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 language | English |
|---|---|
| Pages (from-to) | 907-965 |
| Number of pages | 59 |
| Journal | Archive for Rational Mechanics and Analysis |
| Volume | 215 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2014 |
| Externally published | Yes |
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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 journal › Article
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TY - JOUR
T1 - On Stability of the Spatially Inhomogeneous Navier–Stokes–Boussinesq System with General Nonlinearity
AU - Koba, Hajime
PY - 2014
Y1 - 2014
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84922093209&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84922093209&partnerID=8YFLogxK
U2 - 10.1007/s00205-014-0802-5
DO - 10.1007/s00205-014-0802-5
M3 - Article
AN - SCOPUS:84922093209
VL - 215
SP - 907
EP - 965
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
SN - 0003-9527
IS - 3
ER -