Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

Córdoba, Diego; Luis, Martínez-Zoroa,; Ożański, Wojciech S.

doi:10.48550/arxiv.2210.17458

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Ill-posedness Norm Inflation and Non-uniqueness1

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“…spaces where the norm scales like the Lipschitz norm of the velocity field. See, for example, the works of Bourgain and Li [5,6], Elgindi and Masmoudi [21], Elgindi and Jeong [20], Jeong [27] and Córdoba, Martínez-Zoroa and Ozaǹski [9].…”

Section: Ill-posedness Norm Inflation and Non-uniquenessmentioning

confidence: 99%

Finite-Time Singularity Formation for Strong Solutions to the Boussinesq System

Elgindi

Jeong

2020

Ann. PDE

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We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.

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