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Strong ill-posedness for SQG in critical Sobolev spaces

In-Jee Jeong and Junha Kim

Vol. 17 (2024), No. 1, 133–170
Abstract

We prove that the inviscid surface quasigeostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data H2(𝕋2) without any solutions in LtH2 . Moreover, we prove strong critical norm inflation for C-smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with the two-dimensional incompressible Euler equations.

Keywords
ill-posedness, norm inflation, critical space, surface quasigeostrophic equation, fluid dynamics, nonexistence
Mathematical Subject Classification
Primary: 35Q35, 76B47
Secondary: 35R25
Milestones
Received: 8 September 2021
Revised: 17 March 2022
Accepted: 11 July 2022
Published: 5 February 2024
Authors
In-Jee Jeong
Department of Mathematical Sciences and RIM
Seoul National University
Seoul
South Korea
Junha Kim
School of Mathematics
Korea Institute for Advanced Study
Seoul
South Korea

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