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ISRN Mathematical Physics
Volumeย 2012ย (2012), Article IDย 506863, 12 pages
Research Article

Systems of Sticky Particles Governed by Burgers' Equation

Dรฉpartement de Mathรฉmatiques et Informatique, Facultรฉ des Sciences, Universitรฉ des Sciences et Techniques de Masuku, BP 943 Franceville, Gabon

Received 20 October 2011; Accepted 28 November 2011

Academic Editor: G. F.ย Torres del Castillo

Copyright ยฉ 2012 Octave Moutsinga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We show the existence of two sticky particles models with the same velocity function ๐‘ข๐‘ก(๐‘ฅ) which is the entropy solution of the inviscid Burgers' equation. One of them is governed by the set of discontinuity points of ๐‘ข0. Thus, the trajectories ๐‘กโ†ฆ๐‘‹๐‘ก coincide; however one has different mass distributions ๐œ•๐‘ฅ๐‘ข๐‘ก=d๐‘ข0โˆ˜๐‘‹๐‘กโˆ’1 and ๐œ†โˆ˜๐‘‹๐‘กโˆ’1. Here, ๐œ† denotes the Lebesgue measure.

1. Introduction and Main Results

The one-dimensional Burgersโ€™ equation of viscosity ๐œŽโ‰ฅ0 takes the form ๐œ•๐‘ก๐‘ข+๐‘ข๐œ•๐‘ฅ๐‘ข=๐œŽ๐œ•๐‘ฅ๐‘ฅ๐‘ข. It is widely used in the physical literature to model various phenomenon such as shock waves in hydrodynamics turbulence and gas dynamics [1, 2]. A solution of this equation was given by Hopf [3]. When ๐œŽ=0, one gets the inviscid form which is solved by convergence of the solution of the general form when ๐œŽ tends to 0. 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 easy to see the relation between the inviscid Burgersโ€™ equation and the so-called pressure less gas system ๐œ•๐‘ก๐œŒ+๐œ•๐‘ฅ(๐‘ข๐œŒ)=0, ๐œ•๐‘ก(๐‘ข๐œŒ)+๐œ•๐‘ฅ(๐‘ข2๐œŒ)=0 which are simplified forms of the classical Euler equations. Here, ๐‘ข and ๐œŒ are smooth functions. Indeed, a correct derivation of the second equation gives [๐œ•๐‘ก๐‘ข+๐‘ข๐œ•๐‘ฅ๐‘ข]๐œŒ+๐‘ข[๐œ•๐‘ก๐œŒ+๐œ•๐‘ฅ(๐‘ข๐œŒ)]=0. Taking account of the first equation, one gets [๐œ•๐‘ก๐‘ข+๐‘ข๐œ•๐‘ฅ๐‘ข]๐œŒ=0, which leads to ๐œ•๐‘ก๐‘ข+๐‘ข๐œ•๐‘ฅ๐‘ข=0 (of course if ๐œŒโ‰ข0). The above link holds again when ๐œŒ is a field of nonnegative measures, ๐‘ขโˆˆ๐ถ1(โ„ร—โ„โˆ—+) and the derivations are made in the sense of distributions.

In the case of measures, a solution of the gas system was given for example in [4โ€“6], and more recently in [7, 8]. In [5, 7, 8], the pressureless gas system was obtained from the sticky particles model. In these works, ๐‘ข(๐‘ฅ,๐‘ก) is the velocity of the particle which occupies the position ๐‘ฅ at time ๐‘ก, and ๐œŒ(โ‹…,๐‘ก) is the mass distribution of the whole matter at time ๐‘ก. In the latest work of Moutsinga [8], the sticky particle model was constructed when ๐œŒ(โ‹…,0) is any probability measure and ๐‘ข(โ‹…,0) has no positive jump. The author showed that the particles trajectories ๐‘กโ†ฆ๐‘‹๐‘ก(๐‘ฆ) are such that ๐œ•๐‘ก๐‘‹๐‘ก(๐‘ฆ)=๐‘ข(๐‘‹๐‘ก(๐‘ฆ),๐‘ก).

Unfortunately, even if ๐‘ข(โ‹…,0) is continuous, the velocity immediately presents discontinuities in space and time as soon as the first shocks occur. Thus, it is not easy to obtain rigorously Burgersโ€™ equation from the pressure less gas system.

Very recently, Moutsinga [9] showed that if ๐œŒ(โ‹…,0) is the Lebesgue measure ๐œ†, and ๐‘ข(โ‹…,0) has no positive jump, then the velocity field ๐‘ข is the entropy solution of the inviscid Burgersโ€™ equation. The proof was made using the exact expression of the solution given by Hopf [3].

In this this paper, we give again the same result when ๐‘ข(โ‹…,0) is nonincreasing and the initial c.d.f. of the matter is โˆ’๐‘ข(โ‹…,0). We use known results on scalar conservations laws by considering the conservative form of the inviscid Burgersโ€™ equation ๐œ•๐‘ก๐‘ข(๐‘ฅ,๐‘ก)+๐œ•๐‘ฅ๎‚ต๐‘ข2(๐‘ฅ,๐‘ก)2๎‚ถ=0,๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),(๐‘ฅ,๐‘ก)โˆˆโ„ร—โ„+,(1.1) where ๐‘ข0 is a smooth function.

We show that the set of discontinuity points of ๐‘ข0 governs its own sticky particles model whose velocity function is again ๐‘ข(๐‘ฅ,๐‘ก).

It is well known that discontinuity lines of ๐‘ข start on discontinuity points ๐‘ฆ such that ๐‘ข0(๐‘ฆโˆ’0)>๐‘ข0(๐‘ฆ+0) which are the atoms of the measure d๐‘ข0 (see [1] and the illustrations of Section 2.1). For this reason we consider a nonincreasing function ๐‘ข0 and we define โˆ’d๐‘ข0 as the mass initial distribution of a system of particles. We consider each ๐‘ฆโˆˆโ„ as the position of a particle which starts with the mass ๐‘ข0(๐‘ฆโˆ’0)โˆ’๐‘ข0(๐‘ฆ+0) and the velocity ๐‘ฃ0(๐‘ฆ)โˆถ=(๐‘ข0(๐‘ฆโˆ’0)+๐‘ข0(๐‘ฆ+0))2โˆ’1. This allows to interpret ๐‘ข(๐‘ฅ,๐‘ก) as the velocity of a cluster situated at position ๐‘ฅ at time ๐‘ก with the mass ๐‘ข(๐‘ฅโˆ’0,๐‘ก)โˆ’๐‘ข(๐‘ฅ+0,๐‘ก). This fact is the main result of this paper. We recall that the measure d๐‘ข0 is a Radon measure well-defined byd๐‘ข0]((๐‘Ž,๐‘)=๐‘ข0(๐‘+0)โˆ’๐‘ข0(๐‘Ž+0),โˆ€๐‘Ž<๐‘.(1.2)

Theorem 1.1. Let (๐‘ฅ,๐‘ก)โ†ฆ๐‘ข(๐‘ฅ,๐‘ก) be the entropy solution of Burgersโ€™ equation of initial data ๐‘ข0 like above.(1) The measure โˆ’๐œ•๐‘ฅ๐‘ข(โ‹…,๐‘ก) and the function ๐‘ข(โ‹…,๐‘ก) represent, respectively, at time ๐‘ก, the mass distribution and the velocity function of the sticky particles model whose initial mass distribution and velocity function are given by โˆ’d๐‘ข0 and ๐‘ฃ0.(2) There exists a forward flow (๐‘Œ๐‘ ,๐‘ก,๐‘ก,๐‘ โ‰ฅ0) defined on the measure space (โ„,โˆ’d๐‘ข0) such that ๐‘Œ๐‘ ,๐‘ก(๐‘ฅ) is the position, after having browsed the time ๐‘ก of the particle that occupied the position ๐‘ฅ at time ๐‘ . Moreover,d๐‘ข0โˆ˜๐‘Œโˆ’10,๐‘ก=๐œ•๐‘ฅ๐‘Œ๐‘ข(โ‹…,๐‘ก);โˆ€๐‘ก,0,๐‘ +๐‘ก(๐‘ฅ)=๐‘Œ๐‘ ,๐‘ก๎€ท๐‘Œ0,๐‘ ๎€ธ(๐‘ฅ);โˆ€(๐‘ฅ,๐‘ ,๐‘ก)โˆˆโ„ร—โ„+ร—โ„+,๐œ•๐‘Œ๐œ•๐‘ก๐‘ ,๐‘ก๎€ท๐‘Œ=๐‘ข๐‘ ,๐‘ก๎€ธ.;๐‘ +๐‘ก(1.3)(3) Let C๐‘(โ„) be the space of real continuous functions with compact support. For all ๐‘ ,๐‘กโ‰ฅ0,๎€œ๐‘“๎€ท๐‘Œ0,๐‘ก๎€ธ๐‘ข๎€ท๐‘Œ0,๐‘ก๎€ธ,๐‘กd๐‘ข0=๎€œ๐‘“๎€ท๐‘Œ0,๐‘ก๎€ธ๐‘ข0๎€ท๐‘Œ0,0๎€ธd๐‘ข0โˆ€๐‘“โˆˆC๐‘(โ„),d๐‘ข0โˆ˜๐‘Œโˆ’10,๐‘ ๎€ท๐‘Œโˆ’a.e.,๐‘ข๐‘ ,๐‘ก๎€ธ๎€บ,๐‘ +๐‘ก=E๐‘ข(โ‹…,๐‘ )โˆฃ๐‘Œ๐‘ ,๐‘ก๎€ปโ€–โ€–๐‘ขif0โ€–โ€–โˆž<โˆž.(1.4)(4) The function (๐‘ฅ,๐‘ก)โ†ฆ๐‘€(๐‘ฅ,๐‘ก)โˆถ=โˆ’๐‘ข(๐‘ฅ,๐‘ก) is the unique entropy solution of the scalar conservation law๐œ•๐‘ก๐‘€+๐œ•๐‘ฅ(๐ด(๐‘€))=0suchthat๐‘€(โ‹…,0)=โˆ’๐‘ข0,๎€œwiththe๏ฌ‚uxโ„โˆ‹๐‘šโ†ฆ๐ด(๐‘š)=0โˆ’๐‘š๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ(๐‘ง)d๐‘ง.(1.5)
Moreover, ๐œ•๐‘ฅ๐ด(โˆ’๐‘ข(๐‘ฅ,๐‘ก))=โˆ’๐‘ข(๐‘ฅ,๐‘ก)๐œ•๐‘ฅ๐‘ข(๐‘ฅ,๐‘ก)for all ๐‘ก.

Remark that contrary to the model of [9], Burgers equation is recovered here from the above scalar conservation law (1.5), although the flux (representing the momentum) is not a trivial function.

The second result of this paper is an interpretation of some equations from [3], in terms of image measures of the Lebesgue measure ๐œ† by applications defined from two different sticky particles models: the latest flow and the one of [9]. Let us come back to the model of [9] which is a generalization of [7, 8] to the case of infinite total mass of the particles. For fixed (๐‘ฅ,๐‘ก), Lagrange coordinates ๐‘ฆโˆ—(๐‘ฅ,๐‘ก) and ๐‘ฆโˆ—(๐‘ฅ,๐‘ก) were used in [3] in order to get the solution ๐‘ข(๐‘ฅ,๐‘ก) of (1.1). In the model of [9], [๐‘ฆโˆ—(๐‘ฅ,๐‘ก),๐‘ฆโˆ—(๐‘ฅ,๐‘ก)] is the set of all the initial particles (a cluster) that occupy the position ๐‘ฅ at time ๐‘ก, with the mass ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โˆ’๐‘ฆโˆ—(๐‘ฅ,๐‘ก). Moreover,๐‘ข(๐‘ฅโˆ’0,๐‘ก)โˆถ=lim๐‘ฅโ€ฒโ†’๐‘ฅโˆ’๐‘ข๎€ท๐‘ฅ๎…ž๎€ธ,๐‘ก=๐‘กโˆ’1๎€ท๐‘ฅโˆ’๐‘ฆโˆ—๎€ธ,๐‘ข(๐‘ฅ,๐‘ก)(๐‘ฅ+0,๐‘ก)โˆถ=lim๐‘ฅโ€ฒโ†’๐‘ฅ+๐‘ข๎€ท๐‘ฅ๎…ž๎€ธ,๐‘ก=๐‘กโˆ’1๎€ท๐‘ฅโˆ’๐‘ฆโˆ—๎€ธ,(๐‘ฅ,๐‘ก)๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข(๐‘ฅโˆ’0,๐‘ก)+๐‘ข(๐‘ฅ+0,๐‘ก)2=๎€ท๐‘ฆ2๐‘ฅโˆ’โˆ—(๐‘ฅ,๐‘ก)+๐‘ฆโˆ—๎€ธ(๐‘ฅ,๐‘ก).2๐‘ก(1.6) Hence๐‘ฆโˆ—(๐‘ฅ,๐‘ก)+๐‘ฆโˆ—(๐‘ฅ,๐‘ก)2๐œ•+๐‘ก๐‘ข(๐‘ฅ,๐‘ก)=๐‘ฅโˆ€(๐‘ฅ,๐‘ก),๐‘ฅ๐‘ฆโˆ—(โ‹…,๐‘ก)+๐‘ฆโˆ—(โ‹…,๐‘ก)2+๐‘ก๐œ•๐‘ฅ๐‘ข(โ‹…,๐‘ก)=d๐‘ฅโˆถ=๐œ†โˆ€๐‘ก.(1.7)

