Calculus of Variations and Geometric Measure Theory

M. Bernot - A. Figalli - F. Santambrogio

Generalized solutions for the Euler equations in one and two dimensions

created by santambro on 03 Jun 2008
modified on 01 Nov 2008


Accepted Paper

Inserted: 3 jun 2008
Last Updated: 1 nov 2008

Journal: J. Math. Pures Appl.
Year: 2008


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 equations, incompressible fluids