Existence, uniqueness and regularity of the solution of the time-fractional Fokker

Previous Article
Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume
CPAA Home
This Issue
Next Article
Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

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

Kim-Ngan Le ^1, , William McLean ^2, and Martin Stynes ^3,

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.

Keywords: Fokker–Planck equation, Riemann–Liouville derivative, time-fractional, regularity of solution.

Mathematics Subject Classification: 35R11.

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.

show all references

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.

[1]	Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 741-754. doi: 10.3934/dcdss.2020041
[2]	Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007
[3]	Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025
[4]	Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895
[5]	Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016
[6]	Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
[7]	Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[8]	Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[9]	Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[10]	José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[11]	Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[12]	Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681
[13]	Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[14]	Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[15]	Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169
[16]	Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[17]	Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185
[18]	Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837
[19]	Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[20]	Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687

American Institute of Mathematical Sciences