Blowup Solutions for the De Gregorio Model after Jiajie Chen, Tom Hou, and De Huang
Speaker: Elias ManuelidesDate 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:
- In the linear stability analysis, one can perform a series of reductions to obtain an optimisation problem involving a finite-rank linear operator, which is very compatible with a computer.
- The numerics of the problem are very closely intertwined with the analysis. The linear estimates produce constant factors that influence the nonlinear analysis, which in turn produces more constants that need to be juiced to get the final result.