Self-similar blow-up solutions of $d$-dimensional incompressible Euler equations with $C^{1,\left(1-2/d\right)-}$ velocity

Abstract

We investigate self-similar blow-up solutions to the $d$-dimensional axisymmetric incompressible Euler equations without swirl for $d\ge 3$. For any $\alpha \in (0,{\alpha }_d)$ with ${\alpha }_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity field satisfies $u_0\in C_{\mathrm{loc}}^{1,\alpha }({\mathbb{R}}^d)\cap C^{\infty }({\mathbb{R}}^d\setminus \{0\})$. Our construction relies on a fixed-point argument formulated for the self-similar profile equations, which form a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, while the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge consists in choosing appropriate function spaces that remain invariant under such nonlinear compositions and that simultaneously capture the exact singular behavior near the origin and the symmetry axis. Furthermore, we establish a finite-codimensional stability result for the self-similar profiles obtained above. As a consequence, after suitable truncation and correction of finitely many unstable modes, we obtain finite-energy blow-up solutions with initial velocity in $C^{1,\alpha }({\mathbb{R}}^d)\cap C^{\infty }({\mathbb{R}}^d\setminus \{0\})\cap L^2({\mathbb{R}}^d)$ and compactly supported initial vorticity. These solutions are asymptotically self-similar near the blow-up time.

103 pages, 1 figure. We add the proof of the finite-codimensional stability of our self-similar profiles, and thus we obtain finite-energy blow-up solutions with initial velocity in $C^{1,\alpha }({\mathbb{R}}^d)\cap C^{\infty }({\mathbb{R}}^d\setminus \{0\})\cap L^2({\mathbb{R}}^d)$ and compactly supported initial vorticity