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Approche probabiliste des particules collantes et système de gaz sans pression

Abstract : At each time $t$, our construction of sticky particle dynamics, with initial mass repartition function $F_0$ and velocities $u_0$, is given by the convex hull $H(\cdot,t)$ of the function $m\in (0, 1)\mapsto \int_a^m\big( F_0^(-1)(z) + tu_0\big(F_0^(-1)(z)\big)\big)dz$. Here, $F_0^(-1)$ is one of two inverse functions of $F_0$. We show that the processes $X_t^-(m)= \partial_m^-H(m,t),\; X_t^+(m) = \partial_m^+H(m,t)$, defined on the probability space $([0, 1], (\cal B), \lambda)$, are not distinguishable, and they model the particle trajectories. The process $X_t=:X_t^-=X_t^+$ is a solution of the equation $(SDE):\;\frac(dX_t)(dt) =\E[ u_0(X_0)/X_t]$, such that $P(X_0 \leq x) = F_0(x)\,\,\forall x$. The inverse $M_t:= M(\cdot,t)$ of the map $m\mapsto \partial_mH(m,t)$ is the mass repartition function at time $t$. It is also the repartition function of the random variable $X_t$. We show the existence of a forward flow map $\big(\phi(x,t,M_s, u_s),\,s < t\big)$ satisfying $X_t = \phi(X_s,t,M_s,u_s)$, where $u_s(x) = \E[ u_0(X_0)/X_s = x]$ is the velocity map of the particles at time $s$. If $\frac(dF_0^n)(dx)$ converges weakly towards $\frac(dF_0)(dx)$, then the flow $\phi(\cdot,\cdot,F_0^n,u_0)$ converges uniformly, on every compact subset of $\R\times\R_+$, towards $\phi(\cdot,\cdot,F_0,u_0)$. We then recover and generalise some results about partial differential equations : the map $(x,t)\mapsto M(x,t)$ is the entropy solution of a scalar conservation law with $F_0$ as initial data, and the family $\big(\rho(dx,t) = P(X_t\in dx),\, u(x,t) = \E[ u_0(X_0)/X_t = x],\,t >0\big)$ is a weak solution of the pressure-less gas system of equations with initial datum $\frac(dF_0(x))(dx), u_0$. This thesis also presents other solutions of the above stochastic differential equation $(SDE)$.
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https://tel.archives-ouvertes.fr/tel-00008721
Contributor : Octave Moutsinga <>
Submitted on : Friday, March 11, 2005 - 10:54:21 AM
Last modification on : Thursday, February 21, 2019 - 10:34:03 AM
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Octave Moutsinga. Approche probabiliste des particules collantes et système de gaz sans pression. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2003. Français. ⟨tel-00008721v2⟩

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