Non-existence and Non-uniqueness for Multidimensional Sticky Particle Systems

The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.


Introduction
We consider the initial value problem for the equations of pressureless gases in several space dimensions: ∂ t ρ + div (ρv) = 0, ∂ t (ρv) + div ρv ⊗ v = 0, t ∈]0, T [ , x ∈ IR n , (1.1) (1.2) The system (1.1) was first studied by Zeldovich [13] in the one-dimensional case to model the evolution of a sticky particle system. An example of a measure-valued solution is provided by a finite collection of particles moving with constant speed in the absence of forces. Whenever two or more particles collide, they stick to each other as a single compound particle. The mass of the new particle is equal to the total mass of the particles involved in the collision, while its velocity is determined by the conservation of momentum. The sticky particle system has been investigated extensively by many authors and is well understood in dimension n = 1, see [2,3,4,6,7,8,9,10]. In this case it is known that, for any initial data (ρ,v) with bounded total mass and energy, the Cauchy problem (1.1)-(1.2) has a unique global entropy-admissible weak solution (ρ, v) (see [8] and [10,Theorem 1.3]).
The present paper is concerned with the initial value problem associated with (1.1) in space dimension n ≥ 2. We are interested in weak solutions to (1.1) obeying the sticky particle or adhesion dynamics principle, which are the most relevant from a physical point of view. For initial data containing finitely many particles, it is easy to see that a unique global solution exists, but it does not depend continuously on the initial data. In the case of countably many particles, we show that both uniqueness and existence can fail. Indeed, we construct a Cauchy problem having exactly two solutions, and a second Cauchy problem where no solution exists, not even locally in time.
Both these examples can be adapted to the case of L ∞ initial data.
The remainder of the paper is organized as follows. In Section 2 we give precise definitions of "weak solution" and "sticky solution" for initial data containing countably many point masses and also for continuous mass distributions, following [11]. Section 3 contains an example showing the non-uniqueness of sticky solutions. In Section 4 we describe a Cauchy problem without any local solution and explain how this counterexample relates to the (erroneous) proof of global existence of sticky solutions proposed in [11]. Finally, in Section 5 we extend the analysis to initial data having smooth density and continuous velocity. Even in this case we show that local existence and uniqueness do not hold, in general.

Dynamics of sticky particles
We consider a system containing countably many sticky particles, moving in n-dimensional space. Let In a Lagrangian formulation, the state of the system is described by countably many ODEs for the variables x i . Let be the initial position and the initial velocity of the i-th particle. It is natural to assume that, when particles are at a same location, they stick together traveling with a common speed determined by the conservation of momentum. At any time t ≥ 0, the speed of the i-th particle should thus beẋ Notice that the right hand side of (2.2) is well defined provided that the total mass M = i m i and the initial energy E .
is a weak solution of the equations (2.2) with initial data (2.1) if, for every i ≥ 1 and t ≥ 0, one has In addition, we say that the solution is energy admissible if the corresponding energy is a bounded, non-increasing function of time. In general, even in the case of two particles, the energy-admissible weak solution need not be unique. To achieve uniqueness (at least for finitely many particles) one more condition must be imposed.
Definition 2. We say that a weak solution {x i (·) ; i ≥ 1} is a sticky solution if it satisfies the additional property Example 1. Given any initial data (2.1), the family of functions always provides a weak, energy admissible solution. Indeed, for any i, the set of times is at most countable, therefore it has measure zero. This implies J i (t) = {i} for a.e. t ≥ 0, hence (2.4) trivially holds.
Example 2. Consider two particles moving in the plane, with masses m 1 = m 2 = 1 and with initial positions and velocities given bȳ As in Example 1, the maps provide an energy-admissible weak solution, which however does not satisfies the stickiness assumption (SP). The unique solution that satisfies (SP) is given by (2.8) for t ∈ [0, 1], while We remark that the weak solution (2.8) depends continuously on the initial data, but the sticky solution does not. Indeed, if we slightly perturb the initial data, say by takingx 1 = 1 ε with ε = 0, then the two particles do not collide and (2.8) provides the unique sticky solution to the Cauchy problem.
We also observe that the above Cauchy problem has infinitely many weak solutions which are not energy admissible (and not sticky). Indeed, for any given time T ≥ 1, a weak solution is defined by (2.8) for t ∈ [0, 1], by (2.9) for t ∈ [1, T ], and by Remark 1. When only finitely many particles are present, is it an easy matter to prove the global existence and uniqueness of a sticky solution to the Cauchy problem (2.1). The proof can be achieved by induction on the number N of particles. When N = 1 the result is trivial. Next, assume that the result is true whenever the initial number of particles is < N . Consider an initial data consisting of exactly N particles. Let x i (t) . =x i + tv i and define the first interaction time τ .
If τ = +∞, then describes the unique sticky solution. If τ < ∞, then at time τ two or more colliding particles are lumped together in a single compound particle with speed determined by the conservation of momentum. This yields a new Cauchy problem, where the initial data at t = τ contains a number of particles strictly less than N . The result follows by induction.
For initial data where the density can be an arbitrary measure, a general notion of sticky weak solutions for the system (1.1) was introduced by Sever in [11]. This definition is reviewed here, and will be later used in Section 5. In the following we assume that the initial density and the initial velocity of the pressureless gas satisfȳ Here P 2 (IR n ) is the set of all probability measures ρ such that Let D be an open set in IR n with L n (D) = 1, where L n is the Lebesgue measure on IR n . Regarding y ∈ D as a Lagrangian coordinate, a flow will be described by a mapping X : In addition to the usual Lebesgue spaces L 2 (D) and L 2 [0, T ] × D), we shall also need the following function spaces: It was shown in [11,Lemma 3 and v(t, ·) J,t ≤ g L 2 . (2.12) In addition, the following orthogonality relation holds: We can now recall the definition of sticky weak solution introduced in [11].

