Let $W\in C^2(\mathbb R)$ be even, have at most quadratic growth, and satisfy $\|W''\|_\infty<+\infty$. For $\mu\in\mathcal P_2(\mathbb R)$, define the interaction force map $\partial_x(W*\mu):\mathbb R\to\mathbb R$ by
Call a curve admissible if it is a narrowly continuous map $\rho:[0,+\infty)\to\mathcal P_2(\mathbb R)$ and, with the velocity field $v:[0,+\infty)\times\mathbb R\to\mathbb R$ defined by $v(t,x):=-\partial_x(W*\rho_t)(x)$, one has
for every $\varphi\in C_c^\infty((0,+\infty)\times\mathbb R)$. Call $\bar\rho\in\mathcal P_2(\mathbb R)$ stationary if the constant curve $t\mapsto\bar\rho$ is admissible.
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Suppose that there exist compactly supported stationary measures $\bar\rho_0,\bar\rho_1\in\mathcal P_2(\mathbb R)$ with
Suppose further that there exist subsets $U_0,U_1\subset\mathcal P_2(\mathbb R)$ such that, for each $i\in\{0,1\}$ and each $\sigma\in U_i$, there exists at least one admissible solution starting from $\sigma$, and every admissible solution $\rho:[0,+\infty)\to\mathcal P_2(\mathbb R)$ with $\rho_0\in U_i$ satisfies
Then the aggregation flow is not globally contractive in $W_2$: there is no rule assigning to every initial measure an admissible solution such that, for all chosen solutions $\alpha,\beta:[0,+\infty)\to\mathcal P_2(\mathbb R)$ and all $t\ge 0$,