Stability of the two-dimensional point vortices in Euler flows

TL;DR

This paper studies the stability of point vortices in 2D Euler flows, showing that initial vortex configurations remain concentrated near moving points over time, with quantitative descriptions and conditions for long-term stability.

Contribution

It provides a rigorous analysis of vortex concentration stability, including a decomposition of solutions and conditions under which vortices stay close to initial points.

Findings

Vortex solutions decompose into concentrated and small perturbation parts.

Concentration persists for times up to logarithmic scale in perturbation size.

Quantitative description of vortex concentration without compact support assumption.

Abstract

We consider the two-dimensional incompressible Euler equation \[\begin{cases} \partial_t \omega + u\cdot \nabla \omega=0 \\ \omega(0,x)=\omega_0(x). \end{cases}\] We are interested in the cases when the initial vorticity has the form $\omega_0=\omega_{0,\epsilon}+\omega_{0p,\epsilon}$, where $\omega_{0,\epsilon}$ is concentrated near $M$ disjoint points $p_m^0$ and $\omega_{0p,\epsilon}$ is a small perturbation term. First, we prove that for such initial vorticities, the solution $\omega(x,t)$ admits a decomposition $\omega(x,t)=\omega_{\epsilon}(x,t)+\omega_{p,\epsilon}(x,t)$, where $\omega_{\epsilon}(x,t)$ remains concentrated near $M$ points $p_m(t)$ and $\omega_{p,\epsilon}(x,t)$ remains small for $t \in [0,T]$. Second, we give a quantitative description when the initial vorticity has the form $\omega_0(x)=\sum_{m=1}^M \frac{\gamma_m}{\epsilon^2}\eta(\frac{x-p_m^0}{\epsilon})$, where we do not assume $\eta$ to have compact support. Finally, we prove that if $p_m(t)$ remains separated for all $t\in[0,+\infty)$, then $\omega(x,t)$ remains concentrated near $M$ points at least for $t \le c_0 |\log A_{\epsilon}|$, where $A_{\epsilon}$ is small and converges to $0$ as $\epsilon \to 0$.