On Continuity Properties of Solution Maps of the Generalized SQG Family

We study a family of active scalar equations which interpolate between the 2D incompressible Euler equations and the (inviscid) surface quasi-geostrophic equation. We derive the entire family as the Euler–Arnold equations and show that the Eulerian data-to-solution maps fail to be uniformly continuo...

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Bibliographic Details
Published in:	Vietnam journal of mathematics Vol. 52; no. 3; pp. 689 - 698
Main Authors:	Misiołek, Gerard, Vu, Xuan-Truong
Format:	Journal Article
Language:	English
Published:	Singapore Springer Nature Singapore 01-07-2024 Springer Nature B.V
Subjects:	Euler-Lagrange equation Mathematics Mathematics and Statistics Original Article Topology Data-to-solution maps Non-uniform continuity 35Q31 35B30 SQG equations Euler equations
Online Access:	Get full text
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Summary:	We study a family of active scalar equations which interpolate between the 2D incompressible Euler equations and the (inviscid) surface quasi-geostrophic equation. We derive the entire family as the Euler–Arnold equations and show that the Eulerian data-to-solution maps fail to be uniformly continuous on bounded sets in Sobolev topology.
ISSN:	2305-221X 2305-2228
DOI:	10.1007/s10013-023-00647-x