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Communications IΠ

Commun. Math. Phys. 177, 349 - 380 (1996)

Mathematical Physics © Springer-Verlag 1996

Generalized Variational Principles, Global Weak Solutions and Behavior with Random Initial Data for Systems of Conservation Laws Arising in Adhesion Particle Dynamics Weinan E 1 , Yu.G. Rykov2, Ya.G. Sinai3 1

Courant Institute of Mathematical Sciences, New York, NY 10012, USA Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia 3 Mathematics Department, Princeton University, Princeton, NJ 08544, USA and Landau Institute of Theoretical Physics, Moscow, Russia 2

Received: 30 December 1994/in revised form: 1 May 1995

Abstract: We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2 x 2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.

1. Introduction This paper has two main goals: The first is to give an explicit construction of weak solutions for the initial value problem of the systems of conservation laws:

:°o and f (pu)x = 0 (pu2)x = -pgx 9χχ

(1.2)

= P

The second is to study the qualitative behavior of such weak solutions when initial data are random. We prove that for a wide class of probability distributions for the

350

W. E, Yu.G. Rykov, Ya.G. Sinai

initial data, almost every weak solution has the following structure: At any positive time t > 0, p( , t) becomes a purely singular measure even though it may be continuous at t = 0. Moreover, this singular measure is supported on a dense set which can also be considered as the shock set of u. We will also study a variant of (1.2) in which a weighted velocity instead of momentum is conserved at the collisions [GMS, VFDN]: =0

Our construction of weak solutions for (1.1) is based on a connection between (1.1) and the "sticky particle model" of Zeldovich (see [Z] and also [CPY]). There is a similar connection between (1.2) and the gravitationally interacting sticky particles. Consider a system of particles on Rι with initial velocities {vj}, locations {x?} and masses {rrή}, j G Z. The particles move with constant velocities unless they collide. At collisions the colliding particles stick and form a new massive particle. The mass and velocity of this new particle are given by the laws of conservation of mass and momentum. This model was proposed by Zeldovich [SZ], and developed further by Kofman, Shandarin, et al. (see [GMS,KPS], and the survey paper [VDFN]) to explain the formation of large scale structures in the universe. In this context it is also referred to as the model of "adhesion dynamics." One main result of this paper is that the adhesion dynamics of free particles is in a sense integrable, and this gives rise to weak solutions of (1.1). A similar connection exists between (1.2) and the gravitationally interacting sticky particles. The Hamiltonian governing the dynamics between collisions is given by 2 H(p,x) = Σ p~ + Σ mmj\xi - xj\ .

(1.4)

We will assume that Σ / mι < °° When particles collide, again they form a new particle with mass and velocity given by the conservation of mass and momentum. The gravitational force acting on a particle is proportional to the difference between the total masses from the right and from the left of the particle. This system is also integrable in the same sense and leads to weak solutions of (1.2). For smooth solutions, (1.1) is equivalent to the Burgers equation (L5)

j)=0 together with a scalar transport equation

(1.6)

= 0.

Given the initial data {po9^o}> the solution of (1.5)—(1.6) can be easily found via ι ι the method of characteristics. Define the forward flow map φt : R —» R by x = Ψt(y) = y + tuo(y).

(1.7)

For small t, this map is usually invertible, and we have ,-1 t

ι

where y = φ^ (x)

(x)),

p(xj) = po(

defines the backward flow map.

dx~X dy

(1.8)

Conservation Laws in Adhesion Particle Dynamics

351

It is well-known [L] that this construction ceases to be valid after some critical time Γ* at which the solution of (1.5) develops shocks. In general (1.1) and (1.5)—(1.6) also cease to be equivalent after Γ*. In analogy with fluid mechanics, we call y the Lagrangian coordinate and φt(y) the Eulerian coordinate at time t. After Γ* the mapping y —> φt(y) is no longer one-to-one, and no longer defined by (1.7): a whole interval can be mapped to a single point which is the location of a shock. However, in all cases φt defines a partition ξt of Rι where elements of the partition are given by ®(x)

= {φ-ι(x),

xβR1}.

