On the Lagrangian description of absolutely continuous curves in the Wasserstein space on the line; well-posedness for the Continuity Equation Mohamed AmsaadA. Tudorascu 34A1234A3435A0235A2435F2035Q3560E0576A02Continuity EquationLagrangian FlowOptimal TransportWasserstein metricWasserstein space The Lagrangian description of absolutely continuous curves of probability measures on the real line is analyzed. Whereas each such curve admits a Lagrangian description as a well-defined flow of its velocity field, further conditions on the curve and/or its velocity are necessary for uniqueness. We identify two seemingly unrelated such conditions that ensure that the only flow map associated with the curve consists of a time-independent rearrangement of the generalized inverses of the cumulative distribution functions of the measures on the curve. At the same time, our methods of proof yield uniqueness within a certain class for the curve associated with a given velocity; that is, they provide uniqueness for the solution of the continuity equation within a certain class of curves. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5727 10.1512/iumj.2015.64.5727 en Indiana Univ. Math. J. 64 (2015) 1835 - 1877 state-of-the-art mathematics http://iumj.org/access/