# American Institute of Mathematical Sciences

October  2019, 39(10): 5707-5727. doi: 10.3934/dcds.2019250

## An operator splitting scheme for the fractional kinetic Fokker-Planck equation

 1 School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK 2 Department of Mathematics, Duke University, Durham NC 27708, USA

* Corresponding author: Yulong Lu

Received  June 2018 Revised  March 2019 Published  July 2019

Fund Project: MHD was supported by ERC Starting Grant 335120

In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. The first phase is solved exactly using the convolution operator while the second one is solved approximately using a variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. We prove the convergence of the scheme to a weak solution to FKFPE. As a by-product of our analysis, we also establish a variational formulation for a kinetic transport equation that is relevant in the second phase. Finally, we discuss some extensions of our analysis to more complex systems.

Citation: Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
##### References:
 [1] P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, SIAM J. Math. Anal., 51 (2019), 469-488. doi: 10.1137/17M1152073. Google Scholar [2] M. Agueh, Local existence of weak solutions to kinetic models of granular media, Arch. Ration. Mech. Anal., 221 (2016), 917-959. doi: 10.1007/s00205-016-0975-1. Google Scholar [3] N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z. Google Scholar [4] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics. ETH Zürich, Birkhauser, Basel, 2008. Google Scholar [5] D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. Google Scholar [6] S. Arnrich, A. Mielke, M. A. Peletier, G. Savaré and M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calc. Var. Partial Differential Equations, 44 (2012), 419-454. doi: 10.1007/s00526-011-0440-9. Google Scholar [7] T. Bodineau and R. Lefevere, Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats, J. Stat. Phys., 133 (2008), 1-27. doi: 10.1007/s10955-008-9601-4. Google Scholar [8] M. Bowles and M. Agueh, Weak solutions to a fractional Fokker–Planck equation via splitting and Wfasserstein gradient flow, Applied Mathematics Letters, 42 (2015), 30–35, URL http://www.sciencedirect.com/science/article/pii/S0893965914003346. doi: 10.1016/j.aml.2014.10.008. Google Scholar [9] E. A. Carlen and W. Gangbo, Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric, Arch. Ration. Mech. Anal., 172 (2004), 21-64. doi: 10.1007/s00205-003-0296-z. Google Scholar [10] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376. Google Scholar [11] L. Cesbron, A. Mellet and K. Trivisa, Anomalous transport of particles in plasma physics, Applied Mathematics Letters, 25 (2012), 2344-2348, URL http://www.sciencedirect.com/science/article/pii/S0893965912003163. doi: 10.1016/j.aml.2012.06.029. Google Scholar [12] L. Cesbron, Anomalous diffusion limit of kinetic equations in spatially bounded domains, Comm. Math. Phys., 364 (2018), 233-286. doi: 10.1007/s00220-018-3158-0. Google Scholar [13] Z.-Q. Chen and X. Zhang, Propagation of regularity in $l^p$-spaces for Kolmogorov type hypoelliptic operators, Journal of Evolution Equations, 2019, 1–29, arXiv: 1706.02181. doi: 10.1007/s00028-019-00505-9. Google Scholar [14] Z.-Q. Chen and X. Zhang, $L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators, J. Math. Pures Appl. (9), 116 (2018), 52-87. doi: 10.1016/j.matpur.2017.10.003. Google Scholar [15] F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577–1630, URL http://dx.doi.org/10.1016/j.jfa.2010.05.002. doi: 10.1016/j.jfa.2010.05.002. Google Scholar [16] J. Droniou, T. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499–521, Dedicated to Philippe Bénilan. doi: 10.1007/s00028-003-0503-1. Google Scholar [17] M. H. Duong, Long time behaviour and particle approximation of a generalised Vlasov dynamic, Nonlinear Anal., 127 (2015), 1-16. doi: 10.1016/j.na.2015.06.018. Google Scholar [18] M. H. Duong, A. Lamacz, M. A. Peletier and U. Sharma, Variational approach to coarse-graining of generalized gradient flows, Calc. Var. Partial Differential Equations, 56 (2017), Art. 100, 65pp. doi: 10.1007/s00526-017-1186-9. Google Scholar [19] M. H. Duong, M. A. Peletier and J. Zimmer, Conservative-dissipative approximation schemes for a generalized kramers equation, Mathematical Methods in the Applied Sciences, 37 (2014), 2517-2540. doi: 10.1002/mma.2994. Google Scholar [20] M. H. Duong and H. M. Tran, Analysis of the mean squared derivative cost function, Mathematical Methods in the Applied Sciences, 40 (2017), 5222-5240. doi: 10.1002/mma.4382. Google Scholar [21] M. H. Duong and H. M. Tran, On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type, Discrete Contin. Dyn. Syst., 38 (2018), 3407-3438. doi: 10.3934/dcds.2018146. Google Scholar [22] B. Düring, D. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935. Google Scholar [23] J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. Math. Phys., 212 (2000), 105-164. doi: 10.1007/s002200000216. Google Scholar [24] M. Erbar, Gradient flows of the entropy for jump processes, Ann. Inst. H. Poincare Probab. Statist., 50 (2014), 920-945. doi: 10.1214/12-AIHP537. Google Scholar [25] W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), 1041-1073. doi: 10.1080/03605300902892345. Google Scholar [26] P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Modern Phys., 62 (1990), 251-341. doi: 10.1103/RevModPhys.62.251. Google Scholar [27] H. Holden, K. H. Karlsen, K.-A. Lie and N. H. Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010, Analysis and MATLAB programs. doi: 10.4171/078. Google Scholar [28] C. Huang, A variational principle for the Kramers equation with unbounded external forces, J. Math. Anal. Appl., 250 (2000), 333-367. doi: 10.1006/jmaa.2000.7109. Google Scholar [29] L. Huang, S. Menozzi and E. Priola, $L^p$ estimates for degenerate non-local Kolmogorov operators, J. Math. Pures Appl. (9), 121 (2019), 162-215. doi: 10.1016/j.matpur.2017.12.008. Google Scholar [30] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar [31] D. Kinderlehrer and A. Tudorascu, Transport via mass transportation, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 311-338. doi: 10.3934/dcdsb.2006.6.311. Google Scholar [32] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (1940), 284-304. doi: 10.1016/S0031-8914(40)90098-2. Google Scholar [33] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51. doi: 10.1515/fca-2017-0002. Google Scholar [34] L. Lafleche, Fractional fokker-planck equation with general confinement force, 2018.Google Scholar [35] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, N.J., 1967. Google Scholar [36] M. Ottobre and G. A. Pavliotis, Asymptotic analysis for the generalized Langevin equation, Nonlinearity, 24 (2011), 1629-1653. doi: 10.1088/0951-7715/24/5/013. Google Scholar [37] A. Pascucci, Kolmogorov equations in physics and in finance, Elliptic and Parabolic Problems, Birkhäuser Basel, Basel, 63 (2005), 353–364. doi: 10.1007/3-7643-7384-9_35. Google Scholar [38] H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, Springer Series in Synergetics, 18. Springer-Verlag, Berlin, 1984. URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/354061530X. doi: 10.1007/978-3-642-96807-5. Google Scholar [39] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046. Google Scholar [40] U. Stefanelli, The Brezis–Ekeland principle for doubly nonlinear equations, SIAM Journal on Control and Optimization, 47 (2008), 1615-1642. doi: 10.1137/070684574. Google Scholar [41] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, Nonlinear Partial Differential Equations, Springer, Heidelberg, 7 (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15. Google Scholar [42] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857. Google Scholar [43] J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, vol. 2186 of Lecture Notes in Math., Springer, Cham, 2017,205–278. Google Scholar [44] C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. Google Scholar