(1.9)

We should stress that the solutions are assumed to be continuous from the right. The elements of ξt can be either single points, or segments. More importantly, knowing ξt, we can reconstruct φt and «(•,£) from the two conservation laws:

where Ct{y) denotes the element of the partition ξt containing y9 and %(x) = φΊ~x(x). In the more general case when the initial distribution of mass is given by a nonnegative Borel measure Po, (1.10) takes the form

Both (1.10) and (1.11) state that φt(y) is now the position of the center of mass

of GOOWe are left with the key step of defining {ξt}t>o- Let us first consider the simpler case of a finite number of particles with initial data {Xj9Vj,mj}, 1 ^ j S N. A crucial observation is that the necessary and sufficient condition for N particles to collide and form a single particle before, or at time t, is that

Σί=i (χj + rf)mj ^ Σ,w+i (ή + tή>\ m

i=\ )

Σj=J+\

m

j

holds for all J, 1 ^ J ^ N — 1. Indeed assume that (1.12) holds, yet there are more than one cluster of particles at time t. Without loss of generality, let us assume that / 1 there are two such clusters, {1,2,...,/ } and {J + 1,...,/V} located at X\(t) and X2{t) with Xλ(t) < X2(t). Then we have

since the conservation of mass and momentum dictates that the cluster has to be located at the center of mass. Equation (1.13) contradicts the assumption that (1.12) holds for all j . On the other hand, assume that (1.12) is violated for some J = J\ then the group of particles {I,...,/'} will never catch up with the group {Jr + 1,...,/V} before time t. For details, see Sect. 3. Before going into the continuous case, let us state the conditions we will impose on the initial data.

352

W. E, Yu.G. Rykov, Ya.G. Sinai ι

Let Po, /o G M: the space of Radon measures on R ,Po ^ 0. ι

(Al) Po(A) < oo for any compact A C R and PQ is either discrete or absolutely continuous with respect to the Lebesgue measure. In the latter case, we assume that density po(x) > 0, for x G Supp(Po) If Supp^o) is unbounded, we assume additionally f sdP0(s) —• + o c as \x\ —> + o o . o (A2) The initial distribution of momentum /o is absolutely continuous with respect to PQ. The Radon-Nikodym derivative w( , 0 ) = -^- is the initial velocity. In the case when PQ is absolutely continuous, we assume that w( , 0 ) is also continuous. (A3) For any z > 0 sup |wo(*)| ^ bo(z) and lim - bo(z) = 0 .

|x|^z

|z|-κ»Z

The first essential result of this paper is the following principle for the construction of ξt using the initial data. Generalized Variational Principle (GVP): y G Rι is the left endpoint of an element of ξt iff for any y~, y+ G Rι, such that y~ < y < y+, the following holds: /

ι JL

ill J

(η + tu(η; 0))dP0(η) L

i

l

l




We can also formulate GVP for right endpoints of elements of ξt, but we will omit this since we do not need it. Having {ξt}t>o, we define φt via (1.11) and the density and momentum distributions at time t,Pt and It, by Pt{Δ) = P0{φγ\Δ)),

(1.15)

It(Δ) = h{φγ\Δ))

ι

for A C R . In the case of continuous u(x;0) the mapping φt is also continuous. It is clear that It is absolutely continuous with respect to Pt, and we can introduce the Radon-Nikodym derivative (1.16)

u(x,t)=^-(x) art

which is the velocity at (x,t). We will use the following definition of weak solutions. Definition 1. Let (PtJt) be a family of Borel measures, weakly continuous with respect to t, such that It is absolutely continuous with respect to Pt for each fixed t. Define u via (1.16). (PtJt,u)t^o is a weak solution of (1.1) if for any f,g£ CQ(R1), the space of C 1 -functions on Rι with compact support, and 0 < t\ < t2, (Dl)

(D2)

ff(η)dPt2(η)

- J f(η)dPh(η)

= Jdτ f ff(η)dlτ(η)

,

Jg(η)dlί2(η) - J g(η)dlh(η) = J dτ J g\η)u(η,τ)dlτ(η) .

