Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems

We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions, introduced by Brenier, Gangbo, Savar\'{e} and Westdickenberg, and entropy solutions, studied by Nguyen and Tudorascu for the Euler--Poisson system, are equivalent. For the Euler--Poisson system this can be seen as a generalization to second-order systems of the equivalence between $L^2$-gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier. The key observation is an equivalence between Ole\u{\i}nik's E-condition for conservation laws and a characterization due to Natile and Savar\'{e} of the normal cone for $L^2$-gradient flows. This new equivalence allows us to define unique solutions after blow-up for classical solutions of the Euler--Poisson system with quadratic confinement due to Carrillo, Choi and Zatorska, as well as to describe their asymptotic behavior.