Convex hulls, Sticky particle dynamics and Pressure-less gas system
Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 1, pp. 57-80.

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities ${u}_{0}$ with negative jumps. We show the existence of a stochastic process and a forward flow $\phi$ satisfying ${X}_{s+t}=\phi \left({X}_{s},t,{P}_{s},{u}_{s}\right)$ and $\mathrm{d}{X}_{t}=\mathrm{E}\left[{u}_{0}\left({X}_{0}\right)/{X}_{t}\right]\mathrm{d}t$, where ${P}_{s}=P{X}_{s}^{-1}$ is the law of ${X}_{s}$ and ${u}_{s}\left(x\right)=\mathrm{E}\left[{u}_{0}\left({X}_{0}\right)/{X}_{s}=x\right]$ is the velocity of particle $x$ at time $s\ge 0$. Results on the flow characterization and Lipschitz continuity are also given.

Moreover, the map $\left(x,t\right)↦M\left(x,t\right):=P\left({X}_{t}\le x\right)$ is the entropy solution of a scalar conservation law ${\partial }_{t}M+{\partial }_{x}\left(A\left(M\right)\right)=0$ where the flux $A$ represents the particles momentum, and $\left({P}_{t},\phantom{\rule{0.166667em}{0ex}}{u}_{t},\phantom{\rule{0.277778em}{0ex}}t>0\right)$ is a weak solution of the pressure-less gas system of equations of initial datum ${P}_{0},{u}_{0}$.

DOI: 10.5802/ambp.239
Classification: 52A10,  52A22,  60G44,  60H10,  60H30
Keywords: Convex hull, sticky particles, forward flow, stochastic differential equation, scalar conservation law, pressure-less gas system, Hamilton-Jacobi equation
Octave Moutsinga 1

1 Université des Sciences et Techniques de Masuku Faculté des Sciences - Dpt Mathématiques et Informatique BP 943 Franceville, Gabon.
@article{AMBP_2008__15_1_57_0,
author = {Octave Moutsinga},
title = {Convex hulls, {Sticky} particle dynamics and {Pressure-less} gas system},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {57--80},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {15},
number = {1},
year = {2008},
doi = {10.5802/ambp.239},
mrnumber = {2418013},
zbl = {1153.76062},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.239/}
}
TY  - JOUR
TI  - Convex hulls, Sticky particle dynamics and Pressure-less gas system
JO  - Annales mathématiques Blaise Pascal
PY  - 2008
DA  - 2008///
SP  - 57
EP  - 80
VL  - 15
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.239/
UR  - https://www.ams.org/mathscinet-getitem?mr=2418013
UR  - https://zbmath.org/?q=an%3A1153.76062
UR  - https://doi.org/10.5802/ambp.239
DO  - 10.5802/ambp.239
LA  - en
ID  - AMBP_2008__15_1_57_0
ER  - 
%0 Journal Article
%T Convex hulls, Sticky particle dynamics and Pressure-less gas system
%J Annales mathématiques Blaise Pascal
%D 2008
%P 57-80
%V 15
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.239
%R 10.5802/ambp.239
%G en
%F AMBP_2008__15_1_57_0
Octave Moutsinga. Convex hulls, Sticky particle dynamics and Pressure-less gas system. Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 1, pp. 57-80. doi : 10.5802/ambp.239. https://ambp.centre-mersenne.org/articles/10.5802/ambp.239/

[1] Y. Brenier; E. Grenier Sticky particles and scalar conservation laws, Siam. J. Numer. Anal., Volume 35 (1998), pp. 2317-2328 (No 6) | DOI | MR | Zbl

[2] C. M. Dafermos Polygonal approximations of solutions of the initial value problem for a conservation law, Journal of Mathematical Analysis and Appl., Volume 38 (1972), pp. 33-41 | DOI | MR | Zbl

[3] A. Dermoune Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics, C. R. Acad. Sci. Paris, Volume tome 5 (1998), pp. 595-599 | MR | Zbl

[4] A. Dermoune; O. Moutsinga Generalized variational principles, Séminaire de Probabilités XXXVI, Lect. Notes in Math., Volume 1801 (2003), pp. 183-193 | EuDML | Numdam | MR | Zbl

[5] W. E; Yu. G. Rykov; Ya. G. Sinai Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Com. Math. Phys., Volume 177 (1996), pp. 349-380 | DOI | MR | Zbl

[6] O. Moutsinga Equations de gaz sans pression avec une distribution initiale de Radon (2002) (Technical report)

[7] O. Moutsinga Probabilistic approch of sticky particles and pressure-less gas system, Univ. Sciences Tech. Lille (2003) (Ph. D. Thesis)

[8] Ya. B. Zeldovich Gravitational instability; an approximation theory for large density perturbations, Astron. Astrophys, Volume 5 (1970), pp. 84-89

Cited by Sources: