# Systems of Sticky Particles Governed by Burgers' Equation

Authors: Octave Moutsinga
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

We show the existence of two sticky particles models with the same
velocity functionut(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.


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## Body

## 1. Introduction and Main Results

The one-dimensional Burgers’ equation of viscosityσ≥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]. A solution of this equation was given by Hopf [3]. When σ=0, one gets the inviscid form which is solved by convergence of the solution of the general form when σ 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 Burgers’ equation and the so-called pressure less gas system∂tρ+∂x(uρ)=0, ∂t(uρ)+∂x(u2ρ)=0 which are simplified forms of the classical Euler equations. Here, u and ρ are smooth functions. Indeed, a correct derivation of the second equation gives [∂tu+u∂xu]ρ+u[∂tρ+∂x(uρ)]=0. Taking account of the first equation, one gets [∂tu+u∂xu]ρ=0, which leads to ∂tu+u∂xu=0 (of course if ρ≢0). The above link holds again when ρ is a field of nonnegative measures, u∈C1(ℝ×ℝ+*) 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 [4–6], and more recently in [7, 8]. In [5, 7, 8], the pressureless gas system was obtained from the sticky particles model. In these works, u(x,t) is the velocity of the particle which occupies the position x at time t, and ρ(·,t) is the mass distribution of the whole matter at time t. In the latest work of Moutsinga [8], the sticky particle model was constructed when ρ(·,0) is any probability measure and u(·,0) has no positive jump. The author showed that the particles trajectories t↦Xt(y) are such that ∂tXt(y)=u(Xt(y),t).Unfortunately, even ifu(·,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 Burgers’ equation from the pressure less gas system.Very recently, Moutsinga [9] showed that if ρ(·,0) is the Lebesgue measure λ, and u(·,0) has no positive jump, then the velocity field u is the entropy solution of the inviscid Burgers’ 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 whenu(·,0) is nonincreasing and the initial c.d.f. of the matter is -u(·,0). We use known results on scalar conservations laws by considering the conservative form of the inviscid Burgers’ equation (1.1)∂tu(x,t)+∂x(u2(x,t)2)=0,u(x,0)=u0(x),(x,t)∈R×R+,
where u0 is a smooth function.We show that the set of discontinuity points ofu0 governs its own sticky particles model whose velocity function is again u(x,t).It is well known that discontinuity lines ofu start on discontinuity points y such that u0(y-0)>u0(y+0) which are the atoms of the measure du0 (see [1] and the illustrations of Section 2.1). For this reason we consider a nonincreasing function u0 and we define -du0 as the mass initial distribution of a system of particles. We consider each y∈ℝ as the position of a particle which starts with the mass u0(y-0)-u0(y+0) and the velocity v0(y):=(u0(y-0)+u0(y+0))2-1. This allows to interpret u(x,t) as the velocity of a cluster situated at position x at time t with the mass u(x-0,t)-u(x+0,t). This fact is the main result of this paper. We recall that the measure du0 is a Radon measure well-defined by(1.2)du0((a,b])=u0(b+0)-u0(a+0),∀a<b.Theorem 1.1.
Let(x,t)↦u(x,t) be the entropy solution of Burgers’ equation of initial data u0 like above.(1)
The measure-∂xu(·,t) and the function u(·,t) represent, respectively, at time t, the mass distribution and the velocity function of the sticky particles model whose initial mass distribution and velocity function are given by -du0 and v0.(2)
There exists a forward flow(Ys,t,t,s≥0) defined on the measure space (ℝ,-du0) such that Ys,t(x) is the position, after having browsed the time t of the particle that occupied the position x at time s. Moreover,(1.3)du0∘Y0,t-1=∂xu(⋅,t);∀t,Y0,s+t(x)=Ys,t(Y0,s(x));∀(x,s,t)∈R×R+×R+,∂∂tYs,t=u(Ys,t;s+t).(3)
LetCc(ℝ) be the space of real continuous functions with compact support. For all s,t≥0,(1.4)∫f(Y0,t)u(Y0,t,t)du0=∫f(Y0,t)u0(Y0,0)du0∀f∈Cc(R),du0∘Y0,s-1-a.e.,u(Ys,t,s+t)=E[u(⋅,s)∣Ys,t]if‖u0‖∞<∞.(4)
The function(x,t)↦M(x,t):=-u(x,t) is the unique entropy solution of the scalar conservation law(1.5)∂tM+∂x(A(M))=0suchthatM(⋅,0)=-u0,withthefluxR∋m↦A(m)=∫-m0v0(u0-1(z))dz.
Moreover,∂xA(-u(x,t))=-u(x,t)∂xu(x,t)for all t.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 λ 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 (x,t), Lagrange coordinates y*(x,t) and y*(x,t) were used in [3] in order to get the solution u(x,t) of (1.1). In the model of [9], [y*(x,t),y*(x,t)] is the set of all the initial particles (a cluster) that occupy the position x at time t, with the mass y*(x,t)-y*(x,t). Moreover,(1.6)u(x-0,t):=lim⁡x′→x-u(x′,t)=t-1(x-y*(x,t)),u(x+0,t):=lim⁡x′→x+u(x′,t)=t-1(x-y*(x,t)),u(x,t)=u(x-0,t)+u(x+0,t)2=2x-(y*(x,t)+y*(x,t))2t.
Hence(1.7)y*(x,t)+y*(x,t)2+tu(x,t)=x∀(x,t),∂xy*(⋅,t)+y*(⋅,t)2+t∂xu(⋅,t)=dx:=λ∀t.Proposition 1.2.
Suppose thatu0 is nonincreasing and lim|x|→∞u0(x)x-1=0. There exists two forward flows (x,s,t)∈ℝ×ℝ+×ℝ+↦Xs,t(x),Ys,t(x) which model the trajectories of two different sticky particles models, such that for all t≥0,(1)
∂x(y*(·,t)+y*(·,t))2-1=λ∘X0,t-1;(2)
∂xu(·,t)=du0∘Y0,t-1;(3)
ifu0 is not constant in any neighborhood of x, then for all t≥0, X0,t(x)=Y0,t(x). So λ∘X0,t-1+tdu0∘X0,t-1=λ, which is equivalent to
(1.8)λ∘X0,t-1-tλ∘(Z0,t)-1=λ∀t,
with Z0,t:=Y0,t((-u0)-1).For any nonnegative Radon measureP and any smooth enough function u, it was defined in [9] the sticky particles model of initial mass distribution P and initial velocity function u. The state of the particles at each time t≥0 is given by the function y↦ϕ(y,t,P,u). In fact, the flow X was already given in [9] by(1.9)Xs,t(y)=ϕ(y,t,λ∘X0,s-1,u(⋅,s))∀(y,s,t)∈R×R+×R+.
Then, the first equation of assertion (1) is one of the properties of X since y*(x,t)=y*(x,t) a.e; the second equation is given in Theorem 1.1. The end of assertion (3) comes from the fact that -du0 is the image measure of λ by the generalized inverse (-u0)-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).Let𝒮 be the support of du0. We will see in fact that Y is given by(1.10)Y0,t(y)=ϕ(y,t,-du0,v0)∀(y,t)∈S×R+,Ys,t(y)=ϕ(y,t,-du0∘Y0,s-1,u(⋅,s))∀(y,s,t)∈Y0,s(S)×R+*×R+.Before the proof of the Theorem1.1, we study the link between X and Y.


