On the De Gregorio modification of the Constantin-Lax-Majda Model

TL;DR

This paper investigates the De Gregorio modification of the Constantin-Lax-Majda model, demonstrating through analysis that solutions with smooth initial data tend to equilibria over time, indicating global existence and convergence.

Contribution

The paper provides a mathematical proof that solutions of the De Gregorio model near equilibria exist globally and exhibit convergence, extending understanding of this fluid dynamics model.

Findings

Solutions with smooth initial data exist globally.

Solutions tend to equilibria as time approaches infinity.

Behavior resembles inviscid damping.

Abstract

We study a modification due to De Gregorio of the Constantin-Lax-Majda (CLM) model $\omega_t = \omega H\omega$ on the unit circle. The De Gregorio equation is $\omega_t+u \omega_x-u_x\omega =0, u_x = H\omega.$ In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when $t\to\pm\infty$, in a way resembling inviscid damping. We prove that such a behavior takes place near a manifold of equilibria.