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Convex hulls, sticky particle dynamics and pressure-less gas system. (English) Zbl 1153.76062
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 φ satisfying X s+t =φ(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 s0. Results on the flow characterization and Lipschitz continuity are also given. Moreover, the map (x,t)M(x,t):=P(X t x) is the entropy solution of a scalar conservation law t M+ 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:
76T15Dusty-gas two-phase flows
76M35Stochastic analysis (fluid mechanics)
60H30Applications of stochastic analysis
60H15Stochastic partial differential equations