Date | 2024-04-24 |
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Speaker | Vincent Martinez |

Dept. | CUNY Graduate Center |

Room | 27-317 |

Time | 16:00-17:00 |

We study dissipative perturbations of the 2D generalized surface quasi-geostrophic (gSQG) equations. This family contains the 2D Euler equations in vorticity form at one endpoint, the SQG equation at its midpoint, and the Okhitani equation at the most singular endpoint of the family. Recent work of Bourgain/Li, Elgindi/Masmoudi, Cordoba/Zoroa-Martinez, and Jeong/Kim have established ill-posedness in this family at critical regularity.

This talk will consider “mild" dissipative perturbations of the gSQG equation which recover well-posedness in a setting of critical regularity, but additionally confer a mild degree of regularity instantaneously. These perturbations belong to an intermediate regime that is in between “strongly” dissipative perturbations, which instantaneously gives Gevrey regularity and recover well-posedness at critical regularity (Jolly/Kumar/M 2021), and inviscid regularizations, which do not dissipate energy or instantaneously give additional regularity, but nevertheless recover local well-posedness in critical regularity settings (Chae/Wu 2010).

We show that in this intermediate regime, one may recover local well-posedness in borderline Sobolev regularity settings, as well as maintain a global existence theory at the 2D Euler endpoint. Moreover, we provide a general local existence theory for an entire class of such perturbations that is conjecturally sharp in light of the recent ill-posedness results mentioned above.

This is joint work with A. Kumar (Florida State University).

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