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April  2021, 14(2): 199-209. doi: 10.3934/krm.2021002

Captivity of the solution to the granular media equation

Julian Tugaut ,

Univ Lyon, Université Jean Monnet, CNRS UMR 5208, Institut Camille Jordan, Maison de l'Université, 10 rue Tréfilerie, CS 82301, 42023 Saint-Átienne Cedex 2, France

* Corresponding author: Julian Tugaut

Received  June 2020 Revised  October 2020 Published  April 2021 Early access  December 2020

The goal of the current paper is to provide assumptions under which the limiting probability of the granular media equation is known when there are several stable states. Indeed, it has been proved in our previous works [17,18] that there is convergence. However, very few is known about the limiting probability, even with a small diffusion coefficient.

Keywords: Granular media equation, Self-stabilizing diffusion, Non uniqueness of the invariant probabilities, Limiting probability.
Mathematics Subject Classification: Primary: 35K55, 60J60; Secondary: 26D10, 60E15, 60F10.
Citation: Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
References:
[1]

D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560.  Google Scholar

[2]

S. Benachour, B. Roynette, D. Talay and P. Vallois, Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75 (1998), 173-201.  doi: 10.1016/S0304-4149(98)00018-0.  Google Scholar

[3]

S. Benachour, B. Roynette and P. Vallois, Nonlinear self-stabilizing processes. II. Convergence to invariant probability, Stochastic Process. Appl., 75 (1998), 203-224.  doi: 10.1016/S0304-4149(98)00019-2.  Google Scholar

[4]

P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields, 140 (2008), 19-40.  doi: 10.1007/s00440-007-0056-3.  Google Scholar

[5]

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

[6]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[7]

Samuel Herrmann, Peter Imkeller and Dierk Peithmann, Large deviations and a Kramers' type law for self-stabilizing diffusions, Ann. Appl. Probab., 18 (2008), 1379-1423.  doi: 10.1214/07-AAP489.  Google Scholar

[8]

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009.  Google Scholar

[9]

S. Herrmann and J. Tugaut, Stationary measures for self-stabilizing processes: Asymptotic analysis in the small noise limit, Electron. J. Probab., 15 (2010), 2087-2116.  doi: 10.1214/EJP.v15-842.  Google Scholar

[10]

S. Herrmann and J. Tugaut, Self-stabilizing processes: Uniqueness problem for stationary measures and convergence rate in the small noise limit, ESAIM Probability and statistics, 16 (2012), 277-305.  doi: 10.1051/ps/2011152.  Google Scholar

[11]

M. Kac, Probability and Related Topics in Physical Sciences, With special lectures by G. E. Uhlenbeck, A. R. Hibbs and B. van der Pol. Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Colo., 1957, Vol. I Interscience Publishers, London-New York 1959.  Google Scholar

[12]

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3.  Google Scholar

[13]

Fl orent Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560.  doi: 10.1214/aoap/1050689593.  Google Scholar

[14]

H. P. McKean. Jr, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[15]

H. P. McKean. Jr, Propagation of chaos for a class of nonlinear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., (1967), 41–57.  Google Scholar

[16]

Yo zo Tamura, On asymptotic behaviors of the solution of a nonlinear diffusion equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1984), 195-221.   Google Scholar

[17]

J. Tugaut, Convergence to the equilibria for self-stabilizing processes in double-well landscape, Ann. Probab., 41 (2013), 1427-1460.  doi: 10.1214/12-AOP749.  Google Scholar

[18]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence, Stochastic Processes and Their Applications, 123 (2013), 1780-1801.  doi: 10.1016/j.spa.2012.12.003.  Google Scholar

[19]

J. Tugaut, Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.  doi: 10.1080/17442508.2013.775287.  Google Scholar

[20]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Invariant probabilities, J. Theoret. Probab., 27 (2014), 57-79.  doi: 10.1007/s10959-012-0435-2.  Google Scholar

[21]

J. Tugaut, Exit-problem of McKean-Vlasov diffusions in double-well landscape, J. Theoret. Probab., 31 (2018), 1013-1023.  doi: 10.1007/s10959-016-0737-x.  Google Scholar

show all references

References:
[1]

D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560.  Google Scholar

[2]

S. Benachour, B. Roynette, D. Talay and P. Vallois, Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75 (1998), 173-201.  doi: 10.1016/S0304-4149(98)00018-0.  Google Scholar

[3]

S. Benachour, B. Roynette and P. Vallois, Nonlinear self-stabilizing processes. II. Convergence to invariant probability, Stochastic Process. Appl., 75 (1998), 203-224.  doi: 10.1016/S0304-4149(98)00019-2.  Google Scholar

[4]

P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields, 140 (2008), 19-40.  doi: 10.1007/s00440-007-0056-3.  Google Scholar

[5]

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

[6]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[7]

Samuel Herrmann, Peter Imkeller and Dierk Peithmann, Large deviations and a Kramers' type law for self-stabilizing diffusions, Ann. Appl. Probab., 18 (2008), 1379-1423.  doi: 10.1214/07-AAP489.  Google Scholar

[8]

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009.  Google Scholar

[9]

S. Herrmann and J. Tugaut, Stationary measures for self-stabilizing processes: Asymptotic analysis in the small noise limit, Electron. J. Probab., 15 (2010), 2087-2116.  doi: 10.1214/EJP.v15-842.  Google Scholar

[10]

S. Herrmann and J. Tugaut, Self-stabilizing processes: Uniqueness problem for stationary measures and convergence rate in the small noise limit, ESAIM Probability and statistics, 16 (2012), 277-305.  doi: 10.1051/ps/2011152.  Google Scholar

[11]

M. Kac, Probability and Related Topics in Physical Sciences, With special lectures by G. E. Uhlenbeck, A. R. Hibbs and B. van der Pol. Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Colo., 1957, Vol. I Interscience Publishers, London-New York 1959.  Google Scholar

[12]

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3.  Google Scholar

[13]

Fl orent Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560.  doi: 10.1214/aoap/1050689593.  Google Scholar

[14]

H. P. McKean. Jr, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[15]

H. P. McKean. Jr, Propagation of chaos for a class of nonlinear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., (1967), 41–57.  Google Scholar

[16]

Yo zo Tamura, On asymptotic behaviors of the solution of a nonlinear diffusion equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1984), 195-221.   Google Scholar

[17]

J. Tugaut, Convergence to the equilibria for self-stabilizing processes in double-well landscape, Ann. Probab., 41 (2013), 1427-1460.  doi: 10.1214/12-AOP749.  Google Scholar

[18]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence, Stochastic Processes and Their Applications, 123 (2013), 1780-1801.  doi: 10.1016/j.spa.2012.12.003.  Google Scholar

[19]

J. Tugaut, Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.  doi: 10.1080/17442508.2013.775287.  Google Scholar

[20]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Invariant probabilities, J. Theoret. Probab., 27 (2014), 57-79.  doi: 10.1007/s10959-012-0435-2.  Google Scholar

[21]

J. Tugaut, Exit-problem of McKean-Vlasov diffusions in double-well landscape, J. Theoret. Probab., 31 (2018), 1013-1023.  doi: 10.1007/s10959-016-0737-x.  Google Scholar

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