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September 2019, 18(5): 2765-2787. doi: 10.3934/cpaa.2019124

## Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing

 1 School of Mathematical Sciences, Monash University, VIC 3800, Australia 2 School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia 3 Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China

Received  November 2018 Revised  November 2018 Published  April 2019

Fund Project: The research of the first author is supported in part by the Australian Government through the Australian Research Council Discovery Projects funding scheme (project number DP170100605). The research of the third author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401

A time-fractional Fokker–Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u -{\bf{F}}\partial_t^{1-\alpha}u)$, where $0<\alpha <1$. The forcing function ${\bf{F}} = {\bf{F}}(t,x)$, which is more difficult to analyse than the case ${\bf{F}} = {\bf{F}}(x)$ investigated previously by other authors. The spatial domain $\Omega \subset\mathbb{R}^d$, where $d\ge 1$, has a smooth boundary. Existence, uniqueness and regularity of a mild solution $u$ is proved under the hypothesis that the initial data $u_0$ lies in $L^2(\Omega)$. For $1/2<\alpha<1$ and $u_0\in H^2(\Omega)\cap H_0^1(\Omega)$, it is shown that $u$ becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived—these are known to be needed in numerical analyses of this problem.

Citation: Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124
##### References:
 [1] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-5975-0. [2] K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-14574-2. [3] S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002. [4] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [5] C. Huang, K. N. Le and M. Stynes, A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing, IMA J. Numer. Anal. (to appear). [6] K. N. Le, W. McLean and K. Mustapha, Numerical solution of the time-fractional Fokker-Planck equation with general forcing, SIAM J. Numer. Anal., 54 (2016), 1763-1784. doi: 10.1137/15M1031734. [7] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018. [8] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Computers & Mathematics with Applications, 59 (2010), 1766-1772. doi: 10.1016/j.camwa.2009.08.015. [9] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138. doi: 10.1017/S1446181111000617. [10] W. McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput., 34 (2012), A3039-A3056. doi: 10.1137/120870505. [11] W. McLean, K. Mustapha, R. Ali and O. Knio, Well-posedness of time-fractional advection-diffusion-reaction equations, arXiv e-prints. [12] W. McLean, K. Mustapha, R. Ali and O. Knio, Regularity theory for time-fractional, advection-diffusion-reaction equations, arXiv e-prints. [13] J. Mu, B. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Computers & Mathematics with Applications, 73 (2017), 985-996. doi: 10.1016/j.camwa.2016.04.039. [14] K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA Journal of Numerical Analysis, 34 (2014), 1426-1446. doi: 10.1093/imanum/drt048. [15] L. Pinto and E. Sousa, Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion, Commun. Nonlinear Sci. Numer. Simul., 50 (2017), 211-228. doi: 10.1016/j.cnsns.2017.03.004. [16] R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048. [17] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061.

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##### References:
 [1] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-5975-0. [2] K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-14574-2. [3] S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002. [4] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [5] C. Huang, K. N. Le and M. Stynes, A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing, IMA J. Numer. Anal. (to appear). [6] K. N. Le, W. McLean and K. Mustapha, Numerical solution of the time-fractional Fokker-Planck equation with general forcing, SIAM J. Numer. Anal., 54 (2016), 1763-1784. doi: 10.1137/15M1031734. [7] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018. [8] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Computers & Mathematics with Applications, 59 (2010), 1766-1772. doi: 10.1016/j.camwa.2009.08.015. [9] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138. doi: 10.1017/S1446181111000617. [10] W. McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput., 34 (2012), A3039-A3056. doi: 10.1137/120870505. [11] W. McLean, K. Mustapha, R. Ali and O. Knio, Well-posedness of time-fractional advection-diffusion-reaction equations, arXiv e-prints. [12] W. McLean, K. Mustapha, R. Ali and O. Knio, Regularity theory for time-fractional, advection-diffusion-reaction equations, arXiv e-prints. [13] J. Mu, B. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Computers & Mathematics with Applications, 73 (2017), 985-996. doi: 10.1016/j.camwa.2016.04.039. [14] K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA Journal of Numerical Analysis, 34 (2014), 1426-1446. doi: 10.1093/imanum/drt048. [15] L. Pinto and E. Sousa, Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion, Commun. Nonlinear Sci. Numer. Simul., 50 (2017), 211-228. doi: 10.1016/j.cnsns.2017.03.004. [16] R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048. [17] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061.
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