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On the De Gregorio modification of the Constantin-Lax-Majda Model
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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.
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The spectral picture of self-similar collapse in the Constantin-Lax-Majda equation
On the origin-H2 realization, the CLM collapse linearization has essential spectrum Re λ = -1/2 and point spectrum {0,1}, hence a spectral gap 1/2; weaker L2 realizations fill the whole strip.
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