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Zbl 1153.76062
Moutsinga, Octave
Convex hulls, sticky particle dynamics and pressure-less gas system. (English)
[J] Ann. Math. Blaise Pascal 15, No. 1, 57-80 (2008). ISSN 1259-1734

Summary: 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 $dX_{t} = E[u_{0}(X_{0})/X_{t}]dt$, where $P_{s}=PX_{s}^{-1}$ is the law of $X_{s}$ and $u_{s}(x) = E[u_{0}(X_{0})/X_{s} = x]$ is the velocity of particle $x$ at time $s \geq 0$. Results on the flow characterization and Lipschitz continuity are also given. Moreover, the map $(x,t) \mapsto M(x,t) := P(X_{t} \leq x)$ is the entropy solution of a scalar conservation law $\partial _{t}M + \partial _{x}(A(M)) = 0$, where the flux $A$ represents the particles momentum, and $P_{t} , u_{t} , t > 0$ is a weak solution of the pressure-less gas system of equations of initial datum $P_{0},u_{0}$.

MSC 2000:: *76T15 Dusty-gas two-phase flows
76M35 Stochastic analysis
60H30 Appl. of stochastic analysis
60H15 Stochastic partial differential equations

Keywords: stochastic differential equation; scalar conservation law; Hamilton-Jacobi equation

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