## 2. Comparison of the Two Models

### 2.1. When a Diffuse Dynamics Hides a Discrete One

In this part,u0 is nonincreasing and stair size.


#### 2.1.1. Example of a Single Shock Wave

We begin with an initial velocity of the type(2.1)u0(y)={1,ify<a0,ify≥a.
In this case, 𝒮={a} and Y0,t(a)=a+t/2 for all t≥0. What about X0,t(a)? Let us define x=Y0,t(a). The cluster {y:X0,t(y)=x}=[y*(x,t),y*(x,t)] is the set on which the function y↦G1(y,x,t)=∫0y[z+tu0(z)]dz reaches its absolute lower bound. This is necessarily done in at least one y which is a continuity point of u0, and y+tu0(y)=x. Suppose that y>a. Then y=x, and a simple computation gives(2.2)G1(y,x,t)-G1(y1,x,t)=-(y1+t-x)22,∀y1<a
which shows that the lower bound is also reached in y1:=x-t<a. If y<a, then y=x-t and a simple computation gives(2.3)G1(y2,x,t)-G1(y,x,t)=(y2-x)22,∀y2>a
which shows that the lower bound is also reached in y2:=x. Thus, y*(x,t)=x-t<a<x=y*(x,t). Hence,(2.4)X0,t(y)=Y0,t(a)=x=a+t2,∀y∈[a-t2,a+t2].For the dynamics modeled byX, this means that the trajectory of a is a straight line a(t)=a+t/2 along which all the shocks occur continuously. At time t, the particle a is in a cluster of center a and of diameter t, with the velocity 1/2.


#### 2.1.2. Stairsize Velocity

By similar arguments as above, one gets the following results whenu0 is a stair size function of the type(2.5)u0=∑i=0Nwi1[yi,yi+1)with-∞=y0<y1<⋯<yN+1=+∞,wi>wi+1∀i.
Let w be the velocity field of the discrete dynamics constructed from -du0,v0.Proposition 2.1.
For allx∈Y0,t(𝒮), there exists a unique couple (n,m) such that n<m,yn<x-twn=y*(x,t)<yn+1≤ym<x-twm=y*(x,t)<ym+1 and for all (y,i)∈[x-twn,x-twm]×[n+1,m]:(2.6)x=X0,t(y)=Y0,t(yi)=1wn-wm∑j=n+1m[wj-1-wj][yj+twj+wj-12],u(x,t)=w(x,t)=wn+wm2.
If x∉Y0,t(𝒮),∃!(y,n)such that y∈(yn,yn+1) and
(2.7)x=X0,t(y)=y+twn,u(x,t)=u0(y)=wn.These results can be also seen as a simple consequence of the following general case.


### 2.2. The General Case

The functionu0 is nonincreasing and lim|x|→∞u0(x)x-1=0.From u0 and v0(x)=(u0(x-0)+u0(x+0))2-1, we have two sticky particles models:(i)
eachy∈ℝ is the position of an initial particle which occupies the position ϕ(y,t,λ,u0)=:x at time t≥0, with the velocity u(x,t) and the mass y*(x,t)-y*(x,t);(ii)
eachy∈𝒮 is the position of an initial particle which occupies the position ϕ(y,t,-du0,v0):=x′ at time t≥0, with the velocity w(x′,t) and the mass u0(α-0)-u0(β+0), with [α,β]={y′:ϕ(y′,t,-du0,v0)=x′}.For allt,ϕ(ℝ,t,λ,u0)=ℝ. Then, for all x∈ϕ(𝒮,t,-du0,v0), there exist clusters 𝒞1(x,t)=[y*(x,t),y*(x,t)],𝒞2(x,t) such that(2.8)ϕ(y1,t,λ,u0)=ϕ(y2,t,-du0,v0)=x∀(y1,y2)∈C1(x,t)×C2(x,t).
For t>0, if we suppose that u=w, then both the functions s↦y1(s):=ϕ(y1,t,λ,u0), y2(s):=ϕ(y2,t,-du0,v0) are solutions of(2.9)dy(s)=u(y(s),s)ds,0<t≤s,y(t)=x.
As u satisfies the Œlenick type entropy condition (see [9])(2.10)u(x2,s)-u(x1,s)x2-x1≤1s,∀x1,x2,∀s>0,
it follows from the results of Fillipov [10] that y1(s)=y2(s), for all s≥t>0.Unfortunately, this method fails whent=0. In the sequel, using other arguments, we show indeed that u(·,t)=w(·,t) for t>0, and that the trajectories coincide for all t≥0. In fact, we show that 𝒞2(x,t)⊂𝒞1(x,t).Remark that the link can be established only for the initial positionsy∈𝒮,thatis,u0 is not constant in any neighborhood of y. We recall that(2.11)u(x-0,t)=t-1(x-y*(x,t)),u(x+0,t)=t-1(x-y*(x,t)),∀(x,t),x=y*(x,t)+tu0(y*(x,t))=y*(x,t)+tu0(y*(x,t)).Theorem 2.2.
Letu be the entropy solution of Burgers’ equation with a nonincreasing initial data u0 such that lim|x|→∞u0(x)x-1=0. For all (x,t), let us define 𝒞(x,t)={y′:ϕ(y′,t,-du0,v0)=x}=:[α,β]∩𝒮.(1)
One hasu(x-0,t)≠u(x+0,t) if and only if(2.12)u0(α-0)>u(x,t)=w(x,t)=u0(α-0)+u0(β+0)2>u0(β+0),y*(x,t)=x-tu0(α-0)≤α≤β≤y*(x,t)=x-tu0(β+0).(2)
Ifu(x-0,t)=u(x+0,t), one has two possibilities:
(i)
∅≠𝒞(x,t) satisfies y*(x,t)=y*(x,t)=α=β,
(2.13)u(x,t)=w(x,t)=u0(α-0)=u0(β+0)=u0(α).(ii)
𝒞(x,t)=∅ and there exist a<b such that(2.14)ϕ(a,t,-du0,v0)<ϕ(b,t,-du0,v0),u0(a+0)=u0(b-0),a<y*(x,t)=y*(x,t)=x-tu0(b-0)<b.Before giving the proof, we remark that ifu0 is continuous in z1,z2, then (2.15)∫z1z2[η+tu0(η)]dη=t∫z1z2[η+tu0(η)](-du0)(η)+(z2+tu0(z2))2-(z1+tu0(z1))22.∀t
If u0 is not continuous in zi, the formula must be corrected with the right term u0(zi-0) or u0(zi+0), depending on the fact that the integration is closed or not in zi. This formula is due to the integration by parts(2.16)-∫z1z2ηdu0(η)=-z2u0(z2)+z1u0(z1)+∫z1z2u0(η)dη,-∫z1z2v0du0=∫u0(z1)u0(z2)v0(u0-1(-z))dz=-∫u0(z1)u0(z2)zdz=u0(z1)2-u0(z2)22.So we get(2.17)∫z1z2[η+tv0(η)](-du0)(η)=∫z1z2u0(η)dη-z2u0(z2)+z1u0(z1)+tu0(z1)2-u0(z2)22=∫z1z2u0(η)dη+(z1+tu0(z1))2-(z2+tu0(z2))2-z12+z222t=1t∫z1z2[η+tu0(η)]dη+(z1+tu0(z1))2-(z2+tu0(z2))22t.Proof of Theorem2.2.
(1) When u is discontinuous in (x,t), we have y*(x,t)<y*(x,t), and u(x,t) is defined as the velocity of the cluster [y*(x,t),y*(x,t)]:(2.18)u(x,t)=∫y*(x,t)y*(x,t)u0(η)dηy*(x,t)-y*(x,t).
For x∈ϕ(𝒮,t,-du0,v0), the cluster [α,β]∩𝒮:={y′:ϕ(y′,t,-du0,v0)=x} satisfies
(2.19)β+tu0(β+0)≤x≤α+tu0(α-0).
Let z1:=x-tu0(α-0) and suppose that y*(x,t)≠z1. First, we remark that x=z1+u0(z1-0). It is obvious if z1=α. If z1≠α, we have x<α+tu0(α-0). This occurs only when α is not an accumulation of particles from the left; then there exists another cluster [a,b] such that b<α and
(2.20)du0(b,α)=0=u0(α-0)-u0(b+0),b+tu0(b+0)≤ϕ(b,t,-du0,v0)<x<α+tu0(α-0).
Thus x-tu0(α-0)=z1∈(b,α). As u0 is constant in (b,α), we get z1+tu0(z1)=x. On the other hand, it is easy to see that u0 is continuous in y*(x,t),y*(x,t). Then, we have necessarily y*(x,t)=x-tu0(y*(x,t))∉[b,α]. If y*(x,t)<b, z2:=y*(x,t) gives in (2.15):
(2.21)-t∫y*(x,t)α-0[η+tv0(η)]du0(η)=-t∫y*(x,t)z1-0[η+tv0(η)]du0(η)=∫y*(x,t)z1[η+tu0(η)]dη=G(x,z1,t)-G(x,y*(x,t),t)+x(z1-y*(x,t)).
As
(2.22)z1-y*(x,t)=t[u0(y*(x,t))-u0(α-0)]=t(-du0)([y*(x,t),α)),
we get
(2.23)∫y*(x,t)α-0[η+tv0(η)-x](-du0)(η)=G(x,z1,t)-G(x,y*(x,t),t)t.
From the definition of α and y*(x,t), the fist term is negative and the second is nonnegative. This is absurd, as well as the case y*(x,t)>α. We can conclude that y*(x,t)=z1=x-tu0(α-0). In the same way, y*(x,t)=x-tu0(β+0). Thus
(2.24)y*(x,t)<y*(x,t)⟺u0(α-0)>u0(β+0),
and it comes from (2.15) that
(2.25)u(x,t)=t-1(x-y*(x,t)+y*(x,t)2)=u0(α-0)+u0(β+0)2=∫α-0β+0v0(η)(-du0)(η)u0(α-0)-u0(β+0):=w(x,t).
(2.i)
(2.26)y*(x,t)=y*(x,t)=α=β⟺u0(α-0)=u0(β+0)=u0(α),
and in this case
(2.27)u(x,t)=t-1(x-α)=u0(α)=v0(α)≔w(x,t).
(2.ii) For x∉ϕ(𝒮,t,-du0,v0), there exist x1,x2∈ϕ(𝒮,t,-du0,v0) and clusters
(2.28)[αi,βi]∩S:={y′:ϕ(y′,t,-du0,v0)=xi}
such that x1<x<x2 and
(2.29)du0(β1,α2)=0=u0(α2-0)-u0(β1+0),β1<α2.
As the function x′↦y(x′,t) does not decrease, we have
(2.30)y*(x1,t)≤y*(x,t)≤y*(x,t)≤y*(x2,t),β1≤x1-tu0(β1+0)=y*(x1,t)≤y*(x2,t)=x2-tu0(α2-0)≤α2,
and this leads to y*(x,t),y*(x,t)∈[β1,α2]. From the fact that u0(α2-0)=u0(β1+0) and y*(x,t)+tu0(y*(x,t))=x=y*(x,t)+tu0(y*(x,t)), we conclude that
(2.31)β1<y*(x,t)=y*(x,t)=x-tu0(β1+0)<α2.We have shown that ally∈𝒮 is the position of two particles which move following two different dynamics given, at time t, by their respective positions, velocities, and masses xi(y,t),vi(y,t),mi(y,t) (with i=1,2) such that(2.32)x1(y,t):=ϕ(y,t,λ,u0)=ϕ(y,t,-du0,v0)=:x2(y,t),∀t>0,v1(y,t)=v2(y,t),m1(y,t)=tm2(y,t).
However, we had at time zero,(2.33)m1(y,0)=0,m2(y,0)=u0(y-0)-u0(y+0),v1(y,0)=u0(y),v2(y,0)=u0(y-0)+u0(y+0)2.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 (if𝒮 is discrete).Now we show the connection between the second sticky particles model and the inviscid Burgers’ equation.Proof of Theorem1.1.
(1) and (4) The function x↦v0(x)=(u0(x-0)+u0(x+0))2-1 is nonincreasing. As in [8] we can define the sticky particles model of initial mass distribution and velocity function -du0,v0. For all t, the particles position are given by a continuous nondecreasing function (y,t)↦ϕ(y,t,-du0,v0) well defined on the support of du0 and is extended to ℝ by linear interpolation. In the following, we set ϕ(y,t,-du0,v0)=ϕt(y). For any r.v. Y0 having -du0 as law, define Yt=ϕ(Y0,t,-du0,v0). The velocity field of the particles (y,t)↦u(y,t) is such that u(Yt,t)=E[v0(Y0)∣Yt]=(d/dt)Yt. Here, the derivative holds on the right-hand side, for all t.
At timet≥0, consider the c.d.f. Ft(x)=-du0(Yt≤x). One has F0=-u0 and its inverse is given by F0-1(z)=u0-1(-z). On ([0,1],ℬ([0,1]),λ), Ft-1 is a r.v. which is the same law as Yt.
For fixed(x,t), let M(x,t) be the maximum among the abscissas where the function
(2.34)m↦G(x,m,t)=∫0m[u0-1(-z)+tv0(u0-1(-z))-x]dz
reaches its absolute lower bound. Using the results of [9], it is clear that the function M(·,t) is a c.d.f. of Y0,t and it is the entropy solution of(2.35)∂tM+∂x(A(M))=0suchthatM(x,0)=-u0(x),withthefluxR∋m↦A(m)=∫0mv0(u0-1(-z))dz=∫-m0v0(u0-1(z))dz.
Defining the at most countable set D={y∈ℝ:u0(y-0)>u0(y+0)}, and remarking that v0(y)=u0(y)for all y∉D, we have
(2.36)A(m)=∫0m1{u0-1(-z)∈D}v0(u0-1(-z))dz+∫0m1{u0-1(-z)∉D}v0(u0-1(-z))dz=∑yn∈D∫0m1{u0-1(-z)=yn}u0(yn-0)+u0(yn+0)2dz+∫0m1{u0-1(-z)∉D}u0(u0-1(-z))dz.
As this c.d.f. takes its values in -u0(ℝ), one should compute the values of the flux for m∈-u0(ℝ). Without loss of generality, one can suppose that 0 is a value of continuity of u0. Thus, for yn∈Dthere are only two possibilities:
(2.37)[0,m]∩{z:u0-1(-z)=yn}=∅or{z:u0-1(-z)=yn}⊂[0,m].
As {z:u0-1(-z)=yn}=[-u(yn-0),-u(yn+0)], we have
(2.38)∫1{u0-1(-z)=yn}u0(yn-0)+u0(yn+0)2dz=u0(yn-0)2-u0(yn+0)22=-∫1{u0-1(-z)=yn}zdz
and we get
(2.39)A(m)=-∫0m1{u0-1(-z)∈D}zdz-∫0m1{u0-1(-z)∉D}zdz=-∫0mzdz=-m22.
Then, -M(x,t)=u(x,t) is the entropy solution of Burgers’ equation with initial data u0. Thus,-du0∘(Y0,t)-1=∂xM(x,t)=-∂xu(x,t).
The fact thatu(x,t) is the velocity (for t>0) was already given in Theorem 2.2. We give another proof here. For each discontinuity point (x,t) of u, x is the position, at time t, of a massive particle; so its velocity is
(2.40)A(-u(x+0,t))-A(-u(x-0,t))-u(x+0,t)+u(x-0,t)=u(x-0,t)+u(x+0,t)2=u(x,t).
If u is continuous in (x,t) and x is the position at time t, then it is the position of a cluster {a} which moves with its initial velocity v0(a). We have the equalities of masses M(x-0,t)=M(x+0,t)=F0(a-0)=F0(a+0). This means that u(x,t)=u0(a)=v0(a). The velocity of this particle is then w(x,t)=v0(a)=u(x,t).
(2) and (3) These points are given by properties of ϕ (see [9]). The condition ∥u0∥∞<+∞ means u0∈L1(du0).


## 2.1. When a Diffuse Dynamics Hides a Discrete One

In this part,u0 is nonincreasing and stair size.


### 2.1.1. Example of a Single Shock Wave

We begin with an initial velocity of the type(2.1)u0(y)={1,ify<a0,ify≥a.
In this case, 𝒮={a} and Y0,t(a)=a+t/2 for all t≥0. What about X0,t(a)? Let us define x=Y0,t(a). The cluster {y:X0,t(y)=x}=[y*(x,t),y*(x,t)] is the set on which the function y↦G1(y,x,t)=∫0y[z+tu0(z)]dz reaches its absolute lower bound. This is necessarily done in at least one y which is a continuity point of u0, and y+tu0(y)=x. Suppose that y>a. Then y=x, and a simple computation gives(2.2)G1(y,x,t)-G1(y1,x,t)=-(y1+t-x)22,∀y1<a
which shows that the lower bound is also reached in y1:=x-t<a. If y<a, then y=x-t and a simple computation gives(2.3)G1(y2,x,t)-G1(y,x,t)=(y2-x)22,∀y2>a
which shows that the lower bound is also reached in y2:=x. Thus, y*(x,t)=x-t<a<x=y*(x,t). Hence,(2.4)X0,t(y)=Y0,t(a)=x=a+t2,∀y∈[a-t2,a+t2].For the dynamics modeled byX, this means that the trajectory of a is a straight line a(t)=a+t/2 along which all the shocks occur continuously. At time t, the particle a is in a cluster of center a and of diameter t, with the velocity 1/2.


### 2.1.2. Stairsize Velocity

By similar arguments as above, one gets the following results whenu0 is a stair size function of the type(2.5)u0=∑i=0Nwi1[yi,yi+1)with-∞=y0<y1<⋯<yN+1=+∞,wi>wi+1∀i.
Let w be the velocity field of the discrete dynamics constructed from -du0,v0.Proposition 2.1.
For allx∈Y0,t(𝒮), there exists a unique couple (n,m) such that n<m,yn<x-twn=y*(x,t)<yn+1≤ym<x-twm=y*(x,t)<ym+1 and for all (y,i)∈[x-twn,x-twm]×[n+1,m]:(2.6)x=X0,t(y)=Y0,t(yi)=1wn-wm∑j=n+1m[wj-1-wj][yj+twj+wj-12],u(x,t)=w(x,t)=wn+wm2.
If x∉Y0,t(𝒮),∃!(y,n)such that y∈(yn,yn+1) and
(2.7)x=X0,t(y)=y+twn,u(x,t)=u0(y)=wn.These results can be also seen as a simple consequence of the following general case.


