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Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model

Abstract : We investigate the connection between two classical modelsof phase transition phenomena, the(discrete size) stochastic Becker-D ̈oring, a continous time Markov chain model, and the (continu-ous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation.For general coefficients and initial data, we introduce a scaling parameter and prove that the em-pirical measure associated to the stochastic Becker-D ̈oring system converges in law to the weaksolution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previousstudies, we use a weak topology that includes the boundary ofthe state space (i.e.the sizex=0)allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of in-coming characteristics. The condition reads limx→0(a(x)u(t)−b(x))f(t,x)=αu(t)2wherefis thevolume distribution function, solution of the Lifshitz-Slyozov equation,aandbthe aggregationand fragmentation rates,uthe concentration of free particles andαa nucleation constant emergingfrom the microscopic model. It is the main novelty of this work and it answers to a question thathas been conjectured or suggested by both mathematicians and physicists. We emphasize that thisboundary value depends on a particular scaling (as opposed to a modeling choice) and is the resultof a separation of time scale and an averaging of fast (fluctuating) variables.We investigate the connection between two classical modelsof phase transition phenomena, the(discrete size) stochastic Becker-D ̈oring, a continous time Markov chain model, and the (continu-ous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation.For general coefficients and initial data, we introduce a scaling parameter and prove that the em-pirical measure associated to the stochastic Becker-D ̈oring system converges in law to the weaksolution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previousstudies, we use a weak topology that includes the boundary ofthe state space (i.e.the sizex=0)allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of in-coming characteristics. The condition reads limx→0(a(x)u(t)−b(x))f(t,x)=αu(t)2wherefis thevolume distribution function, solution of the Lifshitz-Slyozov equation,aandbthe aggregationand fragmentation rates,uthe concentration of free particles andαa nucleation constant emergingfrom the microscopic model. It is the main novelty of this work and it answers to a question thathas been conjectured or suggested by both mathematicians and physicists. We emphasize that thisboundary value depends on a particular scaling (as opposed to a modeling choice) and is the resultof a separation of time scale and an averaging of fast (fluctuating) variables.
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  • HAL Id : hal-02801324, version 1
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Julien Deschamps, Erwan Hingant, Romain Yvinec. Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model. 2015. ⟨hal-02801324⟩

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