Conservation Laws in Adhesion Particle Dynamics

353

Theorem 1. Under the assumptions (Al-3), the family (Pt,It,u)t^o constructed using GVP gives a weak solution 0/(1.1) with initial data (PoJo) in the sense that urn/, = /o

PQ,

weakly as t —> 0 + .

Next we turn to (1.2). First of all, we remark that the third equation in (1.2) can be interpreted as

(

+OO

X

\

f p(ξ,t)dξ-

f p(ξ,t)dξ)9 (1.17) / i.e. the acceleration at a point is proportional to the difference between the total masses from the right and from the left of that point. For the initial data, in addition to (A1-A3) we will also assume (A4) P0(Rι) < + o c . The characteristics of (1.2) are now given by quadratic functions of t: 2

x(t) = y + uo(y)t + - ao(y)t

(1.18)

.

This has the effect of changing (1.11) to JCl(y)(η

Ψt{y) = — —

,2

o(η))o(η)

f

-=7-7

JΘ

f

+ a(Q(y))-

(1.19)

+a(%{x))t

where a(Q(y)) = Po(I+)-Po(I-). /

(1.20)

+

ι

and / " are respectively the right and left connected component of R \Ct(y). ι As before, the key issue is to construct the family of partitions of R 9 {ξt}t^o. In this case we formulate an analogous Generalized Variational Principle (GVP): y G Rι is the left end point of an element of ξt iff for any y+,y~,y~ < y < y+, the following holds: / A

Σ

(η + tuo(η))dPo(η)

^

+

t

\

(P( 2 t

oo)-Po(-oo,7)).

(1.21)

Having {ξt}t^o, we can construct φu u using (1.19), Pt as before, and It from u and Pt by a simple integration. Definition. Let (PtJt) be a family of Borel measures, weakly continuous with respect to t, such that It is absolutely continuous with respect to Pt for each fixed t. Define u via (1.16). (PtJt,u)t^o is a weak solution of (1.2) iff for any

354

W. E, Yu.G. Rykov, Ya.G. Sinai 1

/, g eC^R ),

(Dl')

andO < tλ < t2,

Jf(η)dPt2(η)

- J f(η)dPtι(η)

= f dτ f f'(η)dlτ(η) ,

(D2') Jg(η)dlt2(η) - Jg(η)dlh(η)

= Jdτ f

g\η)u(n,τ)dlτ(η)

h

+ jdτf

g(η)(PM -oo) - Pτ(-oo, η))dPτ(η) .

h

Theorem 2. Under the assumptions (Al-4), the family (Pt,It,u)t^o constructed using GVP gives a weak solution of (1.2) with initial data (PQJQ) in the sense that \\mPt = Po,

weakly

as t —• 0 + .

Before continuing, let us put these results in the perspective of general hyperbolic conservation laws. For obvious reasons, (1.1) is sometimes referred to as the pressureless gas dynamics equations. However, compared with the isentropic gas dynamics equation =0 (pu)t

+ (pu2 + p(p))x

=0

there are two important differences. First at a technical level, the natural space for (1.1) is M, the space of Radon measures, instead of BV or L°°. Secondly, the standard entropy condition, which in the present case, takes the form (pS(p))t+(upS(p))x

^ 0 ,

(1.23)

where S is convex, is not enough as a uniqueness criterion. Indeed in the context of particle systems, there is a whole family of inelastic collision rules that satisfy (1.1) and (1.23). The adhesion dynamics considered here is an extreme case of these collision rules. It is easy to see that the weak solutions of (1.1) constructed in this paper has the additional property:

ux *

0, all masses concentrate on points (i.e. the absolute continuous part vanishes), and the set of shock locations is dense. This behavior was to some extent anticipated by Zeldovich [Z] in his work on cosmology. In that context, these point masses are interpreted as the galaxies in a one-dimensional universe. Before ending this introduction, let us mention that. (1.1) and (1.2) also have an origin in kinetic equations. Consider // + t>Λ = O.