## 2.1.1. Example of a Single Shock Wave

We begin with an initial velocity of the type(2.1)u0(y)={1,ify<a0,ify≥a.
In this case, 𝒮={a} and Y0,t(a)=a+t/2 for all t≥0. What about X0,t(a)? Let us define x=Y0,t(a). The cluster {y:X0,t(y)=x}=[y*(x,t),y*(x,t)] is the set on which the function y↦G1(y,x,t)=∫0y[z+tu0(z)]dz reaches its absolute lower bound. This is necessarily done in at least one y which is a continuity point of u0, and y+tu0(y)=x. Suppose that y>a. Then y=x, and a simple computation gives(2.2)G1(y,x,t)-G1(y1,x,t)=-(y1+t-x)22,∀y1<a
which shows that the lower bound is also reached in y1:=x-t<a. If y<a, then y=x-t and a simple computation gives(2.3)G1(y2,x,t)-G1(y,x,t)=(y2-x)22,∀y2>a
which shows that the lower bound is also reached in y2:=x. Thus, y*(x,t)=x-t<a<x=y*(x,t). Hence,(2.4)X0,t(y)=Y0,t(a)=x=a+t2,∀y∈[a-t2,a+t2].For the dynamics modeled byX, this means that the trajectory of a is a straight line a(t)=a+t/2 along which all the shocks occur continuously. At time t, the particle a is in a cluster of center a and of diameter t, with the velocity 1/2.


## 2.1.2. Stairsize Velocity

By similar arguments as above, one gets the following results whenu0 is a stair size function of the type(2.5)u0=∑i=0Nwi1[yi,yi+1)with-∞=y0<y1<⋯<yN+1=+∞,wi>wi+1∀i.
Let w be the velocity field of the discrete dynamics constructed from -du0,v0.Proposition 2.1.
For allx∈Y0,t(𝒮), there exists a unique couple (n,m) such that n<m,yn<x-twn=y*(x,t)<yn+1≤ym<x-twm=y*(x,t)<ym+1 and for all (y,i)∈[x-twn,x-twm]×[n+1,m]:(2.6)x=X0,t(y)=Y0,t(yi)=1wn-wm∑j=n+1m[wj-1-wj][yj+twj+wj-12],u(x,t)=w(x,t)=wn+wm2.
If x∉Y0,t(𝒮),∃!(y,n)such that y∈(yn,yn+1) and
(2.7)x=X0,t(y)=y+twn,u(x,t)=u0(y)=wn.These results can be also seen as a simple consequence of the following general case.


