A simple way to tackle fluid mecanic problems is to
introduce and study conserved quantities. The most influencial are the
mass, the energy, the momentum and so on. However, this falls short
when the system at bay is open -meaning that there is fluid entering
and exiting the domain- because the incoming fluid brings mass, energy
and so on with himself. In that situation, one has to think carefully
on how to treat those -now not conserved- quantities.
In this talk, I will present the case of incompressible 2D Euler, and
how we can modify the relative energy to fall on our feet once again
in the proof of uniqueness.