]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015

[15]

Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L tensor coefficient under relaxed ellipticity condition. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021042

[16]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[17]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[18]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[19]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001

[20]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (38)
  • HTML views (131)
  • Cited by (0)

Other articles
by authors

[Back to Top]