On a class of singular solutions to the incompressible 3-D Euler equation

TL;DR

This paper constructs a class of singular solutions to the 3D incompressible Euler equations, demonstrating their existence in three dimensions but not in two, with solutions remaining bounded and smooth until finite-time singularity.

Contribution

It introduces a novel class of singular solutions for the 3D Euler equations and shows their non-existence in 2D, highlighting dimensional differences in solution behavior.

Findings

Constructed singular solutions in 3D Euler equations

Solutions are bounded and smooth until finite-time singularity

No L2 bound on the velocity field despite smoothness

Abstract

A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to finite time where they become singular. Although the solution is smooth and bounded there seems to be no bound in L2 of the velocity field.