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

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 $C^{3,\frac{1}{2}}\cap L^2$ for the time interval $[0,T)$, where $T>0$ is finite, subject to a uniform force in $C^{1,\frac{1}{2}-\epsilon }\cap L^2$. 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 $C^{1,\frac{1}{3}+}$ threshold regularity of axy-symmetric solutions without swirl.

28 pages