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Abstract
We show the existence of two sticky particles models with the same velocity function ut(x) which is the entropy solution of the inviscid Burgers' equation. One of them is governed by the set of discontinuity points of u0. Thus, the trajectories t↦Xt coincide; however one has different mass distributions ∂xut=du0∘Xt-1 and λ∘Xt-1. Here, λ denotes the Lebesgue measure.
Highlights
Introduction and Main ResultsThe one-dimensional Burgers’ equation of viscosity σ ≥ 0 takes the form ∂tu u∂xu σ∂xxu
It is widely used in the physical literature to model various phenomenon such as shock waves in hydrodynamics turbulence and gas dynamics 1, 2
We show that the set of discontinuity points of u0 governs its own sticky particles model whose velocity function is again u x, t
Summary
The one-dimensional Burgers’ equation of viscosity σ ≥ 0 takes the form ∂tu u∂xu σ∂xxu. It is well known that the entropy solution of the inviscid equation is interpreted as the velocity function of some sticky particles model 4, 5 , but this link was shown only for continuous initial data, and the connection with the trajectories is still unknown. It is well known that discontinuity lines of u start on discontinuity points y such that u0 y − 0 > u0 y 0 which are the atoms of the measure du[0] see 1 and the illustrations of Section 2.1 For this reason we consider a nonincreasing function u0 and we define −du[0] as the mass initial distribution of a system of particles. The second result of this paper is an interpretation of some equations from 3 , in terms of image measures of the Lebesgue measure λ by applications defined from two different sticky particles models: the latest flow and the one of 9.
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