(1.27)

If we look for solutions of the form f(x,υ,t) = p(x,t)δ(υ - u(x,t)),

(1.28)

we obtain (1.1) for (p,u). Similarly consider the Vlasov-Poisson-Jeans equation [VDFN] ί ft + υfx - gxfv = 0 { r O 29) [θxx = J f(x,υ9t)dv. If we look for solutions of the form (1.28), we obtain (1.2) for (p,u). The paper has eight sections and one appendix. In Sect. 2, we compare our GVP with the Lax-Oleinik variational principle for scalar quasi-linear equations. In Sect. 3, we consider the discrete version of (1.1) and prove GVP for this case. In Sect. 4, we extend these results to the continuous case and complete the proof of Theorem 1. In Sect. 5, we explain the additional steps needed for the proof of Theorem 2. In Sect. 6, we consider the modified gravitational system (1.3). Part II consists of two sections. In Sect. 7, we consider (1.1) and (1.2) with random initial data. In Sect. 8, we extend these results to (1.3). After this paper was submitted for publication, we received a preprint [BG] by Brenier and Grenier in which existence of weak solutions of (1.1) was proved without resorting to GVP at the continuous level. [BG] also contains some very interesting ideas for the multi-dimensional version of (1.1). We thank Brenier and Grenier for timely communication of their results.

356

W. E, Yu.G. Rykov, Ya.G. Sinai

Part I. Generalized Variational Principles and Global Weak Solutions 2. Preliminaries and Comparison with the Lax-Oleinik Variational Principle In the following we will concentrate on (1.1). The necessary changes for (1.2) will be summarized in Sect. 5. Intuition from adhesion dynamics suggests that the masses cluster more and more, and the accumulated masses will never split apart again. We formulate this as: Lemma 1. The family of partitions {ξt}t>o determined with the help of GVP is decreasing. In other words, for 0 < t' < t, each element of ξt is contained in an element of ξtt. Proof Assume to the contrary that there exists y G dζt, but y£dξt*9 where dξt denotes the collection of points belonging to the boundary of some element of ξt. Without loss of generality, we can assume that y is the left end point of an element in ξt. Then for some y~ < y < jμ+, we should have

(

> [ y , y )

=

Consider two linear functions l\(s)J2(s)

fίy,y+]

dPo(

'

}

of s defined by

ηdPo(η)

For sufficiently small s9 we have l\(s) < I2(s) while l\(tf) ^ I2(tf). Since l\ and l2 are linear, we conclude that l\{t) > I2(t). This contradicts the fact that y e dξt: i.e. y satisfies GVP at time t. Now we will compare GVP with the classical Lax-Oleinik variational principle see [L, O]. We assume that PQ has a density po> a n d 0 < const ^ PoOO < oo. Introduce y

y

Φι(y) = / (η + tuo(η))po(η)dη,

φ2(y) = J Po(η)dη

(2.3)

and C(>;

'^

} =

02(/') - Φ2(/) '

(2

'4)

Conservation Laws in Adhesion Particle Dynamics

357

1

x

02 defines a C^-diffeomorphism of/? . Let y — φ^ {z)9 and define

^

{

)

_

ι

r

(

)

In these notations, (1.14) becomes inf

+

c(y,y ) ^

+

y >y

or, for z = φ2(y),

inf ^

sup c(y~,y)

(2.6)

y- w e w a n t t 0 s n o w t n a t ^/,y-i(0 < Cj,j"(t)> L e t hJiτ Ji be the consecutive elements of & to the left of Xj, and JC°, £ I\ = {xf9 i\ ^ / ;§ z'2} Let Jι,J2,...,Jr be the consecutive elements of ^ to the right of Xj (including the one containing Xj), and x0,, G Jr = {xf, yΊ ^ z ^7*2}- From Lemma 3, we have Λ 2

,

Λ /

φ

(

3

.

8

)

We also have CΛ(0
i€lj

We also have \xij(t) — Xj(t + At)\ ^ Const At since all velocities are bounded. Let f,ge C^(Rι). We first show that ff(η)dPt(η) is a differentiable function of t, and jtJf(η)dPt(η)

= Jf'(η)dlt(η)

.

