# American Institute of Mathematical Sciences

October  2010, 14(3): 907-934. doi: 10.3934/dcdsb.2010.14.907

## Evaporation law in kinetic gravitational systems described by simplified Landau models

 1 IRMAR, Université Rennes 1, Rennes, 35700, France, France, France 2 IRSAMC, Université Paul Sabatier, Toulouse, 31400, France

Received  July 2009 Revised  April 2010 Published  July 2010

This paper is devoted to a mathematical and numerical study of a simplified kinetic model for evaporation phenomena in gravitational systems. This is a first step towards a mathematical understanding of more realistic kinetic models in this area. It is well known in the astrophysics literature that the appropriate kinetic model to describe escape (evaporation) from stars clusters is the so-called Vlasov-Landau-Poisson system with vanishing boundary condition at positive microscopic energies. Since collisions between stars and their self-consistent interactions are both taken into account in this model, its mathematical analysis is difficult, and so far not achieved. Here, as a first step, we focus on a simplified framework of this model and make the following assumptions: i) Only homogenous (space-independent) distributions functions are considered, leading to a collisional kinetic model with a vanishing boundary condition in velocity. ii) The interaction potential involved in the Landau collision operator is of Maxwellian type. iii) The escape velocity (or energy) is supposed to be constant. Using these assumptions, we first establish the well-posedness of the associated Cauchy problem. Then, we focus on the long time behavior of the solution and prove that the energy of the system decreases in time as $O(1/\log(t))$ (logarithmic evaporation), with convergence to a Dirac distribution in velocity when time goes to infinity. Finally, a suitable numerical scheme is constructed for this model and some simulations are performed to illustrate the theoretical study.
Citation: Pierre Carcaud, Pierre-Henri Chavanis, Mohammed Lemou, Florian Méhats. Evaporation law in kinetic gravitational systems described by simplified Landau models. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 907-934. doi: 10.3934/dcdsb.2010.14.907
 [1] Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic & Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333 [2] Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569 [3] Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229 [4] Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049 [5] Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic & Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255 [6] Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations & Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681 [7] Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575 [8] Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109 [9] Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120 [10] D. Blömker, S. Maier-Paape, G. Schneider. The stochastic Landau equation as an amplitude equation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 527-541. doi: 10.3934/dcdsb.2001.1.527 [11] Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 [12] Reiner Henseler, Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. A kinetic model for grain growth. Kinetic & Related Models, 2008, 1 (4) : 591-617. doi: 10.3934/krm.2008.1.591 [13] Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533 [14] Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015 [15] Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic & Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501 [16] Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109 [17] Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895 [18] Hao Zhang, Kai Jiang, Pingwen Zhang. Dynamic transitions for Landau-Brazovskii model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 607-627. doi: 10.3934/dcdsb.2014.19.607 [19] Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715 [20] Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905

2020 Impact Factor: 1.327