Representation of a solution to the problem of evolution of a point vortex in an ideal fluid
type: Item
instance of: scholarly article
title: Representation of a solution to the problem of evolution of a point vortex in an ideal fluid
zbMATH Open document ID: 0859.76011
DOI: 10.1007/BF02104789
published in: Siberian Mathematical Journal
publication date: 1997-04-09T00:00:00Z
review text: The two-dimensional Euler equations for inviscid incompressible fluid are considered to describe the evolution of the vorticity \(\Omega(z,t)\) and the velocity \(u(z,t)\) with the initial data \(\Omega_0(z)=\omega_0(z) + \Gamma\delta(z-a_0)\), \(z\in \mathbb{C}\), concentrated at a point \(a_0\in \mathbb{C}\), where \(\delta\) is the Dirac function, \(\omega_0\) is the regular part of vorticity, \(z=x+iy\) is the complex variable, and \(\Gamma\) is a real constant. An approximation \(\Omega_\varepsilon\) is introduced with regular initial data \(\Omega_{0\varepsilon}(z)= \omega_0(z)+\varepsilon^{-2} \Gamma\chi_\varepsilon (z)\), where \(\chi_\varepsilon\) stands for the characteristic function of a ball with center \(a_0\) and area \(\varepsilon^2\). It is proved that \(\Omega_\varepsilon \to \omega(z,t)+\Gamma\delta(z-a(t)) \equiv\Omega\) weakly, and \(u_\varepsilon\to v(z,t)+{\Gamma\over 2\pi i} {1\over z-a(t)}\equiv u\) strongly as \(\varepsilon\to 0\). The limit functions \(\Omega\) and \(u\) solve the Euler equations in a weak sense, and the function \(a(t)\) solves the Cauchy problem \({d\over dt}a=v(a,t)\), \(a(0)=a_0\) in a weak sense, too.
Mathematics Subject Classification ID: 76B47
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 916371
zbMATH Keywords: singular initial data
zbMATH Keywords: Dirac function
zbMATH Keywords: complex variable
zbMATH Keywords: approximation
zbMATH Keywords: limit functions
zbMATH Keywords: Cauchy problem
author: V. N. Starovoĭtov
reviewed by: V. V. Shelukhin
MaRDI profile type: Publication
cites work: Q3489649
cites work: On classical solutions of the two-dimensional non-stationary Euler equation
cites work: Q3939421
cites work: Euler evolution for singular initial data and vortex theory
cites work: On the evolution of a concentrated vortex in an ideal fluid
cites work: Q4040553
cites work: Viscoelastic state of a three-layer cylindrical shell during impulsive thermomechanical loading
Recommended article: Uniqueness of a solution to the problem of evolution of a point vortex
Recommended article: On the evolution of a concentrated vortex in an ideal fluid
Recommended article: Q3412346
Recommended article: Vortex Line Representation for the Hydrodynamic Type Equations
Recommended article: The point vortex model for the Euler equation
Recommended article: A transformation of the point vortex equations
Recommended article: Point vortex in a viscous incompressible fluid
Recommended article: Representation of the vortex sheets in the perfect fluid
Recommended article: Q3763751
Recommended article: Q4015154
description: scientific article; zbMATH DE number 916371