Proposition 1.2. Suppose that ๐‘ข0 is nonincreasing and lim|๐‘ฅ|โ†’โˆž๐‘ข0(๐‘ฅ)๐‘ฅโˆ’1=0. There exists two forward flows (๐‘ฅ,๐‘ ,๐‘ก)โˆˆโ„ร—โ„+ร—โ„+โ†ฆ๐‘‹๐‘ ,๐‘ก(๐‘ฅ),๐‘Œ๐‘ ,๐‘ก(๐‘ฅ) which model the trajectories of two different sticky particles models, such that for all ๐‘กโ‰ฅ0,(1)๐œ•๐‘ฅ(๐‘ฆโˆ—(โ‹…,๐‘ก)+๐‘ฆโˆ—(โ‹…,๐‘ก))2โˆ’1=๐œ†โˆ˜๐‘‹โˆ’10,๐‘ก;(2)๐œ•๐‘ฅ๐‘ข(โ‹…,๐‘ก)=d๐‘ข0โˆ˜๐‘Œโˆ’10,๐‘ก;(3)if ๐‘ข0 is not constant in any neighborhood of ๐‘ฅ, then for all ๐‘กโ‰ฅ0, ๐‘‹0,๐‘ก(๐‘ฅ)=๐‘Œ0,๐‘ก(๐‘ฅ). So ๐œ†โˆ˜๐‘‹โˆ’10,๐‘ก+๐‘กd๐‘ข0โˆ˜๐‘‹โˆ’10,๐‘ก=๐œ†, which is equivalent to ๐œ†โˆ˜๐‘‹โˆ’10,๐‘ก๎€ท๐‘โˆ’๐‘ก๐œ†โˆ˜0,๐‘ก๎€ธโˆ’1=๐œ†โˆ€๐‘ก,(1.8) with ๐‘0,๐‘กโˆถ=๐‘Œ0,๐‘ก((โˆ’๐‘ข0)โˆ’1).

For any nonnegative Radon measure ๐‘ƒ and any smooth enough function ๐‘ข, it was defined in [9] the sticky particles model of initial mass distribution ๐‘ƒ and initial velocity function ๐‘ข. The state of the particles at each time ๐‘กโ‰ฅ0 is given by the function ๐‘ฆโ†ฆ๐œ™(๐‘ฆ,๐‘ก,๐‘ƒ,๐‘ข). In fact, the flow ๐‘‹ was already given in [9] by๐‘‹๐‘ ,๐‘ก๎‚€(๐‘ฆ)=๐œ™๐‘ฆ,๐‘ก,๐œ†โˆ˜๐‘‹โˆ’10,๐‘ ๎‚,๐‘ข(โ‹…,๐‘ )โˆ€(๐‘ฆ,๐‘ ,๐‘ก)โˆˆโ„ร—โ„+ร—โ„+.(1.9) Then, the first equation of assertion (1) is one of the properties of ๐‘‹ since ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—(๐‘ฅ,๐‘ก) a.e; the second equation is given in Theorem 1.1. The end of assertion (3) comes from the fact that โˆ’d๐‘ข0 is the image measure of ๐œ† by the generalized inverse (โˆ’๐‘ข0)โˆ’1. The coincidence of the two flows is given in Section 2 (Theorem 2.2 where we also precise the link between the two models).

Let ๐’ฎ be the support of d๐‘ข0. We will see in fact that ๐‘Œ is given by๐‘Œ0,๐‘ก๎€ท(๐‘ฆ)=๐œ™๐‘ฆ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธโˆ€(๐‘ฆ,๐‘ก)โˆˆ๐’ฎร—โ„+,๐‘Œ๐‘ ,๐‘ก๎‚€(๐‘ฆ)=๐œ™๐‘ฆ,๐‘ก,โˆ’d๐‘ข0โˆ˜๐‘Œโˆ’10,๐‘ ๎‚,๐‘ข(โ‹…,๐‘ )โˆ€(๐‘ฆ,๐‘ ,๐‘ก)โˆˆ๐‘Œ0,๐‘ (๐’ฎ)ร—โ„โˆ—+ร—โ„+.(1.10)

Before the proof of the Theorem 1.1, we study the link between ๐‘‹ and ๐‘Œ.

2. Comparison of the Two Models

2.1. When a Diffuse Dynamics Hides a Discrete One

In this part, ๐‘ข0 is nonincreasing and stair size.

2.1.1. Example of a Single Shock Wave

We begin with an initial velocity of the type๐‘ข0๎‚ป(๐‘ฆ)=1,if๐‘ฆ<๐‘Ž0,if๐‘ฆโ‰ฅ๐‘Ž.(2.1) In this case, ๐’ฎ={๐‘Ž} and ๐‘Œ0,๐‘ก(๐‘Ž)=๐‘Ž+๐‘ก/2 for all ๐‘กโ‰ฅ0. What about ๐‘‹0,๐‘ก(๐‘Ž)? Let us define ๐‘ฅ=๐‘Œ0,๐‘ก(๐‘Ž). The cluster {๐‘ฆโˆถ๐‘‹0,๐‘ก(๐‘ฆ)=๐‘ฅ}=[๐‘ฆโˆ—(๐‘ฅ,๐‘ก),๐‘ฆโˆ—(๐‘ฅ,๐‘ก)] is the set on which the function ๐‘ฆโ†ฆ๐บ1โˆซ(๐‘ฆ,๐‘ฅ,๐‘ก)=๐‘ฆ0[๐‘ง+๐‘ก๐‘ข0(๐‘ง)]d๐‘ง reaches its absolute lower bound. This is necessarily done in at least one ๐‘ฆ which is a continuity point of ๐‘ข0, and ๐‘ฆ+๐‘ก๐‘ข0(๐‘ฆ)=๐‘ฅ. Suppose that ๐‘ฆ>๐‘Ž. Then ๐‘ฆ=๐‘ฅ, and a simple computation gives๐บ1(๐‘ฆ,๐‘ฅ,๐‘ก)โˆ’๐บ1๎€ท๐‘ฆ1๎€ธ๎€ท๐‘ฆ,๐‘ฅ,๐‘ก=โˆ’1๎€ธ+๐‘กโˆ’๐‘ฅ22,โˆ€๐‘ฆ1<๐‘Ž(2.2) which shows that the lower bound is also reached in ๐‘ฆ1โˆถ=๐‘ฅโˆ’๐‘ก<๐‘Ž. If ๐‘ฆ<๐‘Ž, then ๐‘ฆ=๐‘ฅโˆ’๐‘ก and a simple computation gives๐บ1๎€ท๐‘ฆ2๎€ธ,๐‘ฅ,๐‘กโˆ’๐บ1๎€ท๐‘ฆ(๐‘ฆ,๐‘ฅ,๐‘ก)=2๎€ธโˆ’๐‘ฅ22,โˆ€๐‘ฆ2>๐‘Ž(2.3) which shows that the lower bound is also reached in ๐‘ฆ2โˆถ=๐‘ฅ. Thus, ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก<๐‘Ž<๐‘ฅ=๐‘ฆโˆ—(๐‘ฅ,๐‘ก). Hence,๐‘‹0,๐‘ก(๐‘ฆ)=๐‘Œ0,๐‘ก๐‘ก(๐‘Ž)=๐‘ฅ=๐‘Ž+2๎‚ƒ๐‘ก,โˆ€๐‘ฆโˆˆ๐‘Žโˆ’2๐‘ก,๐‘Ž+2๎‚„.(2.4)

For the dynamics modeled by ๐‘‹, this means that the trajectory of ๐‘Ž is a straight line ๐‘Ž(๐‘ก)=๐‘Ž+๐‘ก/2 along which all the shocks occur continuously. At time ๐‘ก, the particle ๐‘Ž is in a cluster of center ๐‘Ž and of diameter ๐‘ก, with the velocity 1/2.

2.1.2. Stairsize Velocity

By similar arguments as above, one gets the following results when ๐‘ข0 is a stair size function of the type๐‘ข0=๐‘๎“๐‘–=0๐‘ค๐‘–1[๐‘ฆ๐‘–,๐‘ฆ๐‘–+1)withโˆ’โˆž=๐‘ฆ0<๐‘ฆ1<โ‹ฏ<๐‘ฆ๐‘+1=+โˆž,๐‘ค๐‘–>๐‘ค๐‘–+1โˆ€๐‘–.(2.5) Let ๐‘ค be the velocity field of the discrete dynamics constructed from โˆ’d๐‘ข0,๐‘ฃ0.

Proposition 2.1. For all ๐‘ฅโˆˆ๐‘Œ0,๐‘ก(๐’ฎ), there exists a unique couple (๐‘›,๐‘š) such that ๐‘›<๐‘š,๐‘ฆ๐‘›<๐‘ฅโˆ’๐‘ก๐‘ค๐‘›=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)<๐‘ฆ๐‘›+1โ‰ค๐‘ฆ๐‘š<๐‘ฅโˆ’๐‘ก๐‘ค๐‘š=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)<๐‘ฆ๐‘š+1 and for all (๐‘ฆ,๐‘–)โˆˆ[๐‘ฅโˆ’๐‘ก๐‘ค๐‘›,๐‘ฅโˆ’๐‘ก๐‘ค๐‘š]ร—[๐‘›+1,๐‘š]:๐‘ฅ=๐‘‹0,๐‘ก(๐‘ฆ)=๐‘Œ0,๐‘ก๎€ท๐‘ฆ๐‘–๎€ธ=1๐‘ค๐‘›โˆ’๐‘ค๐‘š๐‘š๎“๐‘—=๐‘›+1๎€บ๐‘ค๐‘—โˆ’1โˆ’๐‘ค๐‘—๎€ป๎‚ธ๐‘ฆ๐‘—๐‘ค+๐‘ก๐‘—+๐‘ค๐‘—โˆ’12๎‚น,๐‘ข๐‘ค(๐‘ฅ,๐‘ก)=๐‘ค(๐‘ฅ,๐‘ก)=๐‘›+๐‘ค๐‘š2.(2.6) If ๐‘ฅโˆ‰๐‘Œ0,๐‘ก(๐’ฎ),โˆƒ!(๐‘ฆ,๐‘›)such that ๐‘ฆโˆˆ(๐‘ฆ๐‘›,๐‘ฆ๐‘›+1) and ๐‘ฅ=๐‘‹0,๐‘ก(๐‘ฆ)=๐‘ฆ+๐‘ก๐‘ค๐‘›,๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข0(๐‘ฆ)=๐‘ค๐‘›.(2.7)

These results can be also seen as a simple consequence of the following general case.

2.2. The General Case

The function ๐‘ข0 is nonincreasing and lim|๐‘ฅ|โ†’โˆž๐‘ข0(๐‘ฅ)๐‘ฅโˆ’1=0.From ๐‘ข0 and ๐‘ฃ0(๐‘ฅ)=(๐‘ข0(๐‘ฅโˆ’0)+๐‘ข0(๐‘ฅ+0))2โˆ’1, we have two sticky particles models:(i)each ๐‘ฆโˆˆโ„ is the position of an initial particle which occupies the position ๐œ™(๐‘ฆ,๐‘ก,๐œ†,๐‘ข0)=โˆถ๐‘ฅ at time ๐‘กโ‰ฅ0, with the velocity ๐‘ข(๐‘ฅ,๐‘ก) and the mass ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โˆ’๐‘ฆโˆ—(๐‘ฅ,๐‘ก); (ii)each ๐‘ฆโˆˆ๐’ฎ is the position of an initial particle which occupies the position ๐œ™(๐‘ฆ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0)โˆถ=๐‘ฅ๎…ž at time ๐‘กโ‰ฅ0, with the velocity ๐‘ค(๐‘ฅโ€ฒ,๐‘ก) and the mass ๐‘ข0(๐›ผโˆ’0)โˆ’๐‘ข0(๐›ฝ+0), with [๐›ผ,๐›ฝ]={๐‘ฆโ€ฒโˆถ๐œ™(๐‘ฆโ€ฒ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0)=๐‘ฅ๎…ž}.

For all ๐‘ก,๐œ™(โ„,๐‘ก,๐œ†,๐‘ข0)=โ„. Then, for all ๐‘ฅโˆˆ๐œ™(๐’ฎ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0), there exist clusters ๐’ž1(๐‘ฅ,๐‘ก)=[๐‘ฆโˆ—(๐‘ฅ,๐‘ก),๐‘ฆโˆ—(๐‘ฅ,๐‘ก)],๐’ž2(๐‘ฅ,๐‘ก) such that๐œ™๎€ท๐‘ฆ1,๐‘ก,๐œ†,๐‘ข0๎€ธ๎€ท๐‘ฆ=๐œ™2,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธ๎€ท๐‘ฆ=๐‘ฅโˆ€1,๐‘ฆ2๎€ธโˆˆ๐’ž1(๐‘ฅ,๐‘ก)ร—๐’ž2(๐‘ฅ,๐‘ก).(2.8) For ๐‘ก>0, if we suppose that ๐‘ข=๐‘ค, then both the functions ๐‘ โ†ฆ๐‘ฆ1(๐‘ )โˆถ=๐œ™(๐‘ฆ1,๐‘ก,๐œ†,๐‘ข0), ๐‘ฆ2(๐‘ )โˆถ=๐œ™(๐‘ฆ2,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0) are solutions ofd๐‘ฆ(๐‘ )=๐‘ข(๐‘ฆ(๐‘ ),๐‘ )d๐‘ ,0<๐‘กโ‰ค๐‘ ,๐‘ฆ(๐‘ก)=๐‘ฅ.(2.9) As ๐‘ข satisfies the ล’lenick type entropy condition (see [9])๐‘ข๎€ท๐‘ฅ2๎€ธ๎€ท๐‘ฅ,๐‘ โˆ’๐‘ข1๎€ธ,๐‘ ๐‘ฅ2โˆ’๐‘ฅ1โ‰ค1๐‘ ,โˆ€๐‘ฅ1,๐‘ฅ2,โˆ€๐‘ >0,(2.10) it follows from the results of Fillipov [10] that ๐‘ฆ1(๐‘ )=๐‘ฆ2(๐‘ ), for all ๐‘ โ‰ฅ๐‘ก>0.

Unfortunately, this method fails when ๐‘ก=0. In the sequel, using other arguments, we show indeed that ๐‘ข(โ‹…,๐‘ก)=๐‘ค(โ‹…,๐‘ก) for ๐‘ก>0, and that the trajectories coincide for all ๐‘กโ‰ฅ0. In fact, we show that ๐’ž2(๐‘ฅ,๐‘ก)โŠ‚๐’ž1(๐‘ฅ,๐‘ก).