(3.12)

This implies (Dl). Indeed, we have

= Σ Σ f(xj(.t + Δt) + Xij(t) - xj(t + Δt))mυ(t) j

i

Xij(t)-xj(t + At))mij{t) + o(At) j

ieij

Since Σ, (XijίO - xj(t + At))mij(t) = Σ XijiOmtjit) - Xj(t + ^Owy(ί + ^ 0 /G/y

/

= -Zlί Σ % ( 0 ^ ( 0 = -AtVj(t + zlί)my(ί + J we obtain

Jf(η)dPt(η) = Jf(η)dPt+At(η) - ΔtΣf'(Xj(t + Δt))Vj(t + Δt)mj{t + Δt) + o(Δt) . j

This implies (3.12).

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W. E, Yu.G. Rykov, Ya.G. Sinai 1

In contrast Jg(η)dlt(η) is not C in t because of the inelastic nature of the collisions. To see this, consider just two particles colliding at t = τ. Before collision they are denoted by {x\{t),V\{t),m\{t)),(x2(t),v2{t),m2{t)),t < τ, respectively. After collision they form a single particle (x(t),v(t),m(t)),t ^ τ. We have m(τ) = /«i(τ) + m2{τ\ m(τ)v(τ) = mχ(τ)v\(τ) + m2(τ)v2{τ\ m(τ)x(τ) = m\(τ)x\(τ) + m2(τ)x2(τ). For t < τ, we have Jg(η)dlt(η) = = 0(*(τ))(ι;i(O/wi(O + v2(t)m2(t)) + (t- τ)g'(x(τ))(v\{t)mλ(t) + ^ ( 0 ^ ( 0 ) + O((ί - τ) 2 ) = Jg(η)dlτ(η) + (ί -

For

t > τ, we have Jg(η)dlt(η)

= g{x{t))v(t)m(t) = Jg(η)dlτ(η)

+ (t- τ)g'(x(τ))v2(τ)m(τ)

+ O((t - τ ) 2 ) .

In general, energy decreases at collisions ι; 2 (τ)m!(τ) + v22(τ)m2(τ) +

v2(τ)m(τ).

Hence Jg(η)dlt(η) is not C 1 in ί. Nevertheless, we can prove that (D2) is still valid. We only have to prove this for the case when there is no collision in (t\,t2) and at t = t2, a group of particles, say with indices i\9 i\ + 1,..., i2, are colliding to form a single particle at x{t2). We then have

Jdτjg'{η)u{η,τ)dlτ{η) = J J i f y ^ W ) ^ ) ™ ; Ί

Ί

'g

1

1=1

±

miVi g(Xi(τ)) ' * dτ

ι

'=1

+ j miVi±g(Xi(τ)) ι

ί = h

ι

ι

dτ

+ J2 /=/2+1

A

Σ

TV

miVid(xi(h)) ~ Σ

m v

ii

4. Proof of Theorem 1 We will first give the proof for the case when Po is absolutely continuous, and then for the case when Po is discrete. For initial data satisfying (A1-A3), we construct a decreasing family of partitions {ξt}t>o of Rι according to GVP (1.14). Having {6}/>o> we can define φt,PtJt, and u(x,t). Obviously φt is a non-decreasing function of y for any fixed t.

Conservation Laws in Adhesion Particle Dynamics

361

Furthermore, as a consequence of the assumption that u(x; 0) is continuous, φt is also continuous, and we have

Jf(η)dPt(η) fg(η)dlt(η)

= Jf(φt(η))dP0(η) = Jg(φt(η))dlo(η)

, (4.1)

.

We will prove Theorem 1 via discrete approximations. Take a sequence of meaW) sures PQ concentrated on finite sets {xf\i = 1,...,} such that P^ —> Po weakly. Define IQ to be a signed measure concentrated on the set {x;\i — 1,...,} such that 4n\{x\n)}) = uo(x\n))P{Q\{x\n)}). Then I™ -> 70 weakly. Using GVP, we construct the corresponding families of partitions ξ^ and mappings φ^tn\ Moreover we already showed (Theorem 1') that for f,ge C0(Rι),

(4.2) Here P\n\ήn)

are constructed in Sect. 1. We also have for 0 < t\ < t2,

=J ^

J ^^\

\η,τ)dl^ .

^

(4.3)

h

We can extend the definition of φ(tn) to the whole line by putting φ^iy) = φ^n\x\n)) if X™ ^ y ^ χ«>l9 φ