Some classical and recent results on the Euler equations governing perfect (incompressible
and inviscid) fluid motion are collected and reviewed, with some small novelties scattered …

We review some recent results for a class of fluid mechanics equations called active scalars,
with fractional dissipation. Our main examples are the surface quasi-geostrophic equation …

Proceedings of the International Congress of Mathematicians (ICM 2018) : SMALL SCALES
AND SINGULARITY FORMATION IN FLUID DYNAMIC Page 1 P . I . C . M . – 2018 Rio de …

In recent work of Luo and Hou [10], a new scenario for finite time blow up in solutions of 3D
Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the …

The focus of the course is on small scale formation in solutions of the incompressible Euler
equation of fluid dynamics and associated models. We first review the regularity results and …

We derive a PDE that models the behavior of a boundary layer solution to the
incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D …

Whether the three-dimensional (3D) incompressible Euler equations can develop a finite-
time singularity from smooth initial data with finite energy is a major open problem in partial …

Recently, a new singularity formation scenario for the 3D axi-symmetric Euler equation and
the 2D inviscid Boussinesq system has been proposed by Hou and Luo (Multiscale Model …

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a
major open problem. Several years ago, Luo and Hou [16] proposed a new finite time blow …

The well-posedness problem of Euler equations is one of the most intriguing yet difficult
mathematical problems in fluids. The global existence of classical solutions of 2D Euler …