show all references

##### References:
 [1] P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, SIAM J. Math. Anal., 51 (2019), 469-488. doi: 10.1137/17M1152073. Google Scholar [2] M. Agueh, Local existence of weak solutions to kinetic models of granular media, Arch. Ration. Mech. Anal., 221 (2016), 917-959. doi: 10.1007/s00205-016-0975-1. Google Scholar [3] N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z. Google Scholar [4] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics. ETH Zürich, Birkhauser, Basel, 2008. Google Scholar [5] D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. Google Scholar [6] S. Arnrich, A. Mielke, M. A. Peletier, G. Savaré and M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calc. Var. Partial Differential Equations, 44 (2012), 419-454. doi: 10.1007/s00526-011-0440-9. Google Scholar [7] T. Bodineau and R. Lefevere, Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats, J. Stat. Phys., 133 (2008), 1-27. doi: 10.1007/s10955-008-9601-4. Google Scholar [8] M. Bowles and M. Agueh, Weak solutions to a fractional Fokker–Planck equation via splitting and Wfasserstein gradient flow, Applied Mathematics Letters, 42 (2015), 30–35, URL http://www.sciencedirect.com/science/article/pii/S0893965914003346. doi: 10.1016/j.aml.2014.10.008. Google Scholar [9] E. A. Carlen and W. Gangbo, Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric, Arch. Ration. Mech. Anal., 172 (2004), 21-64. doi: 10.1007/s00205-003-0296-z. Google Scholar [10] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376. Google Scholar [11] L. Cesbron, A. Mellet and K. Trivisa, Anomalous transport of particles in plasma physics, Applied Mathematics Letters, 25 (2012), 2344-2348, URL http://www.sciencedirect.com/science/article/pii/S0893965912003163. doi: 10.1016/j.aml.2012.06.029. Google Scholar [12] L. Cesbron, Anomalous diffusion limit of kinetic equations in spatially bounded domains, Comm. Math. Phys., 364 (2018), 233-286. doi: 10.1007/s00220-018-3158-0. Google Scholar [13] Z.-Q. Chen and X. Zhang, Propagation of regularity in $l^p$-spaces for Kolmogorov type hypoelliptic operators, Journal of Evolution Equations, 2019, 1–29, arXiv: 1706.02181. doi: 10.1007/s00028-019-00505-9. Google Scholar [14] Z.-Q. Chen and X. Zhang, $L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators, J. Math. Pures Appl. (9), 116 (2018), 52-87. doi: 10.1016/j.matpur.2017.10.003. Google Scholar [15] F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577–1630, URL http://dx.doi.org/10.1016/j.jfa.2010.05.002. doi: 10.1016/j.jfa.2010.05.002. Google Scholar [16] J. Droniou, T. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499–521, Dedicated to Philippe Bénilan. doi: 10.1007/s00028-003-0503-1. Google Scholar [17] M. H. Duong, Long time behaviour and particle approximation of a generalised Vlasov dynamic, Nonlinear Anal., 127 (2015), 1-16. doi: 10.1016/j.na.2015.06.018. Google Scholar [18] M. H. Duong, A. Lamacz, M. A. Peletier and U. Sharma, Variational approach to coarse-graining of generalized gradient flows, Calc. Var. Partial Differential Equations, 56 (2017), Art. 100, 65pp. doi: 10.1007/s00526-017-1186-9. Google Scholar [19] M. H. Duong, M. A. Peletier and J. Zimmer, Conservative-dissipative approximation schemes for a generalized kramers equation, Mathematical Methods in the Applied Sciences, 37 (2014), 2517-2540. doi: 10.1002/mma.2994. Google Scholar [20] M. H. Duong and H. M. Tran, Analysis of the mean squared derivative cost function, Mathematical Methods in the Applied Sciences, 40 (2017), 5222-5240. doi: 10.1002/mma.4382. Google Scholar [21] M. H. Duong and H. M. Tran, On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type, Discrete Contin. Dyn. Syst., 38 (2018), 3407-3438. doi: 10.3934/dcds.2018146. Google Scholar [22] B. Düring, D. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935. Google Scholar [23] J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. Math. Phys., 212 (2000), 105-164. doi: 10.1007/s002200000216. Google Scholar [24] M. Erbar, Gradient flows of the entropy for jump processes, Ann. Inst. H. Poincare Probab. Statist., 50 (2014), 920-945. doi: 10.1214/12-AIHP537. Google Scholar [25] W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), 1041-1073. doi: 10.1080/03605300902892345. Google Scholar [26] P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Modern Phys., 62 (1990), 251-341. doi: 10.1103/RevModPhys.62.251. Google Scholar [27] H. Holden, K. H. Karlsen, K.-A. Lie and N. H. Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010, Analysis and MATLAB programs. doi: 10.4171/078. Google Scholar [28] C. Huang, A variational principle for the Kramers equation with unbounded external forces, J. Math. Anal. Appl., 250 (2000), 333-367. doi: 10.1006/jmaa.2000.7109. Google Scholar [29] L. Huang, S. Menozzi and E. Priola, $L^p$ estimates for degenerate non-local Kolmogorov operators, J. Math. Pures Appl. (9), 121 (2019), 162-215. doi: 10.1016/j.matpur.2017.12.008. Google Scholar [30] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar [31] D. Kinderlehrer and A. Tudorascu, Transport via mass transportation, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 311-338. doi: 10.3934/dcdsb.2006.6.311. Google Scholar [32] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (1940), 284-304. doi: 10.1016/S0031-8914(40)90098-2. Google Scholar [33] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51. doi: 10.1515/fca-2017-0002. Google Scholar [34] L. Lafleche, Fractional fokker-planck equation with general confinement force, 2018.Google Scholar [35] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, N.J., 1967. Google Scholar [36] M. Ottobre and G. A. Pavliotis, Asymptotic analysis for the generalized Langevin equation, Nonlinearity, 24 (2011), 1629-1653. doi: 10.1088/0951-7715/24/5/013. Google Scholar [37] A. Pascucci, Kolmogorov equations in physics and in finance, Elliptic and Parabolic Problems, Birkhäuser Basel, Basel, 63 (2005), 353–364. doi: 10.1007/3-7643-7384-9_35. Google Scholar [38] H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, Springer Series in Synergetics, 18. Springer-Verlag, Berlin, 1984. URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/354061530X. doi: 10.1007/978-3-642-96807-5. Google Scholar [39] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046. Google Scholar [40] U. Stefanelli, The Brezis–Ekeland principle for doubly nonlinear equations, SIAM Journal on Control and Optimization, 47 (2008), 1615-1642. doi: 10.1137/070684574. Google Scholar [41] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, Nonlinear Partial Differential Equations, Springer, Heidelberg, 7 (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15. Google Scholar [42] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857. Google Scholar [43] J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, vol. 2186 of Lecture Notes in Math., Springer, Cham, 2017,205–278. Google Scholar [44] C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. Google Scholar
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