# Multiple solutions of the 2-dimensional Euler equations

13.06.2019 11:30 - 13:00

Alberto Bressan (Penn State University)

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.

Organiser:
A. Constantin
Location: