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Blowup Solutions for the De Gregorio Model after Jiajie Chen, Tom Hou, and De Huang

Speaker: Elias Manuelides
Date of Talk: June 2, 2025

The De Gregorio model is a cousin of the well-known Euler equations. One wants to solve the system \[\begin{align*} \omega _t + au \omega _x &= u_x \omega, \\ u_x &= H \omega,\end{align*}\] where \(u, \omega : [0, T]\times \mathbb{R} \to \mathbb{R}\), \(a\in (0, 1]\), and \(H\) is the Hilbert transform.

As usual, one searches for rescalings that can reduce the dimensionality of the problem. Solutions that are on-the-nose equal to these rescalings are self-similar solutions, and one anticipates that solutions ought to converge to these self-similar solutions over time. In this case, running through the computations yields a rescaled self-similar PDE \[(cx + u ) \Omega _x = \left( c + u_x \right) \Omega \] for some blowup profile \(\Omega : \mathbb{R}\to \mathbb{R}\). Alternatively, one can adjust the rate of scaling with time, and instead one obtains the dynamic self-similar equations \[\widetilde{\omega }_ \tau + \left( C_l + \widetilde{u} \right) \widetilde{\omega }_x = \left( C_ \omega + \widetilde{u}_x \right) \widetilde \omega .\]

The theorem proven by Chen et al. is specifically for \(a = 1\):

Theorem 1. (Chen et al., 2021)

When \(a = 1\), there is initial data \(\omega _0 \in C _{c}^{\infty}(\mathbb{R})\) so that the solution with \(a = 1\) develops an (expanding and) asymptotically self-similar singularity in finite time with compactly supported profile \(\Omega \in H^1(\mathbb{R} )\).

The idea is to begin with a \(\overline{\omega }\) that’s nearly self-similar, then provide some linear and nonlinear stability of the dynamic self-similar equation near \(\overline{\omega }\). This ensures the existence of a bona fide self-similar profile “close” to \(\overline{\omega }\).

The nonlinear stability is performed by analysing an “energy” of solutions and showing that, for \(\overline{\omega }\) chosen well and given some tight estimates, the energy must decrease to a stable minimum, corresponding to a self-similar profile. This is where the computer assistance does some heavy lifting.

Two particularly interesting techniques to remark upon: