In this talk I will describe a connection between Arnold's least-action principle for incompressible flows with free boundary and geodesic paths for Wasserstein distance. The least‐action problem for geodesic distance on the `manifold’ of fluid ‐ blob shapes exhibits instability due to microdroplet formation. Using a conformal map formulation we investigate singularity formation in water-wave dynamics neglecting gravity. A connection with fluid mixture models via a variant of Brenier’s relaxed least‐action principle for generalized Euler flows will also be discussed. This is a joint work with Bob Pego and Dejan Slepcev of CMU and Lei Li of SJTU.

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