Time:14:00-15:30, Monday, July 13 2026

Venue: E14-301, Yungu Campus

Speaker: Feng Shao, AMSS

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

Abstract: We investigate self-similar blow-up solutions to the $d$-dimensional axisymmetric incompressible Euler equations without swirl for $d \geq 3$. 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 $u_0 \in C^{1,\alpha}(\mathbb{R}^d) \cap C^\infty(\mathbb{R}^d \setminus \{0\})$. Our construction relies on a fixed-point argument formulated for the self-similar profile 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 the symmetry axis. Furthermore, we prove a finite-codimensional stability result for the self-similar profiles thus obtained. 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 with compactly supported initial vorticity. These solutions are asymptotically self-similar near the blow-up time.

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