Stability of stationary solutions to the Navier–Stokes equations in the Besov space

Hideo Kozono; Senjo Shimizu

doi:10.1002/mana.202100150

Stability of stationary solutions to the Navier–Stokes equations in the Besov space

Hideo Kozono^*, Senjo Shimizu

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space (Formula presented.) for (Formula presented.). It is clarified that if w is small in (Formula presented.) for (Formula presented.) and (Formula presented.), then for every small initial disturbance (Formula presented.) with (Formula presented.) and (Formula presented.) ((Formula presented.)), there exists a unique solution (Formula presented.) of the nonstationary Navier–Stokes equations on (0, ∞) with (Formula presented.) such that (Formula presented.) and (Formula presented.) as (Formula presented.), for (Formula presented.), (Formula presented.), and small (Formula presented.).

Original language	English
Journal	Mathematische Nachrichten
DOIs	https://doi.org/10.1002/mana.202100150
Publication status	Accepted/In press - 2023

Keywords

homogeneous Besov space
Navier–Stokes equations
stability
stationary solution

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1002/mana.202100150

Cite this

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title = "Stability of stationary solutions to the Navier–Stokes equations in the Besov space",

abstract = "We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space (Formula presented.) for (Formula presented.). It is clarified that if w is small in (Formula presented.) for (Formula presented.) and (Formula presented.), then for every small initial disturbance (Formula presented.) with (Formula presented.) and (Formula presented.) ((Formula presented.)), there exists a unique solution (Formula presented.) of the nonstationary Navier–Stokes equations on (0, ∞) with (Formula presented.) such that (Formula presented.) and (Formula presented.) as (Formula presented.), for (Formula presented.), (Formula presented.), and small (Formula presented.).",

keywords = "homogeneous Besov space, Navier–Stokes equations, stability, stationary solution",

author = "Hideo Kozono and Senjo Shimizu",

note = "Funding Information: The authors would like to express their sincere thanks to the referees for their valuable comments. The research of H. Kozono was partially supported by JSPS Grant‐in‐Aid for Scientific Research (S) 16H06339. The research of S. Shimizu was partially supported by JSPS Grant‐in‐Aid for Scientific Research (B) 16H03945. Publisher Copyright: {\textcopyright} 2023 Wiley-VCH GmbH.",

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journal = "Mathematische Nachrichten",

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T1 - Stability of stationary solutions to the Navier–Stokes equations in the Besov space

AU - Kozono, Hideo

AU - Shimizu, Senjo

N1 - Funding Information: The authors would like to express their sincere thanks to the referees for their valuable comments. The research of H. Kozono was partially supported by JSPS Grant‐in‐Aid for Scientific Research (S) 16H06339. The research of S. Shimizu was partially supported by JSPS Grant‐in‐Aid for Scientific Research (B) 16H03945. Publisher Copyright: © 2023 Wiley-VCH GmbH.

PY - 2023

Y1 - 2023

N2 - We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space (Formula presented.) for (Formula presented.). It is clarified that if w is small in (Formula presented.) for (Formula presented.) and (Formula presented.), then for every small initial disturbance (Formula presented.) with (Formula presented.) and (Formula presented.) ((Formula presented.)), there exists a unique solution (Formula presented.) of the nonstationary Navier–Stokes equations on (0, ∞) with (Formula presented.) such that (Formula presented.) and (Formula presented.) as (Formula presented.), for (Formula presented.), (Formula presented.), and small (Formula presented.).

AB - We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space (Formula presented.) for (Formula presented.). It is clarified that if w is small in (Formula presented.) for (Formula presented.) and (Formula presented.), then for every small initial disturbance (Formula presented.) with (Formula presented.) and (Formula presented.) ((Formula presented.)), there exists a unique solution (Formula presented.) of the nonstationary Navier–Stokes equations on (0, ∞) with (Formula presented.) such that (Formula presented.) and (Formula presented.) as (Formula presented.), for (Formula presented.), (Formula presented.), and small (Formula presented.).

KW - homogeneous Besov space

KW - Navier–Stokes equations

KW - stability

KW - stationary solution

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