### Abstract

A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This booklet constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. We call such stationary solutions Ekman layers. This booklet shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, we discuss the uniqueness of weak solutions and compute the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. It is also shown that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.

Original language | English |
---|---|

Pages (from-to) | 1-127 |

Number of pages | 127 |

Journal | Memoirs of the American Mathematical Society |

Volume | 228 |

Issue number | 1073 |

DOIs | |

Publication status | Published - 2014 Mar |

### Fingerprint

### Keywords

- Asymptotic stability
- Boussinesq system
- Coriolis force
- Ekman spiral
- Maximal Lp-regularity
- Perturbation theory
- Real interpolation theory
- Smoothness and regularity
- Stability of Ekman boundary layers
- Stratification effect
- Strong energy equality
- Strong energy inequality
- Strong solutions
- Uniqueness of weak solutions
- Weak solutions

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Memoirs of the American Mathematical Society*,

*228*(1073), 1-127. https://doi.org/10.1090/memo/1073

**Nonlinear stability of Ekman boundary layers in rotating stratified fluids.** / Koba, Hajime.

Research output: Contribution to journal › Article

*Memoirs of the American Mathematical Society*, vol. 228, no. 1073, pp. 1-127. https://doi.org/10.1090/memo/1073

}

TY - JOUR

T1 - Nonlinear stability of Ekman boundary layers in rotating stratified fluids

AU - Koba, Hajime

PY - 2014/3

Y1 - 2014/3

N2 - A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This booklet constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. We call such stationary solutions Ekman layers. This booklet shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, we discuss the uniqueness of weak solutions and compute the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. It is also shown that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.

AB - A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This booklet constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. We call such stationary solutions Ekman layers. This booklet shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, we discuss the uniqueness of weak solutions and compute the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. It is also shown that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.

KW - Asymptotic stability

KW - Boussinesq system

KW - Coriolis force

KW - Ekman spiral

KW - Maximal Lp-regularity

KW - Perturbation theory

KW - Real interpolation theory

KW - Smoothness and regularity

KW - Stability of Ekman boundary layers

KW - Stratification effect

KW - Strong energy equality

KW - Strong energy inequality

KW - Strong solutions

KW - Uniqueness of weak solutions

KW - Weak solutions

UR - http://www.scopus.com/inward/record.url?scp=84891958891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891958891&partnerID=8YFLogxK

U2 - 10.1090/memo/1073

DO - 10.1090/memo/1073

M3 - Article

AN - SCOPUS:84891958891

VL - 228

SP - 1

EP - 127

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 1073

ER -