Remark that the link can be established only for the initial positions ๐‘ฆโˆˆ๐’ฎ,thatis,๐‘ข0 is not constant in any neighborhood of ๐‘ฆ. We recall that๐‘ข(๐‘ฅโˆ’0,๐‘ก)=๐‘กโˆ’1๎€ท๐‘ฅโˆ’๐‘ฆโˆ—๎€ธ(๐‘ฅ,๐‘ก),๐‘ข(๐‘ฅ+0,๐‘ก)=๐‘กโˆ’1๎€ท๐‘ฅโˆ’๐‘ฆโˆ—๎€ธ(๐‘ฅ,๐‘ก),โˆ€(๐‘ฅ,๐‘ก),๐‘ฅ=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)+๐‘ก๐‘ข0๎€ท๐‘ฆโˆ—๎€ธ(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)+๐‘ก๐‘ข0๎€ท๐‘ฆโˆ—๎€ธ.(๐‘ฅ,๐‘ก)(2.11)

Theorem 2.2. Let ๐‘ข be the entropy solution of Burgersโ€™ equation with a nonincreasing initial data ๐‘ข0 such that lim|๐‘ฅ|โ†’โˆž๐‘ข0(๐‘ฅ)๐‘ฅโˆ’1=0. For all (๐‘ฅ,๐‘ก), let us define ๐’ž(๐‘ฅ,๐‘ก)={๐‘ฆ๎…žโˆถ๐œ™(๐‘ฆโ€ฒ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0)=๐‘ฅ}=โˆถ[๐›ผ,๐›ฝ]โˆฉ๐’ฎ.(1) One has ๐‘ข(๐‘ฅโˆ’0,๐‘ก)โ‰ ๐‘ข(๐‘ฅ+0,๐‘ก) if and only if๐‘ข0(๐‘ข๐›ผโˆ’0)>๐‘ข(๐‘ฅ,๐‘ก)=๐‘ค(๐‘ฅ,๐‘ก)=0(๐›ผโˆ’0)+๐‘ข0(๐›ฝ+0)2>๐‘ข0(๐‘ฆ๐›ฝ+0),โˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก๐‘ข0(๐›ผโˆ’0)โ‰ค๐›ผโ‰ค๐›ฝโ‰ค๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก๐‘ข0(๐›ฝ+0).(2.12)(2) If ๐‘ข(๐‘ฅโˆ’0,๐‘ก)=๐‘ข(๐‘ฅ+0,๐‘ก), one has two possibilities:(i)โˆ…โ‰ ๐’ž(๐‘ฅ,๐‘ก) satisfies ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐›ผ=๐›ฝ, ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ค(๐‘ฅ,๐‘ก)=๐‘ข0(๐›ผโˆ’0)=๐‘ข0(๐›ฝ+0)=๐‘ข0(๐›ผ).(2.13)(ii)๐’ž(๐‘ฅ,๐‘ก)=โˆ… and there exist ๐‘Ž<๐‘ such that๐œ™๎€ท๐‘Ž,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธ๎€ท<๐œ™๐‘,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธ,๐‘ข0(๐‘Ž+0)=๐‘ข0(๐‘โˆ’0),๐‘Ž<๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก๐‘ข0(๐‘โˆ’0)<๐‘.(2.14)

Before giving the proof, we remark that if ๐‘ข0 is continuous in ๐‘ง1,๐‘ง2, then ๎€œ๐‘ง2๐‘ง1๎€บ๐œ‚+๐‘ก๐‘ข0๎€ป๎€œ(๐œ‚)๐‘‘๐œ‚=๐‘ก๐‘ง2๐‘ง1๎€บ๐œ‚+๐‘ก๐‘ข0(๐œ‚)๎€ป๎€ทโˆ’๐‘‘๐‘ข0๎€ธ๎€ท๐‘ง(๐œ‚)+2+๐‘ก๐‘ข0(๐‘ง2)๎€ธ2โˆ’๎€ท๐‘ง1+๐‘ก๐‘ข0(๐‘ง1)๎€ธ22.โˆ€๐‘ก(2.15) If ๐‘ข0 is not continuous in ๐‘ง๐‘–, the formula must be corrected with the right term ๐‘ข0(๐‘ง๐‘–โˆ’0) or ๐‘ข0(๐‘ง๐‘–+0), depending on the fact that the integration is closed or not in ๐‘ง๐‘–. This formula is due to the integration by partsโˆ’๎€œ๐‘ง2๐‘ง1๐œ‚d๐‘ข0(๐œ‚)=โˆ’๐‘ง2๐‘ข0๎€ท๐‘ง2๎€ธ+๐‘ง1๐‘ข0๎€ท๐‘ง1๎€ธ+๎€œ๐‘ง2๐‘ง1๐‘ข0โˆ’๎€œ(๐œ‚)d๐œ‚,๐‘ง2๐‘ง1๐‘ฃ0d๐‘ข0=๎€œ๐‘ข0(๐‘ง2)๐‘ข0๎€ท๐‘ง1๎€ธ๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ๎€œ(โˆ’๐‘ง)d๐‘ง=โˆ’๐‘ข0(๐‘ง2)๐‘ข0๎€ท๐‘ง1๎€ธ๐‘ข๐‘งd๐‘ง=0๎€ท๐‘ง1๎€ธ2โˆ’๐‘ข0๎€ท๐‘ง2๎€ธ22.(2.16)

So we get๎€œ๐‘ง2๐‘ง1๎€บ๐œ‚+๐‘ก๐‘ฃ0(๐œ‚)๎€ป๎€ทโˆ’d๐‘ข0๎€ธ๎€œ(๐œ‚)=๐‘ง2๐‘ง1๐‘ข0(๐œ‚)d๐œ‚โˆ’๐‘ง2๐‘ข0๎€ท๐‘ง2๎€ธ+๐‘ง1๐‘ข0๎€ท๐‘ง1๎€ธ๐‘ข+๐‘ก0๎€ท๐‘ง1๎€ธ2โˆ’๐‘ข0๎€ท๐‘ง2๎€ธ22=๎€œ๐‘ง2๐‘ง1๐‘ข0๎€ท๐‘ง(๐œ‚)d๐œ‚+1+๐‘ก๐‘ข0๎€ท๐‘ง1๎€ธ๎€ธ2โˆ’๎€ท๐‘ง2+๐‘ก๐‘ข0๎€ท๐‘ง2๎€ธ๎€ธ2โˆ’๐‘ง21+๐‘ง22=12๐‘ก๐‘ก๎€œ๐‘ง2๐‘ง1๎€บ๐œ‚+๐‘ก๐‘ข0๎€ป๎€ท๐‘ง(๐œ‚)d๐œ‚+1+๐‘ก๐‘ข0๎€ท๐‘ง1๎€ธ๎€ธ2โˆ’๎€ท๐‘ง2+๐‘ก๐‘ข0๎€ท๐‘ง2๎€ธ๎€ธ2.2๐‘ก(2.17)

