ABSTRACT:

The talk is concerned with the classical Euler equations describing
a 2-dimensional inviscid, incompressible fluid.

The goal is to find ``simplest possible" examples of initial data
admitting multiple solutions. We thus consider initial vorticity
concentrated on two wedges, symmetric w.r.t. the origin, with density in $L^p_{loc}$

Recent numerical simulations by Wen Shen (2017) have shown that, by approximating
this same initial data with functions in $\L^\infty$ in two different ways, one obtains
two distinct limit solutions. One contains a single spiraling vortex,
while the other solution contains two vortices.

Toward a rigorous proof of the existence of such solutions, one needs to combine
(i) a posteriori error estimates for the numerical computation,
valid on a bounded domain where the solution is smooth,
with (ii) an analytic construction of the solution in a neighborhood of infinity, and
(iii) an analytic construction of the solution in a neighborhood of one or two spirals' centers
where singularities occur.