On the existence of convex classical solutions to a generalized Prandtl-Batchelor free-boundary problem. II
type: Item
instance of: scholarly article
title: On the existence of convex classical solutions to a generalized Prandtl-Batchelor free-boundary problem. II
zbMATH Open document ID: 1014.35092
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2003-03-27T00:00:00Z
review text: For Part I, cf. [the author, ibid. 49, 1-30 (1998; Zbl 0903.35093)]. We give an analytical proof of the existence of convex classical solutions for the (convex) Prandtl-Batchelor free boundary problem in fluid dynamics. In this problem, a convex vortex core of constant vorticity \(\mu>0\) is embedded in a closed irrotational flow inside a closed, convex vessel in \(\mathbb{R}^2\). The unknown boundary of the vortex core is a closed curve \(\Gamma\) along which \((v^+)^2-(v^-)^2=\Lambda\), where \(v^+\) and \(v^-\) denote, respectively, the exterior and interior flow-speeds along \(\Gamma\), and \(\Lambda\) is a given constant. Our existence results all apply to the natural multidimensional mathematical generalization of the above problem. The present existence theorems are the only ones available for the Prandtl-Batchelor problem for \(\Lambda>0\), because (a) the author's prior existence treatment was restricted to the case where \(\Lambda<0\), and because (b) there is no analytical existence theory available for this problem in the nonconvex case, regardless of the sign of \(\Lambda\).
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76B47
zbMATH DE Number: 1889156
zbMATH Keywords: existence of convex classical solutions
zbMATH Keywords: Prandtl-Batchelor free boundary problems
zbMATH Keywords: vortex core
zbMATH Keywords: convex, steady state vorticity localization in potential flow
zbMATH Keywords: vortex-sheet skin effect
author: Andrew Acker
MaRDI profile type: Publication
full work available at URL: s00033-002-8163-6
OpenAlex ID: W2008042897
DOI: 10.1007/S00033-002-8163-6
description: scientific article; zbMATH DE number 1889156