October  2022, 21(10): 3371-3387. doi: 10.3934/cpaa.2022104

Energy-dissipation for time-fractional phase-field equations

1. 

SUSTech International Center for Mathematics, and Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

2. 

SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, China

*Corresponding author

Received  March 2022 Published  October 2022 Early access  June 2022

Fund Project: The second author is supported by by NSFC Grant 11901281, the Stable Support Plan Program of Shenzhen Natural Science Fund (Program Contract No. 20200925160747003), and Shenzhen Science and Technology Program (Grant No. RCYX20210609104358076)

We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by the Caputo time-fractional derivative. Several novel criterions are examined for showing the positive-definiteness of the associated kernel functions. We deduce strict energy-dissipation for a number of non-local energy functionals, thereby proving fractional energy dissipation laws.

Citation: Dong Li, Chaoyu Quan, Jiao Xu. Energy-dissipation for time-fractional phase-field equations. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3371-3387. doi: 10.3934/cpaa.2022104
References:
[1]

M. Caputo and M. Fabrizio, Damage and fatigue described by a fractional derivative model, J. Comput. Phys., 293 (2015), 400-408.  doi: 10.1016/j.jcp.2014.11.012.

[2]

J. M. Carcione, F. J. Sanchez-Sesma, F. Luzón and J. J. P. Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, J. Phys. A, 46 (2013), 345501, 23 pp. doi: 10.1088/1751-8113/46/34/345501.

[3]

D. del Castillo-Negrete, B. A. Carreras, and V. E. Lynch, Nondiffusive transport in plasma turbulence: A fractional diffusion approach, Phys. Rev. Lett., 94 (2005), 065003, 4 pp. doi: 10.1103/PhysRevLett. 94.065003.

[4]

H. Dong and Y. Liu, Weighted mixed norm estimates for fractional wave equations with VMO coefficients., arXiv: 2102.01136.

[5] R. Durrett, Probability: Theory and Examples, Vol. 49. Cambridge university press, Cambridge, 2019.  doi: 10.1017/9781108591034.
[6]

K.-N. LeW. Mclean and M. Stynes, Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing, Commun. Pure Appl. Analysis, 18 (2019), 2765-2787.  doi: 10.3934/cpaa.2019124.

[7]

M. L. KavvasT. TuA. Ercan and J. Polsinelli, Fractional governing equations of transient groundwater flow in confined aquifers with multi-fractional dimensions in fractional time, Earth Syst. Dynam., 8 (2017), 921-929.  doi: 10.5194/esd-8-921-2017.

[8]

L. LiJ.-G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.

[9]

W. Li and A. J. Salgado, Time fractional gradient flows: Theory and numerics, arxiv: 2101.00541.

[10]

H. LiuA. ChengH. Wang and J. Zhao, Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation., Comput. Math. Appl., 76 (2018), 1876-1892.  doi: 10.1016/j.camwa.2018.07.036.

[11]

Y. Luchko and M. Yamamoto, On the maximum principle for a time-fractional diffusion equation, Fract. Calc. Appl. Anal., 20 (2017), 1131-1145.  doi: 10.1515/fca-2017-0060.

[12]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[13]

C. QuanT. Tang and J. Yang, How to define dissipation-preserving energy for time-fractional phase-field equations, CSIAM Trans. Appl. Math., 1 (2020), 478-490.  doi: 10.4208/csiam-am.2020-0024.

[14]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015) 210–239. doi: 10.1137/130941900.

[15]

V. Vergara and R. Zacher, Stability, instability, and blowup for time fractional and other non-local in time semilinear subdiffusion equations, J. Evol. Equ., 17 (2017), 599-626.  doi: 10.1007/s00028-016-0370-2.

[16]

R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations, Math. Ann., 356 (2013), 99-146.  doi: 10.1007/s00208-012-0834-9.

[17]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

show all references

References:
[1]

M. Caputo and M. Fabrizio, Damage and fatigue described by a fractional derivative model, J. Comput. Phys., 293 (2015), 400-408.  doi: 10.1016/j.jcp.2014.11.012.

[2]

J. M. Carcione, F. J. Sanchez-Sesma, F. Luzón and J. J. P. Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, J. Phys. A, 46 (2013), 345501, 23 pp. doi: 10.1088/1751-8113/46/34/345501.

[3]

D. del Castillo-Negrete, B. A. Carreras, and V. E. Lynch, Nondiffusive transport in plasma turbulence: A fractional diffusion approach, Phys. Rev. Lett., 94 (2005), 065003, 4 pp. doi: 10.1103/PhysRevLett. 94.065003.

[4]

H. Dong and Y. Liu, Weighted mixed norm estimates for fractional wave equations with VMO coefficients., arXiv: 2102.01136.

[5] R. Durrett, Probability: Theory and Examples, Vol. 49. Cambridge university press, Cambridge, 2019.  doi: 10.1017/9781108591034.
[6]

K.-N. LeW. Mclean and M. Stynes, Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing, Commun. Pure Appl. Analysis, 18 (2019), 2765-2787.  doi: 10.3934/cpaa.2019124.

[7]

M. L. KavvasT. TuA. Ercan and J. Polsinelli, Fractional governing equations of transient groundwater flow in confined aquifers with multi-fractional dimensions in fractional time, Earth Syst. Dynam., 8 (2017), 921-929.  doi: 10.5194/esd-8-921-2017.

[8]

L. LiJ.-G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.

[9]

W. Li and A. J. Salgado, Time fractional gradient flows: Theory and numerics, arxiv: 2101.00541.

[10]

H. LiuA. ChengH. Wang and J. Zhao, Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation., Comput. Math. Appl., 76 (2018), 1876-1892.  doi: 10.1016/j.camwa.2018.07.036.

[11]

Y. Luchko and M. Yamamoto, On the maximum principle for a time-fractional diffusion equation, Fract. Calc. Appl. Anal., 20 (2017), 1131-1145.  doi: 10.1515/fca-2017-0060.

[12]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[13]

C. QuanT. Tang and J. Yang, How to define dissipation-preserving energy for time-fractional phase-field equations, CSIAM Trans. Appl. Math., 1 (2020), 478-490.  doi: 10.4208/csiam-am.2020-0024.

[14]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015) 210–239. doi: 10.1137/130941900.

[15]

V. Vergara and R. Zacher, Stability, instability, and blowup for time fractional and other non-local in time semilinear subdiffusion equations, J. Evol. Equ., 17 (2017), 599-626.  doi: 10.1007/s00028-016-0370-2.

[16]

R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations, Math. Ann., 356 (2013), 99-146.  doi: 10.1007/s00208-012-0834-9.

[17]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

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