Braids and the variational construction of solutions of 2-d Euler equations

Flows of ideal incompressible fluid can be regarded as geodesics on the group of volume preserving diffeomorphisms of the flow domain. It is natural to look for the shortest geodesics. However, the minimizers of the length functional deliver merely weak solutions of the Lagrange equations, which are continuous analogues of braids. The main part of the work is the proof of their regularity which means that there exists a velocity field corresponding to the length minimizing braid. This field is a weak solution of the Euler equation.

Then we introduce an extension of the braids named "folded braids". Their minimization results in a weak solution of the Euler equation which tends to zero in a weak sense as $t\to 0$. This can be interpreted as a weak solution which is zero for all $t<0$ giving a new class of examples of the phenomenon discovered by Scheffer.