Quantitative stability of a class of explicit steady Euler flows in a disk
Fatao Wang, Guodong Wang
TL;DR
The article analyzes the $L^2$-orbital stability of a class of explicit steady Euler flows on the unit disk. By exploiting the conserved quantities $E$, $J$, and $I$ through an energy–Casimir functional, together with two sharp Poincaré-type inequalities, the authors derive a quantitative bound on the $L^2$-distance to a rotational orbit $\mathbf O$ inside $\mathbf V=\operatorname{span}\{J_0(\mathsf j r), J_1(\mathsf j r)\cos\theta, J_1(\mathsf j r)\sin\theta\}$, where $\mathsf j=j_{1,1}$. The main result provides explicit estimates: if $\mathrm{dist}_2(\omega_0,\mathbf O)\le \varepsilon$, then for all times $t$ the distance satisfies $\mathrm{dist}_2(\omega_t,\mathbf O)\le {C B^{-1} (A^2+B^2)^{1/2} \varepsilon + C B^{-1}\varepsilon^2}$ when $B\neq 0$, and $\mathrm{dist}_2(\omega_t,\mathbf O)\le {C |A|^{1/2} \varepsilon^{1/2} + C \varepsilon}$ when $B=0$. The result extends from mean-zero perturbations to general perturbations via a rotating-frame reduction, highlighting that greater radial symmetry (larger $A$, smaller $B$) can amplify instability. These findings connect to broader quantitative orbital stability results for fluid flows and clarify the role of symmetry and Casimirs in 2D Euler dynamics.
Abstract
We provide a short proof of the $L^2$-orbital stability of a class of explicit steady Euler flows in a disk by establishing a quantitative estimate. The main idea is to exploit the conserved quantities of the Euler equation, including the kinetic energy, the enstrophy, and the moment of fluid impulse. Our result seems to suggest that more radial symmetry leads to stronger instability.