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.
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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/

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