Special Math Colloquium

We describe a recent construction of self-similar blow-up solutions of the incompressible Euler equation. A consequence of the construction is that there exist finite-energy $C^{1,a}$ solutions to the Euler equation that develop a singularity in finite time for some range of $a>0$. The approach we follow is to isolate a simple non-linear equation that encodes the leading order dynamics of solutions to the Euler equation in some regimes and then prove that the simplified equation has stable self-similar blow-up solutions. This builds off of previous work with I. Jeong and with T. Ghoul and N. Masmoudi.