  We study the 'Up the River' problem formulated by Aldous (2002), where a unit
drift is distributed among a finite collection of Brownian particles on $
\mathbb{R}_+ $, which are annihilated once they reach the origin. Starting $ K
$ particles at $ x=1 $, we prove a conjecture of Aldous (2002) that the
'push-the-laggard' strategy of distributing the drift asymptotically (as $
K\to\infty $) maximizes the total number of surviving particles, with
approximately $ \frac{4}{\sqrt{\pi}} K^{1/2} $ surviving particles. We further
establish the hydrodynamic limit of the particle density, in terms of a
two-phase PDE with a moving boundary, by utilizing certain integral identities
and coupling techniques.