## 2.2. The General Case

The functionu0 is nonincreasing and lim|x|→∞u0(x)x-1=0.From u0 and v0(x)=(u0(x-0)+u0(x+0))2-1, we have two sticky particles models:(i)
eachy∈ℝ is the position of an initial particle which occupies the position ϕ(y,t,λ,u0)=:x at time t≥0, with the velocity u(x,t) and the mass y*(x,t)-y*(x,t);(ii)
eachy∈𝒮 is the position of an initial particle which occupies the position ϕ(y,t,-du0,v0):=x′ at time t≥0, with the velocity w(x′,t) and the mass u0(α-0)-u0(β+0), with [α,β]={y′:ϕ(y′,t,-du0,v0)=x′}.For allt,ϕ(ℝ,t,λ,u0)=ℝ. Then, for all x∈ϕ(𝒮,t,-du0,v0), there exist clusters 𝒞1(x,t)=[y*(x,t),y*(x,t)],𝒞2(x,t) such that(2.8)ϕ(y1,t,λ,u0)=ϕ(y2,t,-du0,v0)=x∀(y1,y2)∈C1(x,t)×C2(x,t).
For t>0, if we suppose that u=w, then both the functions s↦y1(s):=ϕ(y1,t,λ,u0), y2(s):=ϕ(y2,t,-du0,v0) are solutions of(2.9)dy(s)=u(y(s),s)ds,0<t≤s,y(t)=x.
As u satisfies the Œlenick type entropy condition (see [9])(2.10)u(x2,s)-u(x1,s)x2-x1≤1s,∀x1,x2,∀s>0,
it follows from the results of Fillipov [10] that y1(s)=y2(s), for all s≥t>0.Unfortunately, this method fails whent=0. In the sequel, using other arguments, we show indeed that u(·,t)=w(·,t) for t>0, and that the trajectories coincide for all t≥0. In fact, we show that 𝒞2(x,t)⊂𝒞1(x,t).Remark that the link can be established only for the initial positionsy∈𝒮,thatis,u0 is not constant in any neighborhood of y. We recall that(2.11)u(x-0,t)=t-1(x-y*(x,t)),u(x+0,t)=t-1(x-y*(x,t)),∀(x,t),x=y*(x,t)+tu0(y*(x,t))=y*(x,t)+tu0(y*(x,t)).Theorem 2.2.
Letu be the entropy solution of Burgers’ equation with a nonincreasing initial data u0 such that lim|x|→∞u0(x)x-1=0. For all (x,t), let us define 𝒞(x,t)={y′:ϕ(y′,t,-du0,v0)=x}=:[α,β]∩𝒮.(1)
One hasu(x-0,t)≠u(x+0,t) if and only if(2.12)u0(α-0)>u(x,t)=w(x,t)=u0(α-0)+u0(β+0)2>u0(β+0),y*(x,t)=x-tu0(α-0)≤α≤β≤y*(x,t)=x-tu0(β+0).(2)
Ifu(x-0,t)=u(x+0,t), one has two possibilities:
(i)
∅≠𝒞(x,t) satisfies y*(x,t)=y*(x,t)=α=β,
(2.13)u(x,t)=w(x,t)=u0(α-0)=u0(β+0)=u0(α).(ii)
𝒞(x,t)=∅ and there exist a<b such that(2.14)ϕ(a,t,-du0,v0)<ϕ(b,t,-du0,v0),u0(a+0)=u0(b-0),a<y*(x,t)=y*(x,t)=x-tu0(b-0)<b.Before giving the proof, we remark that ifu0 is continuous in z1,z2, then (2.15)∫z1z2[η+tu0(η)]dη=t∫z1z2[η+tu0(η)](-du0)(η)+(z2+tu0(z2))2-(z1+tu0(z1))22.∀t
If u0 is not continuous in zi, the formula must be corrected with the right term u0(zi-0) or u0(zi+0), depending on the fact that the integration is closed or not in zi. This formula is due to the integration by parts(2.16)-∫z1z2ηdu0(η)=-z2u0(z2)+z1u0(z1)+∫z1z2u0(η)dη,-∫z1z2v0du0=∫u0(z1)u0(z2)v0(u0-1(-z))dz=-∫u0(z1)u0(z2)zdz=u0(z1)2-u0(z2)22.So we get(2.17)∫z1z2[η+tv0(η)](-du0)(η)=∫z1z2u0(η)dη-z2u0(z2)+z1u0(z1)+tu0(z1)2-u0(z2)22=∫z1z2u0(η)dη+(z1+tu0(z1))2-(z2+tu0(z2))2-z12+z222t=1t∫z1z2[η+tu0(η)]dη+(z1+tu0(z1))2-(z2+tu0(z2))22t.Proof of Theorem2.2.
(1) When u is discontinuous in (x,t), we have y*(x,t)<y*(x,t), and u(x,t) is defined as the velocity of the cluster [y*(x,t),y*(x,t)]:(2.18)u(x,t)=∫y*(x,t)y*(x,t)u0(η)dηy*(x,t)-y*(x,t).
For x∈ϕ(𝒮,t,-du0,v0), the cluster [α,β]∩𝒮:={y′:ϕ(y′,t,-du0,v0)=x} satisfies
(2.19)β+tu0(β+0)≤x≤α+tu0(α-0).
Let z1:=x-tu0(α-0) and suppose that y*(x,t)≠z1. First, we remark that x=z1+u0(z1-0). It is obvious if z1=α. If z1≠α, we have x<α+tu0(α-0). This occurs only when α is not an accumulation of particles from the left; then there exists another cluster [a,b] such that b<α and
(2.20)du0(b,α)=0=u0(α-0)-u0(b+0),b+tu0(b+0)≤ϕ(b,t,-du0,v0)<x<α+tu0(α-0).
Thus x-tu0(α-0)=z1∈(b,α). As u0 is constant in (b,α), we get z1+tu0(z1)=x. On the other hand, it is easy to see that u0 is continuous in y*(x,t),y*(x,t). Then, we have necessarily y*(x,t)=x-tu0(y*(x,t))∉[b,α]. If y*(x,t)<b, z2:=y*(x,t) gives in (2.15):
(2.21)-t∫y*(x,t)α-0[η+tv0(η)]du0(η)=-t∫y*(x,t)z1-0[η+tv0(η)]du0(η)=∫y*(x,t)z1[η+tu0(η)]dη=G(x,z1,t)-G(x,y*(x,t),t)+x(z1-y*(x,t)).
As
(2.22)z1-y*(x,t)=t[u0(y*(x,t))-u0(α-0)]=t(-du0)([y*(x,t),α)),
we get
(2.23)∫y*(x,t)α-0[η+tv0(η)-x](-du0)(η)=G(x,z1,t)-G(x,y*(x,t),t)t.
From the definition of α and y*(x,t), the fist term is negative and the second is nonnegative. This is absurd, as well as the case y*(x,t)>α. We can conclude that y*(x,t)=z1=x-tu0(α-0). In the same way, y*(x,t)=x-tu0(β+0). Thus
(2.24)y*(x,t)<y*(x,t)⟺u0(α-0)>u0(β+0),
