Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity

TL;DR

This paper constructs self-similar blow-up solutions for the incompressible Euler equations in higher dimensions, revealing new singularity formations with specific regularity properties.

Contribution

It introduces a fixed-point framework for self-similar profiles involving coupled elliptic-transport systems in high dimensions.

Findings

Constructed blow-up solutions with initial velocity in $C^{1,rac{d-2}{d}}$

Used a fixed-point approach for coupled elliptic-transport systems

Revealed singular behavior near the origin and symmetry axis

Abstract

We investigate the axisymmetric incompressible Euler equations without swirl in $\mathbb R^d$ with $d\geq 3$. For any $\alpha\in(0, \alpha_d)$, where $\alpha_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity fields satisfy $u_0\in C^{1,\alpha}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point framework formulated for the self-similar profile system, which takes the form of a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, whereas the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge lies in selecting suitable function spaces that remain invariant under such nonlinear compositions, while simultaneously capturing the exact singular behavior near the origin and symmetry axis.