The variational description of the Euler equation due to Arnold and the description of geodesics in the Wasserstein space of probability measures share a number of similarities. These are well known and have been exploited in the pioneering work of Brenier. We intend to revisit the relation of these objects and gain new insights on the connections.

 

In 1980's Brenier has introduced a weak notion of solutions of the Euler equation which enabled him to show the existence of generalized  action-minimizing paths studied by Arnold.

We intend to study the connections between Wasserstein geodesics and the Brenier solutions of the Euler equation.

The goal is for this connection to provide new information on the nature of  Brenier solutions.