Advances in Differential Equations

The Navier-Stokes equations and weak Herz spaces

Yohei Tsutsui

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Abstract

In this paper, we discuss the Cauchy problem for Navier-Stokes equations in homogeneous weak Herz spaces $W{\dot{K}}^\alpha_{p,q}({\mathbb{R^{\textit{n}}}})$. More precisely, we construct the solution in the class $L^\infty(0,T; W{\dot{K}}^\alpha_{p,q})$ with the initial data in $W{\dot{K}}^\alpha_{p,q}$. Further, we consider the blow-up phenomena of time-local solutions and the uniqueness of global solutions with large initial data in $W{\dot{K}}^\alpha_{p,q}$. Also, we give several embeddings of weak Herz spaces into homogeneous Besov spaces $B^{-\alpha}_{p,\infty}({\mathbb{R^{\textit{n}}}})\ (\alpha >0),$ or $bmo^{-1}({\mathbb{R^{\textit{n}}}})$.

Article information

Source
Adv. Differential Equations Volume 16, Number 11/12 (2011), 1049-1085.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703112

Mathematical Reviews number (MathSciNet)
MR2858524

Zentralblatt MATH identifier
1236.35114

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76D05: Navier-Stokes equations [See also 35Q30]

Citation

Tsutsui, Yohei. The Navier-Stokes equations and weak Herz spaces. Adv. Differential Equations 16 (2011), no. 11/12, 1049--1085. https://projecteuclid.org/euclid.ade/1355703112.


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