and it comes from (2.15) that
(2.25)u(x,t)=t-1(x-y*(x,t)+y*(x,t)2)=u0(α-0)+u0(β+0)2=∫α-0β+0v0(η)(-du0)(η)u0(α-0)-u0(β+0):=w(x,t).
(2.i)
(2.26)y*(x,t)=y*(x,t)=α=β⟺u0(α-0)=u0(β+0)=u0(α),
and in this case
(2.27)u(x,t)=t-1(x-α)=u0(α)=v0(α)≔w(x,t).
(2.ii) For x∉ϕ(𝒮,t,-du0,v0), there exist x1,x2∈ϕ(𝒮,t,-du0,v0) and clusters
(2.28)[αi,βi]∩S:={y′:ϕ(y′,t,-du0,v0)=xi}
such that x1<x<x2 and
(2.29)du0(β1,α2)=0=u0(α2-0)-u0(β1+0),β1<α2.
As the function x′↦y(x′,t) does not decrease, we have
(2.30)y*(x1,t)≤y*(x,t)≤y*(x,t)≤y*(x2,t),β1≤x1-tu0(β1+0)=y*(x1,t)≤y*(x2,t)=x2-tu0(α2-0)≤α2,
and this leads to y*(x,t),y*(x,t)∈[β1,α2]. From the fact that u0(α2-0)=u0(β1+0) and y*(x,t)+tu0(y*(x,t))=x=y*(x,t)+tu0(y*(x,t)), we conclude that
(2.31)β1<y*(x,t)=y*(x,t)=x-tu0(β1+0)<α2.We have shown that ally∈𝒮 is the position of two particles which move following two different dynamics given, at time t, by their respective positions, velocities, and masses xi(y,t),vi(y,t),mi(y,t) (with i=1,2) such that(2.32)x1(y,t):=ϕ(y,t,λ,u0)=ϕ(y,t,-du0,v0)=:x2(y,t),∀t>0,v1(y,t)=v2(y,t),m1(y,t)=tm2(y,t).
However, we had at time zero,(2.33)m1(y,0)=0,m2(y,0)=u0(y-0)-u0(y+0),v1(y,0)=u0(y),v2(y,0)=u0(y-0)+u0(y+0)2.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 (if𝒮 is discrete).Now we show the connection between the second sticky particles model and the inviscid Burgers’ equation.Proof of Theorem1.1.
(1) and (4) The function x↦v0(x)=(u0(x-0)+u0(x+0))2-1 is nonincreasing. As in [8] we can define the sticky particles model of initial mass distribution and velocity function -du0,v0. For all t, the particles position are given by a continuous nondecreasing function (y,t)↦ϕ(y,t,-du0,v0) well defined on the support of du0 and is extended to ℝ by linear interpolation. In the following, we set ϕ(y,t,-du0,v0)=ϕt(y). For any r.v. Y0 having -du0 as law, define Yt=ϕ(Y0,t,-du0,v0). The velocity field of the particles (y,t)↦u(y,t) is such that u(Yt,t)=E[v0(Y0)∣Yt]=(d/dt)Yt. Here, the derivative holds on the right-hand side, for all t.
At timet≥0, consider the c.d.f. Ft(x)=-du0(Yt≤x). One has F0=-u0 and its inverse is given by F0-1(z)=u0-1(-z). On ([0,1],ℬ([0,1]),λ), Ft-1 is a r.v. which is the same law as Yt.
For fixed(x,t), let M(x,t) be the maximum among the abscissas where the function
(2.34)m↦G(x,m,t)=∫0m[u0-1(-z)+tv0(u0-1(-z))-x]dz
reaches its absolute lower bound. Using the results of [9], it is clear that the function M(·,t) is a c.d.f. of Y0,t and it is the entropy solution of(2.35)∂tM+∂x(A(M))=0suchthatM(x,0)=-u0(x),withthefluxR∋m↦A(m)=∫0mv0(u0-1(-z))dz=∫-m0v0(u0-1(z))dz.
Defining the at most countable set D={y∈ℝ:u0(y-0)>u0(y+0)}, and remarking that v0(y)=u0(y)for all y∉D, we have
(2.36)A(m)=∫0m1{u0-1(-z)∈D}v0(u0-1(-z))dz+∫0m1{u0-1(-z)∉D}v0(u0-1(-z))dz=∑yn∈D∫0m1{u0-1(-z)=yn}u0(yn-0)+u0(yn+0)2dz+∫0m1{u0-1(-z)∉D}u0(u0-1(-z))dz.
As this c.d.f. takes its values in -u0(ℝ), one should compute the values of the flux for m∈-u0(ℝ). Without loss of generality, one can suppose that 0 is a value of continuity of u0. Thus, for yn∈Dthere are only two possibilities:
(2.37)[0,m]∩{z:u0-1(-z)=yn}=∅or{z:u0-1(-z)=yn}⊂[0,m].
As {z:u0-1(-z)=yn}=[-u(yn-0),-u(yn+0)], we have
(2.38)∫1{u0-1(-z)=yn}u0(yn-0)+u0(yn+0)2dz=u0(yn-0)2-u0(yn+0)22=-∫1{u0-1(-z)=yn}zdz
and we get
(2.39)A(m)=-∫0m1{u0-1(-z)∈D}zdz-∫0m1{u0-1(-z)∉D}zdz=-∫0mzdz=-m22.
Then, -M(x,t)=u(x,t) is the entropy solution of Burgers’ equation with initial data u0. Thus,-du0∘(Y0,t)-1=∂xM(x,t)=-∂xu(x,t).
The fact thatu(x,t) is the velocity (for t>0) was already given in Theorem 2.2. We give another proof here. For each discontinuity point (x,t) of u, x is the position, at time t, of a massive particle; so its velocity is
(2.40)A(-u(x+0,t))-A(-u(x-0,t))-u(x+0,t)+u(x-0,t)=u(x-0,t)+u(x+0,t)2=u(x,t).
If u is continuous in (x,t) and x is the position at time t, then it is the position of a cluster {a} which moves with its initial velocity v0(a). We have the equalities of masses M(x-0,t)=M(x+0,t)=F0(a-0)=F0(a+0). This means that u(x,t)=u0(a)=v0(a). The velocity of this particle is then w(x,t)=v0(a)=u(x,t).
(2) and (3) These points are given by properties of ϕ (see [9]). The condition ∥u0∥∞<+∞ means u0∈L1(du0).



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