Blow-up for the incompressible 3D-Euler equations with uniform $C^{1,\frac{1}{2}-\epsilon}\cap L^2$ force

TL;DR

This paper demonstrates a finite-time blow-up for solutions to the forced 3D incompressible Euler equations with specific regularity and force conditions, highlighting a new blow-up mechanism without relying on self-similar coordinates.

Contribution

The authors construct solutions exhibiting blow-up in finite time for non-axisymmetric 3D Euler equations with a uniform force, without using self-similar coordinates and beyond certain regularity thresholds.

Findings

Finite-time blow-up of solutions with smoothness except at the origin.

Blow-up characterized by divergence of the integral of the gradient magnitude.

Method applies to solutions beyond the $C^{1,1/3+}$ regularity threshold.

Abstract

This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space $C^{3,\frac12}\cap L^2$ for the time interval $[0, T)$, where $T > 0$ is finite, subject to a uniform force in $C^{1,\frac12 -\epsilon}\cap L^2$. Remarkably, our methodology results in a blow-up: as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity, all while preserving the solution's smoothness throughout, except at the origin. In the process of our blow-up construction, self-similar coordinates are not utilized and we are able to treat solutions beyond the $C^{1,\frac13+}$ threshold regularity of axy-symmetric solutions without swirl.