# American Institute of Mathematical Sciences

April  2019, 39(4): 1821-1889. doi: 10.3934/dcds.2019079

## A non-local problem for the Fokker-Planck equation related to the Becker-Döring model

 1 University of Michigan, Department of Mathematics, Ann Arbor, MI 48109-1109, USA 2 Universität Bonn, Institut für Angewandte Mathematik, Endenicher Allee 60, 53129 Bonn, Germany

* Corresponding author

Received  November 2017 Revised  September 2018 Published  January 2019

This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system.

The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield Hölder regularity of the non-local order parameter, which is improved by an iteration argument.

The asymptotic behavior of solutions depends on some order parameter $\rho$ depending on the initial data. The system shows different behavior depending on a value $\rho_s>0$, determined from the potentials and diffusion coefficient. For $\rho \leq \rho_s$, there exists an equilibrium solution $c^ {{ \rm{eq}}} _{(\rho)}$. If $\rho\le\rho_s$ the solution converges strongly to $c^ {{ \rm{eq}}} _{(\rho)}$, while if $\rho > \rho_s$ the solution converges weakly to $c^ {{ \rm{eq}}} _{(\rho_s)}$. The excess $\rho - \rho_s$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-Döring equation.

The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case $\rho<\rho_s$ the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution.

The close connection of the presented model and the Becker-Döring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.

Citation: Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
##### References:
 [1] L. Ambrosio and G. Buttazzo, Weak lower semicontinuous envelope of functionals defined on a space of measures, Ann. Di Mat. Pura Ed Appl., (4) 150 (1988), 311–339. doi: 10.1007/BF01761473. [2] C. Ané, S. Blachère, D. Chafai, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur Les Inégalités de Sobolev Logarithmiques, Panoramas et Synthèses, Société Mathématique de France, 2000. [3] J. M. Ball, J. Carr and O. Penrose, The Becker Döring cluster equations: Basic properties and asymptotic behavior of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070. [4] F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Stud. Math., 159 (2003), 481-497.  doi: 10.4064/sm159-3-9. [5] F. Barthe and C. Roberto, Modified Logarithmic Sobolev Inequalities on $\mathbb{R}$, Potential Anal., 29 (2008), 167-193.  doi: 10.1007/s11118-008-9093-5. [6] R. Becker and W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen, Ann. Der Phys., 24 (1935), 719-752.  doi: 10.1002/andp.19354160806. [7] P. Billingsley, Convergence of Probability Measures, Wiley, 1968, New York-London. [8] S. G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28.  doi: 10.1006/jfan.1998.3326. [9] V. I. Bogachev, Measure Theory, Springer Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-34514-5. [10] V. Bögelein, F. Duzaar and G. Mingione, The boundary regularity of non-linear parabolic systems Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 201–255. doi: 10.1016/j.anihpc.2009.09.003. [11] F. Bolley and C. Villani, Weighted Csiszar-Kullback-Pinsker inequalities and applications to transportation inequalities, Fac. Des Sci. Toulouse, 14 (2005), 331-352.  doi: 10.5802/afst.1095. [12] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, vol. 207 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. [13] J. A. Cañizo, A. Einav and B. Lods, Trend to equilibrium for the Becker-Döring equations: An analogue of Cercignani's conjecture, Anal. PDE., 10 (2017), 1663-1708.  doi: 10.2140/apde.2017.10.1663. [14] J. A. Cañizo, A. Einav and B. Lods, Uniform moment propagation for the Becker-Döring equation, preprint, arXiv: 1706.03524. [15] J. Cañizo and B. Lods, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, Journ. Diff. Eqns., 255 (2013), 905-950.  doi: 10.1016/j.jde.2013.04.031. [16] J.-F. Collet, Some modelling issues in the theory of fragmentation-coagulation systems, Commun. Math. Sci., 2 (2004), 35-54.  doi: 10.4310/CMS.2004.v2.n5.a3. [17] J.-F. Collet, T. Goudon, F. Poupaud and A. Vasseur, The Becker-Döring system and its Lifshitz-Slyozov limit, SIAM J. Appl. Math., 62 (2002), 1488-1500.  doi: 10.1137/S0036139900378852. [18] J.-F. Collet and T. Goudon, On solutions of the Lifshitz-Slyozov model, Nonlinearity, 13 (2000), 1239-1262.  doi: 10.1088/0951-7715/13/4/314. [19] J.-F. Collet and S. Hariz, A modified version of the Lifshitz-Slyozov model, Appl. Math. Lett., 12 (1999), 81-85.  doi: 10.1016/S0893-9659(2.40)00138-4. [20] J. G. Conlon, On a diffusive version of the Lifschitz–Slyozov–Wagner equation, J. Nonlinear Sci., 20 (2010), 463-521.  doi: 10.1007/s00332-010-9065-y. [21] J. Conlon and M. Guha, Stochastic Variational formulas for linear diffusion equations, Rev. Mat. Iberoam., 30 (2014), 581-666.  doi: 10.4171/RMI/794. [22] D. DeBlassie and R. Smits, The influence of a power law drift on the exit time of Brownian motion from a half-line, Stochastic Process. Appl., 117 (2007), 629-654.  doi: 10.1016/j.spa.2006.09.009. [23] S. Eberle, B. Niethammer and A. Schlichting, Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation, Nonlinear Anal., 158 (2017), 142-167.  