Sticky particles and the pressureless Euler equations in one spatial dimension

Abstract

We consider the dynamics of finite systems of point masses which move along the real line. We suppose the particles interact pairwise and undergo perfectly inelastic collisions when they collide. In particular, once particles collide, they remain stuck together thereafter. Our main result is that if the interaction potential is semi convex, this sticky particle property can be quantified and is preserved upon letting the number of particles tend to infinity. This is used to show that solutions of the pressureless Euler equations exist for given initial conditions and satisfy an entropy inequality.

A Newton’s equations

Here we show that the ODE system (1.7) has a solution on the interval $[0,\infty )$ for prescribed initial conditions. We recall the standing assumptions that $W:\mathbb {R}\rightarrow \mathbb {R}$ is continuously differentiable, W is even, and that (1.3) holds.

Proposition A.1

Suppose $m_1,\dots , m_N> 0$, $x_1,\dots , x_N\in \mathbb {R}$ and $v_1,\dots , v_N\in \mathbb {R}$. There are

$$\begin{aligned} \gamma _1,\dots ,\gamma _N\in C^2([0,\infty )) \end{aligned}$$

satisfying

$$\begin{aligned} \ddot{\gamma }_i(t)=-\sum ^N_{j=1}m_jW'(\gamma _i(t)-\gamma _j(t)) \end{aligned}$$

(A.1)

for $t>0$ and

$$\begin{aligned} \gamma _i(0)=x_i\quad \text{ and }\quad {\dot{\gamma }}_i(0)=v_i \end{aligned}$$

(A.2)

for $i=1,\dots , N$.

Proof

By Peano’s existence theorem, there is a solution $\gamma _1,\dots ,\gamma _N\in C^2([0,T))$ of the ODE (A.1) for some $T\in (0,\infty ]$ which satisfies the initial conditions (A.2). We may assume that [0, T) is the maximal interval of existence so that this solution cannot be continued to a larger interval if $T<\infty$. In this case, it must be that

$$\begin{aligned} \sup _{t\in [0,T)}|{\dot{\gamma }}_j(t)|=\infty \end{aligned}$$

(A.3)

for some $j=1,\dots , N$. Otherwise, $\sup _{t\in [0,T)}| {\dot{\gamma }}_i(t)|<\infty$ and

$$\begin{aligned} \sup _{t\in [0,T)}| \gamma _i(t)|\le |x_i| +T \sup _{t\in [0,T)}| {\dot{\gamma }}_i(t)|<\infty \end{aligned}$$

for all $i=1,\dots , N$, and this solution $\gamma _1,\dots ,\gamma _N$ could then be continued to $[0,T+\epsilon )$ for some $\epsilon >0$ (Chapter 1 of [10]).

Observe

$$\begin{aligned} \frac{1}{2}\sum ^N_{i=1}m_i{\dot{\gamma }}_i(t)^2+ \frac{1}{2}\sum ^N_{i,j=1}m_im_jW(\gamma _i(t)-\gamma _j(t))=\frac{1}{2}\sum ^N_{i=1}m_i v_i^2+ \frac{1}{2}\sum ^N_{i,j=1}m_im_jW(x_i-x_j)\nonumber \\ \end{aligned}$$

(A.4)

for $t\in [0,T)$. This can be verified by differentiating the left hand side of (A.4) and by using that $\gamma _1,\dots , \gamma _N$ solves (A.1). Arguing as we did to prove Corollary 2.9, we find

$$\begin{aligned} \sum ^N_{i=1}m_i{\dot{\gamma }}_i(t)^2ds\le \varphi '(t)\; \left( \sum ^N_{i=1}m_iv_0(x_i)^2 +\frac{1}{2}\sum ^N_{i,j=1}m_im_jW'(x_i-x_j)^2 \right) \end{aligned}$$

(A.5)

for $t\in [0,T)$. As $m_i>0$ for each $i=1,\dots , N$, (A.3) could not hold for any $j=1,\dots , N$. We conclude that $T=\infty$. $\square$