A Cauchy problem with two solutions
We construct here an initial configuration containing countably many particles in the plane, such that the Cauchy problem has two distinct sticky solutions. Example 3. A solution of (2.4) will be constructed by induction, starting from the final configuration and going backward in time (see Figure 2). For notational simplicity, we denote with the same symbol a particle and its mass. We also define the times t i .
• For t ≥ t 1 = 1/2 there is one single particle of mass m * 0 = 1, located at the origin.
• For t ∈ [t 2 , t 1 [ there are two particles with masses m 1 = m * 1 = 1/2, moving toward each other with opposite speed and colliding at the origin at time t = t 1 .
• In general, for t ∈ [t i+1 , t i [ , the particle m * i−1 is replaced by two equal particles, with Let v i be the constant speed of the i-th particle m i during the time interval [0, t i ], i.e. before interaction. Moreover, let v * i be the speed of the lumped particle m * i during the time interval Apart from (3.1), the speeds v i , v * i can be freely chosen. We take advantage of this fact, choosing v i , v * i so that the following non-intersection property holds.
(NIP) For each t ≥ 0, the two points do not coincide with any of the finitely many points By induction on i = 1, 2, . . . we thus obtain a sticky solution to the particle equation (2.2) defined for all t ≥ 0. The initial data consists of countably many particles with masses m i = 2 −i , i ≥ 1. As time progresses, these particles collide and stick to each other. In particular during each time interval [t i , t i−1 [ only i distinct compound particles are present.
We now observe that, because of (NIP), the trivial solution (2.6) withv i .
= v i is a sticky solution as well. This provides a counterexample to uniqueness, for an initial configuration containing countably many particles.

A Cauchy problem with no solution
In this section we shall construct an initial configuration containing countably many particles in the plane, such that the corresponding Cauchy problem has no sticky solution, even locally in time.
As a first step, consider a one-dimensional configuration consisting of countably many particles x k , k ≥ 1 moving along the x axis. The masses m k of these particles, and their initial positions x k and velocitiesv k are chosen to be with 0 < α, β, γ < 1. For notational convenience, we denote by the positions of the free particles. Notice that the collection of particles {x j (t) ; j ≥ k} with masses m j has barycenter located at Call t k−1 the time when this barycenter hits the particle x k−1 . Solving the equation b k (t) = x k−1 (t) we obtain Therefore, x Figure 3: A sticky solution containing countably many particles, moving on the x-axis.
In particular, if we choose 0 < β < γ < 1, the sequence of times t k will be strictly decreasing to zero.
The unique solution to the one-dimensional Cauchy problem can be explicitly described as follows ( Figure 3). For t ∈ [t k , t k−1 [ there are k−1 particles with masses m k−1 , . . . , m 1 , located at x k−1 (t) < x k−2 (t) < · · · < x 1 (t), and one compound particle with mass m * k = j≥k m j , located at b k (t).

