Contents Online
Communications in Mathematical Sciences
Volume 20 (2022)
Number 1
Lake equations with an evanescent or emergent island
Pages: 85 – 122
DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n1.a3
Authors
Abstract
We study the asymptotic dynamics of the lake equations in the following two cases, an island shrinking to a point and an emerging island. For both cases, we derive an asymptotic lake-type equation. In the former case, the asymptotic dynamics includes an additional Dirac mass in the vorticity. The main mathematical difficulty is that the equations are singular when the water depth vanishes. We provide new uniform estimates in weighted spaces for the related stream functions which will imply the compactness result.
Keywords
inviscid lakes, point vortex, weighted Sobolev spaces and capacity
2010 Mathematics Subject Classification
35B25, 35B40, 35Q35, 76B03, 76B47
Received 4 April 2021
Received revised 2 June 2021
Accepted 2 June 2021
Published 10 December 2021