Singular vortex patches in a fluid of low viscosity
type: Item
instance of: scholarly article
title: Singular vortex patches in a fluid of low viscosity
zbMATH Open document ID: 1127.35037
DOI: 10.4171/RMI/464
author: Taoufik Hmidi
published in: Revista Matemática Iberoamericana
publication date: 2007-05-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1161871346
full work available at URL: https://eudml.org/doc/41982
review text: The author studies the motion of a bidimensional, incompressible, small viscous fluid as the Navier-Stokes equations \[ \partial_t v_\nu+ v_\nu\nabla v_\nu- \nu\Delta v_\nu= -\nabla p_\nu,\quad \operatorname{div} v_\nu= 0,\quad v_\nu|_{t= 0}= v^0,\tag{1} \] near the motion of a fluid of null viscosity in the system of Euler equations \[ \partial_t v+v\nabla v\neq -\nabla p,\quad\operatorname{div} v= 0,\quad v|_{t= 0}= v^0, \tag{2} \] The used notations are evident. The motions are in the presence of singular vortex patches. The author considers only small viscosity, because he is interested in the comparison of the results for \(\nu\neq 0\) and \(\nu= 0\). He considers \(\lim_{\nu\to 0} v_\nu\) obtained from (1) and compares it with \(v\) obtained from (2). The singularities of the vortex patches concern the singularities which can appear on the boundary \(\partial\Omega\) of the domain \(\Omega\) occupied by the vortex patch: coin, cusp or other singularities. The history of the presented problem is sketched in the introduction, followed by five chapters and an appendix, where there are analysed the following subjects: the velocity field, the singularities on \(\partial\Omega\), the dynamics of the singular patches and the comparison between the motion in (1) and (2). The paper is very rich in content and requires a high level of mathematical knowledge.
reviewed by: Vasile Ionescu
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D17
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 76B47
zbMATH DE Number: 5152581
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: Euler equations
zbMATH Keywords: vortex patches
zbMATH Keywords: Littlewood-Paley theory
MaRDI profile type: Publication
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OpenAlex ID: W4298137196
description: scientific article; zbMATH DE number 5152581