Consider any subset
Proof. Let k > 1 be given. Using Lemma 1, by continuity we can find τ We claim that with this choice the inequalities (4.5) hold as well. Indeed, . After these preliminaries we can describe our main counterexample. • A sequence of black particles moving horizontally along the x 1 axis. As in (4.1), their masses, initial positions, and initial velocities are defined as (4.7) • A sequence of white particles, moving vertically. Their masses, initial positions, and initial velocities are chosen as Figure 4: Left: The initial configuration of black particles x k moving horizontally and white particles X k moving vertically. Right: if the white particle X k scores a hit, then no other white particle X j with j = k can collide with a black particle along the x 1 axis.
We think of the white particles as bullets, knocking the black particles away from the x 1 axis.
We claim that, with this initial configuration, no sticky solution exists. Indeed, assume that a solution exists, and call S ⊂ IN be the set of all white particles that hit a target, i.e. that collide with a lumped black particle while crossing the x 1 axis. Two cases can be considered, each leading to a contradiction. CASE 1: k ∈ S for some k ≥ 1. We claim that this is possible only if j / ∈ S for all j > k. Indeed, let S ′ k ⊆ S k = {j j ≥ k} denote the set of black particles which are NOT hit by some white particle before time τ k . If S ′ k = S k , then by Lemma 2 the barycenter of these particles and the k-th white particle will not hit its target.
On the other hand, if j / ∈ S for all j > k, then none of the black particles x j with j ≥ k + 1 is hit by white bullets. At time τ k+1 the barycenter of this set of black particles is located at b k+1 (τ k+1 )e 1 = X k+1 (τ k+1 ). As a result, the white particle X k+1 hits its target. We have thus proved the two implications leading to a contradiction.

CASE 2:
The remaining possibility is that S = ∅. But in this case the second implication in (4.9) immediately yields a contradiction.
From the above arguments, it is clear that a sticky solution does not exist, even locally in time.
Remark 2. Following [11], one can construct a family of weak solutions as follows. At each time where an interaction occurs, two particles can either stick together, or continue their separate motion without any change in the velocities. The choice (sticking to each other or not) is made in order to minimize the integral where E(t) is the total energy at time t. It is clear that, as two interacting particles stick together, the momentum is conserved but the energy decreases. Letting ε → 0, in [11] it was claimed (but not proved) that any limit of a sequence of weak solutions which minimize J ε should yield a solution to the sticky particle equations. This is true in the case of finitely many particles, as suggested by intuition, but false in general. In our specific example, for a given ε > 0 a solution which minimizes J ε can be described as follows. The black particles traveling along the x 1 axis always stick to each other after collision. On the other hand, there is an integer N = N (ε) such that: (i) For k > N , at time τ k the white particle X k and the corresponding lumped black particle hit each other at b k (τ k )e 1 , without changing their speed (i.e., without sticking).
(ii) At time τ N , the white particle X N hits the corresponding black particle at b N (τ N )e 1 and sticks to it, knocking it away from the x 1 axis.
(iii) For k < N , the remaining white particles do not hit any black particle.
As ε → 0, since we are putting less and less weight on energy at later times, the single white particle that sticks to its target is X N (ε) , with N (ε) → ∞. In the limit, we obtain a weak solution where all black particles stick to each other, but all white particles hit the black particles without sticking. More precisely, for t ∈ [t k , t k−1 [ this limit solution contains k − 1 black particles with masses m 1 , . . . , m k−1 , located at x j (t) = (x j + tv j )e 1 , j = 1, . . . , k − 1, and one lumped black particle with mass m * k = j≥k m j , located at b k (t)e 1 . In addition, it contains countably many white particles with masses M i , located at This is not a sticky solution.
Remark 3. Given a weak solution consisting of countably many particles Y i (t), on can introduce a measure of "non-stickiness" by setting Here m i denotes the mass of the i-th particle, while describes all couples of particles that hit each other without sticking.
As ε → 0+, for the sequence of weak solutions considered in Remark 2 the measure Φ ε of non-stickiness approaches zero. However, this does not imply that the limit solution should be sticky.

