Existence and regularity of mild solutions to fractional stochastic evolution equations. (English) Zbl 1405.60102
Summary: This study is concerned with the stochastic fractional diffusion and diffusion-wave equations driven by multiplicative noise. We prove the existence and uniqueness of mild solutions to these equations by means of the Picard’s iteration method. With the help of the fractional calculus and stochastic analysis theory, we also establish the pathwise spatial-temporal (Sobolev-Hölder) regularity properties of mild solutions to these types of fractional SPDEs in a semigroup framework. Finally, we relate our results to the selection of appropriate numerical schemes for the solutions of these time-fractional SPDEs.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35Q30 | Navier-Stokes equations |
| 35K55 | Nonlinear parabolic equations |
Keywords:
fractional calculus; stochastic diffusion equations; stochastic diffusion-wave equations; mild solutions; numerical resultsReferences:
| [1] | K. Al-Khaled and S. Momani, An approximate solution for a fractional diffusion-wave equation using the decomposition method. Appl. Math. Comput.165 (2005) 473-483. · Zbl 1071.65135 |
| [2] | A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl. Math. Comput.273 (2016) 948-956. · Zbl 1410.35272 |
| [3] | A. Atangana and K. Ilknur, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals89 (2016) 447-454. · Zbl 1360.34150 · doi:10.1016/j.chaos.2016.02.012 |
| [4] | V. Capasso and D. Morale, Stochastic modelling of tumour-induced angiogenesis. J. Math. Biol.58 (2009) 219-233. · Zbl 1161.92034 · doi:10.1007/s00285-008-0193-z |
| [5] | Z.Q. Chen, K.H. Kim and P. Kim, Fractional time stochastic partial differential equations. Stoch. Proc. Appl.125 (2015) 1470-1499. · Zbl 1322.60106 · doi:10.1016/j.spa.2014.11.005 |
| [6] | R. Du, W.R. Cao and Z.Z. Sun, A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model.34 (2010) 2998-3007. · Zbl 1201.65154 · doi:10.1016/j.apm.2010.01.008 |
| [7] | J. Duan and W. Wang, Effective Dynamics of stochastic Partial Differential Equations. Elsevier (2014). · Zbl 1298.60006 |
| [8] | R. Gorenflo, Y. Luchko and F. Mainardi, Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Comput. Appl. Math.118 (2000) 175-191. · Zbl 0973.35012 · doi:10.1016/S0377-0427(00)00288-0 |
| [9] | B. Guo and D. Huang, 3D stochastic primitive equations of the large-scale ocean: global well-posedness and attractors. Commun. Math. Phys.286 (2009) 697-723. · Zbl 1179.37115 · doi:10.1007/s00220-008-0654-7 |
| [10] | G. Hu, Y. Lou and P.D. Christofides, Dynamic output feedback covariance control of stochastic dissipative partial differential equations. Chem. Eng. Sci.63 (2008) 4531-4542. · doi:10.1016/j.ces.2008.06.026 |
| [11] | R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Springer (2014). · Zbl 1285.60002 · doi:10.1007/978-3-319-02231-4 |
| [12] | C. Li, Z. Zhao and Y.Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl.62 (2011) 855-875. · Zbl 1228.65190 · doi:10.1016/j.camwa.2011.02.045 |
| [13] | W. Liu and M. Rökner, SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal.259 (2010) 2902-2922. · Zbl 1236.60064 · doi:10.1016/j.jfa.2010.05.012 |
| [14] | G. Lv and J. Duan, Impacts of noise on a class of partial differential equations. J. Differ. Equ.258 (2015) 2196-2220. · Zbl 1319.60143 · doi:10.1016/j.jde.2014.12.002 |
| [15] | F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett.9 (1996) 23-28. · Zbl 0879.35036 · doi:10.1016/0893-9659(96)00089-4 |
| [16] | C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations. Springer (2007). · Zbl 1123.60001 |
| [17] | H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier (2006). · Zbl 1092.45003 |
| [18] | R.N. Wang, D.H. Chen and T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ.252 (2012) 202-235. · Zbl 1238.34015 · doi:10.1016/j.jde.2011.08.048 |
| [19] | J.R. Wang, M. Feckan, Y. Zhou, A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal.19 (2016) 806-831. · Zbl 1344.35169 |
| [20] | J.R. Wang, M. Feckan, Y. Zhou, Center stable manifold for planar fractional damped equations. Appl. Math. Comput.296 (2017) 257-269. · Zbl 1411.34021 |
| [21] | E. Weinan, K. Khanin, A. Mazel and Ya. Sinai, Invariant measure for Burgers equation with stochastic forcing. Ann. Math.151 (2000) 877-960. · Zbl 0972.35196 · doi:10.2307/121126 |
| [22] | F. Zeng, Second-order stable finite difference schemes for the time-fractional diffusion-wave equation. J. Sci. Comput.65 (2015) 411-430. · Zbl 1408.65058 · doi:10.1007/s10915-014-9966-2 |
| [23] | Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control. Academic Press (2016). · Zbl 1343.34001 |
| [24] | Y. Zhou, Attractivity for fractional differential equations. Appl. Math. Lett.75 (2018) 1-6. · Zbl 1380.34025 · doi:10.1016/j.aml.2017.06.008 |
| [25] | Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations. Comput. Math. Appl.73 (2017) 874-891. · Zbl 1409.76027 · doi:10.1016/j.camwa.2016.03.026 |
| [26] | Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control. Comput. Math. Appl.73 (2017) 1016-1027. · Zbl 1412.35233 · doi:10.1016/j.camwa.2016.07.007 |
| [27] | Y. Zhou and L. Zhang, Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems. Comput. Math. Appl.73 (2017) 1325-1345. · Zbl 1409.35232 · doi:10.1016/j.camwa.2016.04.041 |
| [28] | Y. Zhou, B. Ahmad and A. Alsaedi, Existence of nonoscillatory solutions for fractional neutral differential equations. Appl. Math. Lett.72 (2017) 70-74. · Zbl 1373.34119 · doi:10.1016/j.aml.2017.04.016 |
| [29] | Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness. Evol. Equ. Control Theor.4 (2015) 507-524. · Zbl 1335.34096 · doi:10.3934/eect.2015.4.507 |
| [30] | G.A. Zou, A Galerkin finite element method for time-fractional stochastic heat equation. (2016). |
| [31] | G. Zou and B. Wang, Stochastic Burgers equation with fractional derivative driven by multiplicative noise. Comput. Math. Appl.74 (2017) 3195-3208. · Zbl 1395.35200 · doi:10.1016/j.camwa.2017.08.023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