doi: 10.1016/j.na.2017.04.009. [24] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Regional Conference Series in Mathematics, Vol 74, Amer. Math. Soc., 1990. doi: 10.1090/cbms/074. [25] L. C. Evans, Partial Differential Equations, Amer. Math. Soc. Graduate Study in Mathematics, 19, 1998, AMS Providence. [26] A. Figalli and N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions, J. Math. Pures Appl., 94 (2010), 107-130.  doi: 10.1016/j.matpur.2009.11.005. [27] M. Friedlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, 260, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3. [28] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., 1964,347 pp. [29] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition, Walter de Gruyter, Berlin, 2011. [30] P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker–Döring equations, J. Differ. Equ., 191 (2003), 518–543. doi: 10.1016/S0022-0396(03)00021-4. [31] P. Laurenccot and S. Mischler, From the Becker–Döring to the Lifshitz–Slyozov–Wagner equations, J. Stat. Phys., 106 (2002), 957-991.  doi: 10.1023/A:1014081619064. [32] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids., 19 (1961), 35-50.  doi: 10.1016/0022-3697(61)90054-3. [33] D. Matthes, A. Jüngel and G. Toscani, Convex Sobolev inequalities derived from entropy dissipation, Arch. Rat. Mech. Anal., 199 (2011), 563-596.  doi: 10.1007/s00205-010-0331-9. [34] J. Morales, A new family of transportation costs with applications to reaction-diffusion and parabolic equations with boundary conditions, J. Math. Pures Appl., (9) 112 (2018), 41–88. doi: 10.1016/j.matpur.2017.12.001. [35] B. Muckenhoupt, Hardy's inequality with weights, Stud. Math., 44 (1972), 31-38.  doi: 10.4064/sm-44-1-31-38. [36] B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), 115-155.  doi: 10.1007/s00332-002-0535-8. [37] B. Niethammer and R. L. Pego, On the initial value problem in the Lifschitz-Slyozov-Wagner theory of Ostwald ripening, SIAM J. Math. Anal., 31 (2000), 467-485.  doi: 10.1137/S0036141098338211. [38] B. Niethammer and R. L. Pego, Well-posedness for measure transport in a family of nonlocal domain coarsening models, Indiana Univ. Math. J., 54 (2005), 499-530.  doi: 10.1512/iumj.2005.54.2598. [39] O. Penrose, The Becker-Döring equations at large times and their connection with the LSW theory of coarsening, J. Stat. Phys., 89 (1997), 305-320.  doi: 10.1007/BF02770767. [40] Y. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen, 1 (1956), 177-238.  doi: 10.1137/1101016. [41] M. Protter and H. Weinberger, Maximum principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5. [42] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅰ. Functional Analysis, Academic Press, New York-London, 1972.  doi: 10.1088/0031-9112/23/12/045. [43] O. S. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal., 64 (1985), 296-313.  doi: 10.1016/0022-1236(2.32)90079-5. [44] A. Schlichting, Macroscopic Limit of the Becker-Döring Equation Via Gradient Flows, ESAIM: COCV, Forthcoming article, 2018. doi: 10.1051/cocv/2018011. [45] S. R. S. Varadhan, Large Deviations and Applications, SIAM, Philadelphia, 1984. doi: 10.1137/1.9781611970241. [46] J. J. L. Velázquez, The Becker-Döring equations and the Lifschitz-Slyozov theory of coarsening, J. Stat. Phys., 92 (1998), 195-236.  doi: 10.1023/A:1023099720145. [47] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, Amer. Math. Soc., Providence R.I., 2003. doi: 10.1090/gsm/058. [48] C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung), Z. Elektrochem., 65 (1961), 581-591. [49] J. A. Walker, Dynamical Systems and Evolution Equations: Theory and Applications., Plenum Press, New York-London, 1980.  doi: 10.1007/978-1-4684-1036-5.