Non-existence and non-uniqueness for continuous initial data
In this last section we extend Example 3 and Example 4 to the case of continuous initial data. Our main goal is to prove: Theorem 1. In any dimension n ≥ 2 there exists an initial datum (ρ,v), such that the system (1.1) does not admit any sticky weak solution in the sense of Definition 3, not even locally in time. Hereρ ∈ C ∞ c (IR n ) is the density of a probability measure, whilev ∈ C c (IR n ) is a continuous initial velocity.
Proof. The main idea is to modify the initial data in Example 4, replacing each point mass, say located atȳ j , by a smooth distribution of mass supported on a small ball B j = {x : |x −ȳ j | ≤ r j }. By choosing an appropriate initial velocity, after a very short time all the mass initially contained inside B j collapses to a point mass and then continues its motion as in the previous example. However, a difficulty arises because the black particles move horizontally along the x 1 axis, while the white particles have velocity −e 2 . This would determine a discontinuity in the velocity field at the origin. To avoid this, we need to add a horizontal component to the velocities of the white particles X k , as shown in Fig. 5. We now describe the construction in greater detail.
1. In this step we replace the black particles x k with a smooth distribution of mass. Choose 0 < β < γ < 1 and consider the initial points and velocities as in (4.7). For each k ≥ 1, we choose a k , r k > 0 and define the smooth function Notice that ψ k ∈ C ∞ c (IR n ), with support contained in the ball B k . = {x ∈ IR n ; |x −x k | ≤ r k }.
As initial velocity we choose a continuous functionv such that Notice that, for t ≥ √ r k all the mass initially located inside the ball B k gets concentrated at the single point The choice of the coefficients a k , r k is made in two stages.
First we choose the sequence of radii r k ↓ 0 decreasing to zero fast enough so that (i) the balls B k , k ≥ 1 are mutually disjoint and (ii) during the time interval [0, √ r k ], particles originating from the ball B k do not interact with any other particles from different balls.
Afterwards, we choose the coefficients a k ↓ 0 decreasing to zero fast enough so that the function ρ(x) .
At this stage we also observe that the conclusion of Lemma 2 remains valid if the particle masses, instead of m k = α k , are given by m k = ψ k (x) dx. Indeed, the only relevant assumption is that m k /m k−1 → 0 fast enough.

2.
In this step we replace the white particles X k with a smooth distribution of mass. For this purpose, it is worth noting that in Example 4 there is a lot of freedom in the choice of the positions and masses of the particles X k . Indeed, the only thing that matters is the identity Let the functions b k (·) and the times τ k be as in (4.5). We can then consider a sequence of particles X k , k ≥ 1, with initial velocity and position given respectively by Observe that (5.10) and (5.12) imply We can now replace the countably many particles X k with a continuous distribution of mass, as in the previous step. For each k ≥ 1, chooseã k , R k > 0 and define the smooth functioñ Notice thatψ k ∈ C ∞ c (IR n ), with support contained inside the ball B k . = {x ∈ IR n ; |x − X k | ≤ R k }.
As initial velocity we choose a continuous functionv : IR n → IR n , with bounded support, that satisfies (5.11) together with Notice that, for t ≥ √ R k all the mass initially located inside the ball B k gets concentrated at the single point The choice of the coefficientsã k , R k is made in two stages.
First we choose the sequence of radii R k ↓ 0 decreasing to zero fast enough so that (i) all the balls B j , B k , j, k ≥ 1 are mutually disjoint and (ii) during the time interval [0, √ R k ], particles originating from the ball B k do not interact with any other particles from different balls.
Afterwards, we choose the coefficientsã k ↓ 0 decreasing to zero fast enough so that the functioñ ρ(x) . = k≥1ψ k (x) is in C ∞ c .
3. By the same argument introduced in Example 4, for the initial data consisting of countably many point massesx k , X k with initial velocitiesv k , V k , no sticky solution exists, not even x x x X X X Figure 5: Modifying the initial data in Figure 4 in order to obtain a continuous distribution of initial velocities. Left: the initial speed of the particles x k , X k approaches the same limit as k → ∞ and x k , X k → 0. Right: a point mass is replaced by a continuous distribution on a ball B k , choosing the initial velocity so that after a short time all the mass is concentrated at one single point. locally in time. By the above construction, the same conclusion holds for an initial distribution of mass with smooth densityρ and continuous initial velocity fieldv(·) Here C > 0 is a normalizing constant, chosen so that ρ(x) dx = 1.
In an entirely similar way, one can modify the initial data in Example 3 and obtain Theorem 2. In any dimension n ≥ 2 there exists an initial datum (ρ,v) such that the system (1.1) admits two distinct sticky weak solutions. Hereρ ∈ C ∞ c (IR n ) is the density of a probability measure, whilev ∈ C c (IR n ) is a continuous initial velocity.
Remark 4. With a more careful construction, counterexamples to the existence and uniqueness could be achieved with an initial velocity distributionv ∈ C 1−ε c (IR n ) which is Hölder continuous, with any exponent strictly smaller than 1. However, this initial velocity field cannot be Lipschitz continuous: in order that all the mass initially inside B k or B k collapse to a point within time t k → 0, the Lipschitz constant ofv in (5.11) and (5.14) must tend to infinity as k → ∞.
On the other hand, if the initial velocity fieldv is continuous with Lipschitz constant L, then it is easy to see that the Cauchy problem (1.1)-(1.2) has a unique local solution defined for 0 ≤ t < L −1 . Indeed, for each 0 ≤ t < L −1 the identity v(t, x + tv(x)) =v(x) uniquely defines a Lipschitz continuous vector field v(t, ·). Inserting this function v(t, ·) in (1.1), one obtains a linear transport equation for ρ, with Lipschitz velocity field and hence with a unique solution.