Inviscid limit for axisymmetric flows without swirl in a critical Besov space
type: Item
instance of: scholarly article
title: Inviscid limit for axisymmetric flows without swirl in a critical Besov space
zbMATH Open document ID: 1203.76039
author: Gang Wu
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2010-06-29T00:00:00Z
review text: The author considers the Cauchy problem for the Navier-Stokes equations assuming zero force and the initial condition axisymmetric without swirl (i.e. in cylindrical coordinates the velocity depends only on \(r\), \(z\) and \(t\), and the angular component is zero). It is well-known that if the initial conditions are sufficiently smooth, the regular axisymmetric solution without swirl exists globally in time. The author studies the problem with initial conditions in the critical Besov space \(B_{2,1}^{5/2}(R^3)\) and shows that the regular solution exists in the space \(C(R^+; B_{2,1}^{5/2}(R^3))\) together with the bound \(\|v(t)\|_{B_{2,1}^{5/2}(R^3)} \leq C_0 e^{\exp(\exp(\exp(C_0 t)))}\). Moreover, if the viscosity \(\nu \to 0^+\), then these solutions converge to \(\overline{v}\), a regular solution to the Euler equations, with the convergence rate \(\|v(t)-\overline{v}(t)\|_{L^2(R^3)} \leq C_0 \nu t e^{\exp(\exp(C_0 t))}\), \(C_0\) independent of \(t\) and \(\nu\).
reviewed by: Milan Pokorný
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 5728541
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: Euler equations
zbMATH Keywords: weak solution
zbMATH Keywords: global-in-time regularity
MaRDI profile type: Publication
full work available at URL: s00033-009-0005-3
OpenAlex ID: W2018803847
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DOI: 10.1007/S00033-009-0005-3
description: scientific article; zbMATH DE number 5728541