Proof of Theorem 2.2. (1) When ๐‘ข is discontinuous in (๐‘ฅ,๐‘ก), we have ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)<๐‘ฆโˆ—(๐‘ฅ,๐‘ก), and ๐‘ข(๐‘ฅ,๐‘ก) is defined as the velocity of the cluster [๐‘ฆโˆ—(๐‘ฅ,๐‘ก),๐‘ฆโˆ—(๐‘ฅ,๐‘ก)]:โˆซ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—๐‘ฆ(๐‘ฅ,๐‘ก)โˆ—(๐‘ฅ,๐‘ก)๐‘ข0(๐œ‚)d๐œ‚๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โˆ’๐‘ฆโˆ—.(๐‘ฅ,๐‘ก)(2.18) For ๐‘ฅโˆˆ๐œ™(๐’ฎ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0), the cluster [๐›ผ,๐›ฝ]โˆฉ๐’ฎโˆถ={๐‘ฆ๎…žโˆถ๐œ™(๐‘ฆโ€ฒ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0)=๐‘ฅ} satisfies ๐›ฝ+๐‘ก๐‘ข0(๐›ฝ+0)โ‰ค๐‘ฅโ‰ค๐›ผ+๐‘ก๐‘ข0(๐›ผโˆ’0).(2.19) Let ๐‘ง1โˆถ=๐‘ฅโˆ’๐‘ก๐‘ข0(๐›ผโˆ’0) and suppose that ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โ‰ ๐‘ง1. First, we remark that ๐‘ฅ=๐‘ง1+๐‘ข0(๐‘ง1โˆ’0). It is obvious if ๐‘ง1=๐›ผ. If ๐‘ง1โ‰ ๐›ผ, we have ๐‘ฅ<๐›ผ+๐‘ก๐‘ข0(๐›ผโˆ’0). This occurs only when ๐›ผ is not an accumulation of particles from the left; then there exists another cluster [๐‘Ž,๐‘] such that ๐‘<๐›ผ and d๐‘ข0(๐‘,๐›ผ)=0=๐‘ข0(๐›ผโˆ’0)โˆ’๐‘ข0(๐‘+0),๐‘+๐‘ก๐‘ข0๎€ท(๐‘+0)โ‰ค๐œ™๐‘,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธ<๐‘ฅ<๐›ผ+๐‘ก๐‘ข0(๐›ผโˆ’0).(2.20) Thus ๐‘ฅโˆ’๐‘ก๐‘ข0(๐›ผโˆ’0)=๐‘ง1โˆˆ(๐‘,๐›ผ). As ๐‘ข0 is constant in (๐‘,๐›ผ), we get ๐‘ง1+๐‘ก๐‘ข0(๐‘ง1)=๐‘ฅ. On the other hand, it is easy to see that ๐‘ข0 is continuous in ๐‘ฆโˆ—(๐‘ฅ,๐‘ก),๐‘ฆโˆ—(๐‘ฅ,๐‘ก). Then, we have necessarily ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก๐‘ข0(๐‘ฆโˆ—(๐‘ฅ,๐‘ก))โˆ‰[๐‘,๐›ผ]. If ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)<๐‘, ๐‘ง2โˆถ=๐‘ฆโˆ—(๐‘ฅ,๐‘ก) gives in (2.15): ๎€œโˆ’๐‘ก๐‘ฆ๐›ผโˆ’0โˆ—(๐‘ฅ,๐‘ก)๎€บ๐œ‚+๐‘ก๐‘ฃ0๎€ป(๐œ‚)d๐‘ข0๎€œ(๐œ‚)=โˆ’๐‘ก๐‘ง1๐‘ฆโˆ’0โˆ—(๐‘ฅ,๐‘ก)๎€บ๐œ‚+๐‘ก๐‘ฃ0๎€ป(๐œ‚)d๐‘ข0=๎€œ(๐œ‚)๐‘ง1๐‘ฆโˆ—(๐‘ฅ,๐‘ก)๎€บ๐œ‚+๐‘ก๐‘ข0๎€ป๎€ท(๐œ‚)d๐œ‚=๐บ๐‘ฅ,๐‘ง1๎€ธ๎€ท,๐‘กโˆ’๐บ๐‘ฅ,๐‘ฆโˆ—๎€ธ๎€ท๐‘ง(๐‘ฅ,๐‘ก),๐‘ก+๐‘ฅ1โˆ’๐‘ฆโˆ—๎€ธ.(๐‘ฅ,๐‘ก)(2.21) As ๐‘ง1โˆ’๐‘ฆโˆ—๎€บ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ก0๎€ท๐‘ฆโˆ—๎€ธ(๐‘ฅ,๐‘ก)โˆ’๐‘ข0๎€ป๎€ท(๐›ผโˆ’0)=๐‘กโˆ’d๐‘ข0๐‘ฆ๎€ธ๎€ท๎€บโˆ—(๐‘ฅ,๐‘ก),๐›ผ๎€ธ๎€ธ,(2.22) we get ๎€œ๐‘ฆ๐›ผโˆ’0โˆ—(๐‘ฅ,๐‘ก)๎€บ๐œ‚+๐‘ก๐‘ฃ0(๐œ‚)โˆ’๐‘ฅ๎€ป๎€ทโˆ’d๐‘ข0๎€ธ๐บ๎€ท(๐œ‚)=๐‘ฅ,๐‘ง1๎€ธ๎€ท,๐‘กโˆ’๐บ๐‘ฅ,๐‘ฆโˆ—๎€ธ(๐‘ฅ,๐‘ก),๐‘ก๐‘ก.(2.23) From the definition of ๐›ผ and ๐‘ฆโˆ—(๐‘ฅ,๐‘ก), the fist term is negative and the second is nonnegative. This is absurd, as well as the case ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)>๐›ผ. We can conclude that ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ง1=๐‘ฅโˆ’๐‘ก๐‘ข0(๐›ผโˆ’0). In the same way, ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก๐‘ข0(๐›ฝ+0). Thus ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)<๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โŸบ๐‘ข0(๐›ผโˆ’0)>๐‘ข0(๐›ฝ+0),(2.24) and it comes from (2.15) that ๐‘ข(๐‘ฅ,๐‘ก)=๐‘กโˆ’1๎‚ต๐‘ฆ๐‘ฅโˆ’โˆ—(๐‘ฅ,๐‘ก)+๐‘ฆโˆ—(๐‘ฅ,๐‘ก)2๎‚ถ=๐‘ข0(๐›ผโˆ’0)+๐‘ข0(๐›ฝ+0)2=โˆซ๐›ฝ+0๐›ผโˆ’0๐‘ฃ0๎€ท(๐œ‚)โˆ’d๐‘ข0๎€ธ(๐œ‚)๐‘ข0(๐›ผโˆ’0)โˆ’๐‘ข0(๐›ฝ+0)โˆถ=๐‘ค(๐‘ฅ,๐‘ก).(2.25) (2.i) ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐›ผ=๐›ฝโŸบ๐‘ข0(๐›ผโˆ’0)=๐‘ข0(๐›ฝ+0)=๐‘ข0(๐›ผ),(2.26) and in this case ๐‘ข(๐‘ฅ,๐‘ก)=๐‘กโˆ’1(๐‘ฅโˆ’๐›ผ)=๐‘ข0(๐›ผ)=๐‘ฃ0(๐›ผ)โˆถ=๐‘ค(๐‘ฅ,๐‘ก).(2.27) (2.ii) For ๐‘ฅโˆ‰๐œ™(๐’ฎ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0), there exist ๐‘ฅ1,๐‘ฅ2โˆˆ๐œ™(๐’ฎ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0) and clusters ๎€บ๐›ผ๐‘–,๐›ฝ๐‘–๎€ป๎‚†๐‘ฆโˆฉ๐’ฎโˆถ=โ€ฒ๎€ท๐‘ฆโˆถ๐œ™๎…ž,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธ=๐‘ฅ๐‘–๎‚‡(2.28) such that ๐‘ฅ1<๐‘ฅ<๐‘ฅ2 and d๐‘ข0๎€ท๐›ฝ1,๐›ผ2๎€ธ=0=๐‘ข0๎€ท๐›ผ2๎€ธโˆ’0โˆ’๐‘ข0๎€ท๐›ฝ1๎€ธ+0,๐›ฝ1<๐›ผ2.(2.29) As the function ๐‘ฅ๎…žโ†ฆ๐‘ฆ(๐‘ฅโ€ฒ,๐‘ก) does not decrease, we have ๐‘ฆโˆ—๎€ท๐‘ฅ1๎€ธ,๐‘กโ‰ค๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โ‰ค๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โ‰ค๐‘ฆโˆ—๎€ท๐‘ฅ2๎€ธ,๐›ฝ,๐‘ก1โ‰ค๐‘ฅ1โˆ’๐‘ก๐‘ข0๎€ท๐›ฝ1๎€ธ+0=๐‘ฆโˆ—๎€ท๐‘ฅ1๎€ธ,๐‘กโ‰ค๐‘ฆโˆ—๎€ท๐‘ฅ2๎€ธ,๐‘ก=๐‘ฅ2โˆ’๐‘ก๐‘ข0๎€ท๐›ผ2๎€ธโˆ’0โ‰ค๐›ผ2,(2.30) and this leads to ๐‘ฆโˆ—(๐‘ฅ,๐‘ก),๐‘ฆโˆ—(๐‘ฅ,๐‘ก)โˆˆ[๐›ฝ1,๐›ผ2]. From the fact that ๐‘ข0(๐›ผ2โˆ’0)=๐‘ข0(๐›ฝ1+0) and ๐‘ฆโˆ—(๐‘ฅ,๐‘ก)+๐‘ก๐‘ข0(๐‘ฆโˆ—(๐‘ฅ,๐‘ก))=๐‘ฅ=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)+๐‘ก๐‘ข0(๐‘ฆโˆ—(๐‘ฅ,๐‘ก)), we conclude that ๐›ฝ1<๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฆโˆ—(๐‘ฅ,๐‘ก)=๐‘ฅโˆ’๐‘ก๐‘ข0๎€ท๐›ฝ1๎€ธ+0<๐›ผ2.(2.31)

