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

Lars Eric Hientzsch (Institut Fourier, Laboratoire de Mathématiques, Université Grenoble Alpes, Grenoble, France)

Christophe Lacave (Institut Fourier, Laboratoire de Mathématiques, Université Grenoble Alpes, Grenoble, France)

Evelyne Miot (Institut Fourier, Laboratoire de Mathématiques, Université Grenoble Alpes, Grenoble, France)

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