A 3D Euler equation solution with 2D sets of singularities and data with Hoelder continuous first order derivatives

TL;DR

This paper constructs a specific solution to the 3D Euler equations that develops finite-time singularities with Hoelder continuous derivatives, demonstrating the formation of 2D singular sets.

Contribution

It provides an explicit example of a 3D Euler solution with Hoelder continuous initial data that develops finite-time singularities on a 2D set, extending beyond the singularity time.

Findings

Finite-time singularity formation in 3D Euler equations.

Existence of solutions with Hoelder continuous derivatives leading to singularities.

Extension of solutions beyond the singularity time on a 2D set.

Abstract

An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder continuous first order spatial derivatives. Such a solution branch can be extended beyond a time section at some positive finite time where a two dimensional set of singularities is located.