We have shown that all ๐‘ฆโˆˆ๐’ฎ is the position of two particles which move following two different dynamics given, at time ๐‘ก, by their respective positions, velocities, and masses ๐‘ฅ๐‘–(๐‘ฆ,๐‘ก),๐‘ฃ๐‘–(๐‘ฆ,๐‘ก),๐‘š๐‘–(๐‘ฆ,๐‘ก) (with ๐‘–=1,2) such that๐‘ฅ1๎€ท(๐‘ฆ,๐‘ก)โˆถ=๐œ™๐‘ฆ,๐‘ก,๐œ†,๐‘ข0๎€ธ๎€ท=๐œ™๐‘ฆ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0๎€ธ=โˆถ๐‘ฅ2๐‘ฃ(๐‘ฆ,๐‘ก),โˆ€๐‘ก>0,1(๐‘ฆ,๐‘ก)=๐‘ฃ2(๐‘ฆ,๐‘ก),๐‘š1(๐‘ฆ,๐‘ก)=๐‘ก๐‘š2(๐‘ฆ,๐‘ก).(2.32) However, we had at time zero,๐‘š1(๐‘ฆ,0)=0,๐‘š2(๐‘ฆ,0)=๐‘ข0(๐‘ฆโˆ’0)โˆ’๐‘ข0๐‘ฃ(๐‘ฆ+0),1(๐‘ฆ,0)=๐‘ข0(๐‘ฆ),๐‘ฃ2(๐‘ข๐‘ฆ,0)=0(๐‘ฆโˆ’0)+๐‘ข0(๐‘ฆ+0)2.(2.33)

A surprising fact is that the two dynamics have exactly the same atoms after time zero. The first dynamics is diffuse, and the second one can be discrete (if ๐’ฎ is discrete).

Now we show the connection between the second sticky particles model and the inviscid Burgersโ€™ equation.

