# Convex hulls, Sticky particle dynamics and Pressure-less gas system

• [1] Université des Sciences et Techniques de Masuku Faculté des Sciences - Dpt Mathématiques et Informatique BP 943 Franceville, Gabon.
• Volume: 15, Issue: 1, page 57-80
• ISSN: 1259-1734

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## Abstract

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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 ${P}_{0},{u}_{0}$.

## How to cite

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Moutsinga, Octave. "Convex hulls, Sticky particle dynamics and Pressure-less gas system." Annales mathématiques Blaise Pascal 15.1 (2008): 57-80. <http://eudml.org/doc/10553>.

@article{Moutsinga2008,
abstract = {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 $X_\{s+t\} = \phi (X_s,t,P_s,u_s)$ and $\mathrm\{d\}X_t =\mathrm\{E\}[ u_0(X_0)/X_t]\mathrm\{d\}t$, where $P_s=PX_s^\{-1\}$ is the law of $X_s$ and $u_s(x) = \mathrm\{E\}[ u_0(X_0)/X_s = x]$ 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 $(x,t)\mapsto M(x,t):= P(X_t\le x)$ is the entropy solution of a scalar conservation law $\partial _tM + \partial _x(A(M)) = 0$ where the flux $A$ represents the particles momentum, and $\big (P_t,\,u_t,\; t &gt;0\big )$ is a weak solution of the pressure-less gas system of equations of initial datum $P_0, u_0$.},
affiliation = {Université des Sciences et Techniques de Masuku Faculté des Sciences - Dpt Mathématiques et Informatique BP 943 Franceville, Gabon.},
author = {Moutsinga, Octave},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Convex hull; sticky particles; forward flow; stochastic differential equation; scalar conservation law; pressure-less gas system; Hamilton-Jacobi equation},
language = {eng},
month = {1},
number = {1},
pages = {57-80},
publisher = {Annales mathématiques Blaise Pascal},
title = {Convex hulls, Sticky particle dynamics and Pressure-less gas system},
url = {http://eudml.org/doc/10553},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Moutsinga, Octave
TI - Convex hulls, Sticky particle dynamics and Pressure-less gas system
JO - Annales mathématiques Blaise Pascal
DA - 2008/1//
PB - Annales mathématiques Blaise Pascal
VL - 15
IS - 1
SP - 57
EP - 80
AB - 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 $X_{s+t} = \phi (X_s,t,P_s,u_s)$ and $\mathrm{d}X_t =\mathrm{E}[ u_0(X_0)/X_t]\mathrm{d}t$, where $P_s=PX_s^{-1}$ is the law of $X_s$ and $u_s(x) = \mathrm{E}[ u_0(X_0)/X_s = x]$ 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 $(x,t)\mapsto M(x,t):= P(X_t\le x)$ is the entropy solution of a scalar conservation law $\partial _tM + \partial _x(A(M)) = 0$ where the flux $A$ represents the particles momentum, and $\big (P_t,\,u_t,\; t &gt;0\big )$ is a weak solution of the pressure-less gas system of equations of initial datum $P_0, u_0$.
LA - eng
KW - Convex hull; sticky particles; forward flow; stochastic differential equation; scalar conservation law; pressure-less gas system; Hamilton-Jacobi equation
UR - http://eudml.org/doc/10553
ER -

## References

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1. Y. Brenier, E. Grenier, Sticky particles and scalar conservation laws, Siam. J. Numer. Anal. 35 (1998), 2317-2328 Zbl0924.35080MR1655848
2. C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, Journal of Mathematical Analysis and Appl. 38 (1972), 33-41 Zbl0233.35014MR303068
3. A. Dermoune, Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics, C. R. Acad. Sci. Paris tome 5 (1998), 595-599 Zbl0920.60087MR1649309
4. A. Dermoune, O. Moutsinga, Generalized variational principles, Séminaire de Probabilités XXXVI, Lect. Notes in Math. 1801 (2003), 183-193 Zbl1038.60045MR1971585
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. 177 (1996), 349-380 Zbl0852.35097MR1384139
6. O. Moutsinga, Equations de gaz sans pression avec une distribution initiale de Radon, (2002)
7. O. Moutsinga, Probabilistic approch of sticky particles and pressure-less gas system, (2003)
8. Ya. B. Zeldovich, Gravitational instability; an approximation theory for large density perturbations, Astron. Astrophys 5 (1970), 84-89

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