Support growth of vorticity for bi-rotational Euler flows in high dimensions

TL;DR

This work analyzes the incompressible Euler equations in high dimensions ($d\ge4$) under bi-rotational symmetry without swirl, reducing the dynamics to a scalar vorticity equation for $w$ on the quadrant $\Pi$. By deriving a coupled scalar system for $w$ and the stream function $\psi$, the authors establish local well-posedness for patch-type data and a blow-up criterion tied to the growth of the support, $L(t)=R(t)+S(t)$. The main contribution is a non-quantitative but rigorous result showing infinite growth of the patch support, i.e., $L(t)\to\infty$ along a time sequence, even when the maximal lifespan may be finite or infinite. This advances understanding of patch dynamics and singularity formation mechanisms in high-dimensional, bi-rotational Euler flows and connects to lake-equation-type depths $d(r,s)=r^n s^m$ in the $(r,s)$-plane.

Abstract

We study incompressible Euler equations in $\mathbb{R}^d$ with $d \ge 4$ under bi-rotational symmetry without swirl, which reduces the Euler equations to a scalar vorticity advection in the first quadrant. We show that patch type initial vorticities exhibit infinite growth of the support diameter.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Remark 2.1
• Proposition 3.1
• proof
• proof : Proof of Theorem \ref{['thm:0']}
• Definition 4.1
• Lemma 4.2