Proof of Theorem 1.1. (1) and (4) The function ๐‘ฅโ†ฆ๐‘ฃ0(๐‘ฅ)=(๐‘ข0(๐‘ฅโˆ’0)+๐‘ข0(๐‘ฅ+0))2โˆ’1 is nonincreasing. As in [8] we can define the sticky particles model of initial mass distribution and velocity function โˆ’d๐‘ข0,๐‘ฃ0. For all ๐‘ก, the particles position are given by a continuous nondecreasing function (๐‘ฆ,๐‘ก)โ†ฆ๐œ™(๐‘ฆ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0) well defined on the support of d๐‘ข0 and is extended to โ„ by linear interpolation. In the following, we set ๐œ™(๐‘ฆ,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0)=๐œ™๐‘ก(๐‘ฆ). For any r.v. ๐‘Œ0 having โˆ’d๐‘ข0 as law, define ๐‘Œ๐‘ก=๐œ™(๐‘Œ0,๐‘ก,โˆ’d๐‘ข0,๐‘ฃ0). The velocity field of the particles (๐‘ฆ,๐‘ก)โ†ฆ๐‘ข(๐‘ฆ,๐‘ก) is such that ๐‘ข(๐‘Œ๐‘ก,๐‘ก)=E[๐‘ฃ0(๐‘Œ0)โˆฃ๐‘Œ๐‘ก]=(d/d๐‘ก)๐‘Œ๐‘ก. Here, the derivative holds on the right-hand side, for all ๐‘ก.
At time ๐‘กโ‰ฅ0, consider the c.d.f. ๐น๐‘ก(๐‘ฅ)=โˆ’d๐‘ข0(๐‘Œ๐‘กโ‰ค๐‘ฅ). One has ๐น0=โˆ’๐‘ข0 and its inverse is given by ๐น0โˆ’1(๐‘ง)=๐‘ข0โˆ’1(โˆ’๐‘ง). On ([0,1],โ„ฌ([0,1]),๐œ†), ๐น๐‘กโˆ’1 is a r.v. which is the same law as ๐‘Œ๐‘ก.
For fixed (๐‘ฅ,๐‘ก), let ๐‘€(๐‘ฅ,๐‘ก) be the maximum among the abscissas where the function ๎€œ๐‘šโ†ฆ๐บ(๐‘ฅ,๐‘š,๐‘ก)=๐‘š0๎€บ๐‘ข0โˆ’1(โˆ’๐‘ง)+๐‘ก๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ๎€ป(โˆ’๐‘ง)โˆ’๐‘ฅd๐‘ง(2.34) reaches its absolute lower bound. Using the results of [9], it is clear that the function ๐‘€(โ‹…,๐‘ก) is a c.d.f. of ๐‘Œ0,๐‘ก and it is the entropy solution of๐œ•๐‘ก๐‘€+๐œ•๐‘ฅ(๐ด(๐‘€))=0suchthat๐‘€(๐‘ฅ,0)=โˆ’๐‘ข0๎€œ(๐‘ฅ),withthe๏ฌ‚uxโ„โˆ‹๐‘šโ†ฆ๐ด(๐‘š)=๐‘š0๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ๎€œ(โˆ’๐‘ง)d๐‘ง=0โˆ’๐‘š๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ(๐‘ง)d๐‘ง.(2.35) Defining the at most countable set ๐ท={๐‘ฆโˆˆโ„โˆถ๐‘ข0(๐‘ฆโˆ’0)>๐‘ข0(๐‘ฆ+0)}, and remarking that ๐‘ฃ0(๐‘ฆ)=๐‘ข0(๐‘ฆ)for all ๐‘ฆโˆ‰๐ท, we have ๎€œ๐ด(๐‘š)=๐‘š01{๐‘ข0โˆ’1(โˆ’๐‘ง)โˆˆ๐ท}๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ๎€œ(โˆ’๐‘ง)d๐‘ง+๐‘š01{๐‘ข0โˆ’1(โˆ’๐‘ง)โˆ‰๐ท}๐‘ฃ0๎€ท๐‘ข0โˆ’1๎€ธ=๎“(โˆ’๐‘ง)d๐‘ง๐‘ฆ๐‘›โˆˆ๐ท๎€œ๐‘š01{๐‘ข0โˆ’1(โˆ’๐‘ง)=๐‘ฆ๐‘›}๐‘ข0๎€ท๐‘ฆ๐‘›๎€ธโˆ’0+๐‘ข0๎€ท๐‘ฆ๐‘›๎€ธ+02๎€œd๐‘ง+๐‘š01{๐‘ข0โˆ’1(โˆ’๐‘ง)โˆ‰๐ท}๐‘ข0๎€ท๐‘ข0โˆ’1๎€ธ(โˆ’๐‘ง)d๐‘ง.(2.36) As this c.d.f. takes its values in โˆ’๐‘ข0(โ„), one should compute the values of the flux for ๐‘šโˆˆโˆ’๐‘ข0(โ„). Without loss of generality, one can suppose that 0 is a value of continuity of ๐‘ข0. Thus, for ๐‘ฆ๐‘›โˆˆ๐ทthere are only two possibilities: []โˆฉ๎€ฝ0,๐‘š๐‘งโˆถ๐‘ข0โˆ’1(โˆ’๐‘ง)=๐‘ฆ๐‘›๎€พ๎€ฝ=โˆ…or๐‘งโˆถ๐‘ข0โˆ’1(โˆ’๐‘ง)=๐‘ฆ๐‘›๎€พโŠ‚[].0,๐‘š(2.37) As {๐‘งโˆถ๐‘ข0โˆ’1(โˆ’๐‘ง)=๐‘ฆ๐‘›}=[โˆ’๐‘ข(๐‘ฆ๐‘›โˆ’0),โˆ’๐‘ข(๐‘ฆ๐‘›+0)], we have ๎€œ1{๐‘ข0โˆ’1(โˆ’๐‘ง)=๐‘ฆ๐‘›}๐‘ข0๎€ท๐‘ฆ๐‘›๎€ธโˆ’0+๐‘ข0๎€ท๐‘ฆ๐‘›๎€ธ+02๐‘ขd๐‘ง=0๎€ท๐‘ฆ๐‘›๎€ธโˆ’02โˆ’๐‘ข0๎€ท๐‘ฆ๐‘›๎€ธ+022๎€œ1=โˆ’{๐‘ข0โˆ’1(โˆ’๐‘ง)=๐‘ฆ๐‘›}๐‘งd๐‘ง(2.38) and we get ๎€œ๐ด(๐‘š)=โˆ’๐‘š01{๐‘ข0โˆ’1(โˆ’๐‘ง)โˆˆ๐ท}๎€œ๐‘งd๐‘งโˆ’๐‘š01{๐‘ข0โˆ’1(โˆ’๐‘ง)โˆ‰๐ท}๎€œ๐‘งd๐‘ง=โˆ’๐‘š0๐‘š๐‘งd๐‘ง=โˆ’22.(2.39) Then, โˆ’๐‘€(๐‘ฅ,๐‘ก)=๐‘ข(๐‘ฅ,๐‘ก) is the entropy solution of Burgersโ€™ equation with initial data ๐‘ข0. Thus,โˆ’d๐‘ข0โˆ˜(๐‘Œ0,๐‘ก)โˆ’1=๐œ•๐‘ฅ๐‘€(๐‘ฅ,๐‘ก)=โˆ’๐œ•๐‘ฅ๐‘ข(๐‘ฅ,๐‘ก).
The fact that ๐‘ข(๐‘ฅ,๐‘ก) is the velocity (for ๐‘ก>0) was already given in Theorem 2.2. We give another proof here. For each discontinuity point (๐‘ฅ,๐‘ก) of ๐‘ข, ๐‘ฅ is the position, at time ๐‘ก, of a massive particle; so its velocity is ๐ด(โˆ’๐‘ข(๐‘ฅ+0,๐‘ก))โˆ’๐ด(โˆ’๐‘ข(๐‘ฅโˆ’0,๐‘ก))=โˆ’๐‘ข(๐‘ฅ+0,๐‘ก)+๐‘ข(๐‘ฅโˆ’0,๐‘ก)๐‘ข(๐‘ฅโˆ’0,๐‘ก)+๐‘ข(๐‘ฅ+0,๐‘ก)2=๐‘ข(๐‘ฅ,๐‘ก).(2.40) If ๐‘ข is continuous in (๐‘ฅ,๐‘ก) and ๐‘ฅ is the position at time ๐‘ก, then it is the position of a cluster {๐‘Ž} which moves with its initial velocity ๐‘ฃ0(๐‘Ž). We have the equalities of masses ๐‘€(๐‘ฅโˆ’0,๐‘ก)=๐‘€(๐‘ฅ+0,๐‘ก)=๐น0(๐‘Žโˆ’0)=๐น0(๐‘Ž+0). This means that ๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข0(๐‘Ž)=๐‘ฃ0(๐‘Ž). The velocity of this particle is then ๐‘ค(๐‘ฅ,๐‘ก)=๐‘ฃ0(๐‘Ž)=๐‘ข(๐‘ฅ,๐‘ก).
(2) and (3) These points are given by properties of ๐œ™ (see [9]). The condition โ€–๐‘ข0โ€–โˆž<+โˆž means ๐‘ข0โˆˆ๐ฟ1(d๐‘ข0).


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