On Stability of the Spatially Inhomogeneous Navier–Stokes–Boussinesq System with General Nonlinearity

Hajime Koba

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This paper considers L2-asymptotic stability of the spatially inhomogeneous Navier–Stokes–Boussinesq system with general nonlinearity including both power nonlinear terms and convective terms. We construct a local-in-time strong solution of the system by applying semigroup theory on Hilbert spaces and fractional powers of the Stokes–Laplace operator. It is also shown that under some assumptions on an energy inequality the system has a unique global-in-time strong solution when the initial datum is sufficiently small. Furthermore, we investigate the asymptotic stability of the global-in-time strong solution by using an energy inequality, maximal Lp-in-time regularity for Hilbert space-valued functions, and fractional powers of linear operators in a solenoidal L2-space. We introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal Lp-in-time regularity for Hilbert space-valued functions. Our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear.

Original languageEnglish
Pages (from-to)907-965
Number of pages59
JournalArchive for Rational Mechanics and Analysis
Volume215
Issue number3
DOIs
Publication statusPublished - 2014
Externally publishedYes

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Asymptotic stability
Hilbert spaces
Energy Inequality
Asymptotic Stability
Nonlinearity
Strong Solution
Fractional Powers
Hilbert space
Mathematical operators
Regularity
Semigroup Theory
Term
Stationary Solutions
Viscous Fluid
Incompressible Fluid
Linear Operator
Fluids
Operator
Energy

ASJC Scopus subject areas

  • Analysis
  • Mechanical Engineering
  • Mathematics (miscellaneous)

Cite this

On Stability of the Spatially Inhomogeneous Navier–Stokes–Boussinesq System with General Nonlinearity. / Koba, Hajime.

In: Archive for Rational Mechanics and Analysis, Vol. 215, No. 3, 2014, p. 907-965.

Research output: Contribution to journalArticle

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