Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type
type: Item
instance of: scholarly article
title: Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type
zbMATH Open document ID: 1172.35062
author: Ryo Takada
published in: Journal of Evolution Equations
publication date: 2009-06-08T00:00:00Z
review text: The author introduce Besov type function spaces, based on the weak \(L^p\) spaces instead of the standard \(L^p\) spaces. Some important properties of those space are proved. In particular, embedding theorems and Fourier multiplier theorem are presented. The Cauchy problem to the Euler equations is studied in Besov type function spaces. \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=0,\quad \text{div}\,v=0, \quad x\in \mathbb R^n,\;t\in (0,\infty),\\ &v(x,0)=v_0(x), \quad x\in \mathbb R^n. \end{aligned} \] Here \(v(x,t)=(v_1,\dots,v_n)\) is the velocity field, \(p\,(x,t)\) is the pressure, \(v_0(x)\) is the given initial velocity satisfying div\,\(v_0=0\), \(n\geq 3\). A local-in-time unique existence and blow-up criterion of solutions to the problem are proved in Besov spaces of weak type.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76B03
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5562658
zbMATH Keywords: Euler equation
zbMATH Keywords: local existence
zbMATH Keywords: blow-up criterion
zbMATH Keywords: besov spaces of weak type
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: s00028-008-0403-6
OpenAlex ID: W1972340932
DOI: 10.1007/S00028-008-0403-6
description: scientific article; zbMATH DE number 5562658