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

Abstract

We investigate the axisymmetric incompressible Euler equations without swirl in $\mathbb R^d$ with $d\geq 3$. For any $α\in(0, α_d)$, where $α_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity fields satisfy $u_0\in C^{1,α}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. 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.