-
Previous Article
Diffusion limit of the Vlasov-Poisson-Boltzmann system
- KRM Home
- This Issue
- Next Article
Captivity of the solution to the granular media equation
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 |
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 [
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [19] |
J. Tugaut,
Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.
doi: 10.1080/17442508.2013.775287. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [19] |
J. Tugaut,
Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.
doi: 10.1080/17442508.2013.775287. |
| [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. |
| [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. |
| [1] |
Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 |
| [2] |
Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic and Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201 |
| [3] |
Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151 |
| [4] |
Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 |
| [5] |
Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617 |
| [6] |
Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 |
| [7] |
Fabio Camilli, Lars Grüne. Characterizing attraction probabilities via the stochastic Zubov equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 457-468. doi: 10.3934/dcdsb.2003.3.457 |
| [8] |
Peng Gao. Limiting dynamics for stochastic nonclassical diffusion equations. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021288 |
| [9] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400 |
| [10] |
Yusen Lin, Dingshi Li. Limiting behavior of invariant measures of highly nonlinear stochastic retarded lattice systems. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022054 |
| [11] |
Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007 |
| [12] |
Xiaohua Jing, Masahiro Yamamoto. Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022019 |
| [13] |
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic and Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919 |
| [14] |
Martin Bauer, Thomas Fidler, Markus Grasmair. Local uniqueness of the circular integral invariant. Inverse Problems and Imaging, 2013, 7 (1) : 107-122. doi: 10.3934/ipi.2013.7.107 |
| [15] |
Md. Ibrahim Kholil, Ziqi Sun. A uniqueness theorem for inverse problems in quasilinear anisotropic media. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022008 |
| [16] |
Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56 |
| [17] |
Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 |
| [18] |
Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47 |
| [19] |
Aníbal Rodríguez-Bernal, Alejandro Vidal–López. Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 537-567. doi: 10.3934/dcds.2007.18.537 |
| [20] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]