Generalized solutions for the Euler equations in one and two dimensions

Marc Bernot (1) , Alessio Figalli (2) , Filippo Santambrogio (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 JAD - Laboratoire Jean Alexandre Dieudonné
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper we study generalized solutions (in the Brenier's sense) for the Euler equations. We prove that uniqueness holds in dimension one whenever the pressure field is smooth, while we show that in dimension two uniqueness is far from being true. In the case of the two-dimensional disc we study solutions to Euler equations where particles located at a point $x$ go to $-x$ in a time $\pi$, and we give a quite general description of the (large) set of such solutions. As a byproduct, we can construct a new class of classical solutions to Euler equations in the disc.

Keywords: Euler incompressible

Domains: Analysis of PDEs [math.AP]

Marc Bernot, Alessio Figalli, Filippo Santambrogio. Generalized solutions for the Euler equations in one and two dimensions. Journal de Mathématiques Pures et Appliquées, 2009, 91 (2), pp.137-155. ⟨hal-00284725⟩