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##### References:
 [1] L. Ambrosio and G. Buttazzo, Weak lower semicontinuous envelope of functionals defined on a space of measures, Ann. Di Mat. Pura Ed Appl., (4) 150 (1988), 311–339. doi: 10.1007/BF01761473. [2] C. Ané, S. Blachère, D. Chafai, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur Les Inégalités de Sobolev Logarithmiques, Panoramas et Synthèses, Société Mathématique de France, 2000. [3] J. M. Ball, J. Carr and O. Penrose, The Becker Döring cluster equations: Basic properties and asymptotic behavior of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070. [4] F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Stud. Math., 159 (2003), 481-497.  doi: 10.4064/sm159-3-9. [5] F. Barthe and C. Roberto, Modified Logarithmic Sobolev Inequalities on $\mathbb{R}$, Potential Anal., 29 (2008), 167-193.  doi: 10.1007/s11118-008-9093-5. [6] R. Becker and W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen, Ann. Der Phys., 24 (1935), 719-752.  doi: 10.1002/andp.19354160806. [7] P. Billingsley, Convergence of Probability Measures, Wiley, 1968, New York-London. [8] S. G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28.  doi: 10.1006/jfan.1998.3326. [9] V. I. Bogachev, Measure Theory, Springer Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-34514-5. [10] V. Bögelein, F. Duzaar and G. Mingione, The boundary regularity of non-linear parabolic systems Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 201–255. doi: 10.1016/j.anihpc.2009.09.003. [11] F. Bolley and C. Villani, Weighted Csiszar-Kullback-Pinsker inequalities and applications to transportation inequalities, Fac. Des Sci. Toulouse, 14 (2005), 331-352.  doi: 10.5802/afst.1095. [12] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, vol. 207 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. [13] J. A. Cañizo, A. Einav and B. Lods, Trend to equilibrium for the Becker-Döring equations: An analogue of Cercignani's conjecture, Anal. PDE., 10 (2017), 1663-1708.  doi: 10.2140/apde.2017.10.1663. [14] J. A. Cañizo, A. Einav and B. Lods, Uniform moment propagation for the Becker-Döring equation, preprint, arXiv: 1706.03524. [15] J. Cañizo and B. Lods, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, Journ. Diff. Eqns., 255 (2013), 905-950.  doi: 10.1016/j.jde.2013.04.031. [16] J.-F. Collet, Some modelling issues in the theory of fragmentation-coagulation systems, Commun. Math. Sci., 2 (2004), 35-54.  doi: 10.4310/CMS.2004.v2.n5.a3. [17] J.-F. Collet, T. Goudon, F. Poupaud and A. Vasseur, The Becker-Döring system and its Lifshitz-Slyozov limit, SIAM J. Appl. Math., 62 (2002), 1488-1500.  doi: 10.1137/S0036139900378852. [18] J.-F. Collet and T. Goudon, On solutions of the Lifshitz-Slyozov model, Nonlinearity, 13 (2000), 1239-1262.  doi: 10.1088/0951-7715/13/4/314. [19] J.-F. Collet and S. Hariz, A modified version of the Lifshitz-Slyozov model, Appl. Math. Lett., 12 (1999), 81-85.  doi: 10.1016/S0893-9659(2.40)00138-4. [20] J. G. Conlon, On a diffusive version of the Lifschitz–Slyozov–Wagner equation, J. Nonlinear Sci., 20 (2010), 463-521.  doi: 10.1007/s00332-010-9065-y. [21] J. Conlon and M. Guha, Stochastic Variational formulas for linear diffusion equations, Rev. Mat. Iberoam., 30 (2014), 581-666.  doi: 10.4171/RMI/794. [22] D. DeBlassie and R. Smits, The influence of a power law drift on the exit time of Brownian motion from a half-line, Stochastic Process. Appl., 117 (2007), 629-654.  doi: 10.1016/j.spa.2006.09.009. [23] S. Eberle, B. Niethammer and A. Schlichting, Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation, Nonlinear Anal., 158 (2017), 142-167.  doi: 10.1016/j.na.2017.04.009. [24] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Regional Conference Series in Mathematics, Vol 74, Amer. Math. Soc., 1990. doi: 10.1090/cbms/074. [25] L. C. Evans, Partial Differential Equations, Amer. Math. Soc. Graduate Study in Mathematics, 19, 1998, AMS Providence. [26] A. Figalli and N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions, J. Math. Pures Appl., 94 (2010), 107-130.  doi: 10.1016/j.matpur.2009.11.005. [27] M. Friedlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, 260, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3. [28] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., 1964,347 pp. [29] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition, Walter de Gruyter, Berlin, 2011. [30] P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker–Döring equations, J. Differ. Equ., 191 (2003), 518–543. doi: 10.1016/S0022-0396(03)00021-4. [31] P. Laurenccot and S. Mischler, From the Becker–Döring to the Lifshitz–Slyozov–Wagner equations, J. Stat. Phys., 106 (2002), 957-991.  doi: 10.1023/A:1014081619064. [32] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids., 19 (1961), 35-50.  doi: 10.1016/0022-3697(61)90054-3. [33] D. Matthes, A. Jüngel and G. Toscani, Convex Sobolev inequalities derived from entropy dissipation, Arch. Rat. Mech. Anal., 199 (2011), 563-596.  doi: 10.1007/s00205-010-0331-9. [34] J. Morales, A new family of transportation costs with applications to reaction-diffusion and parabolic equations with boundary conditions, J. Math. Pures Appl., (9) 112 (2018), 41–88. doi: 10.1016/j.matpur.2017.12.001. [35] B. Muckenhoupt, Hardy's inequality with weights, Stud. Math., 44 (1972), 31-38.  doi: 10.4064/sm-44-1-31-38. [36] B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), 115-155.  doi: 10.1007/s00332-002-0535-8. [37] B. Niethammer and R. L. Pego, On the initial value problem in the Lifschitz-Slyozov-Wagner theory of Ostwald ripening, SIAM J. Math. Anal., 31 (2000), 467-485.  doi: 10.1137/S0036141098338211. [38] B. Niethammer and R. L. Pego, Well-posedness for measure transport in a family of nonlocal domain coarsening models, Indiana Univ. Math. J., 54 (2005), 499-530.  doi: 10.1512/iumj.2005.54.2598. [39] O. Penrose, The Becker-Döring equations at large times and their connection with the LSW theory of coarsening, J. Stat. Phys., 89 (1997), 305-320.  doi: 10.1007/BF02770767. [40] Y. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen, 1 (1956), 177-238.  doi: 10.1137/1101016. [41] M. Protter and H. Weinberger, Maximum principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5. [42] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅰ. Functional Analysis, Academic Press, New York-London, 1972.  doi: 10.1088/0031-9112/23/12/045. [43] O. S. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal., 64 (1985), 296-313.  doi: 10.1016/0022-1236(2.32)90079-5. [44] A. Schlichting, Macroscopic Limit of the Becker-Döring Equation Via Gradient Flows, ESAIM: COCV, Forthcoming article, 2018. doi: 10.1051/cocv/2018011. [45] S. R. S. Varadhan, Large Deviations and Applications, SIAM, Philadelphia, 1984. doi: 10.1137/1.9781611970241. [46] J. J. L. Velázquez, The Becker-Döring equations and the Lifschitz-Slyozov theory of coarsening, J. Stat. Phys., 92 (1998), 195-236.  doi: 10.1023/A:1023099720145. [47] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, Amer. Math. Soc., Providence R.I., 2003. doi: 10.1090/gsm/058. [48] C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung), Z. Elektrochem., 65 (1961), 581-591. [49] J. A. Walker, Dynamical Systems and Evolution Equations: Theory and Applications., Plenum Press, New York-London, 1980.  doi: 10.1007/978-1-4684-1036-5.
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