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Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids
Hajime Koba, Waseda University, Tokyo, Japan

Memoirs of the American Mathematical Society
2014; 127 pp; softcover
Volume: 228
ISBN-10: 0-8218-9133-2
ISBN-13: 978-0-8218-9133-9
List Price: US$79
Individual Members: US$47.40
Institutional Members: US$63.20
Order Code: MEMO/228/1073
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A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This book 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. The author calls such stationary solutions Ekman layers. This book shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, the author discusses the uniqueness of weak solutions and computes the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. The author also shows 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.

Table of Contents

  • Introduction
  • Formulation and main results
  • Linearized problem
  • Existence of global weak solutions
  • Uniqueness of weak solutions
  • Nonlinear stability
  • Smoothness of weak solutions
  • Some extensions of the theory
  • Appendix A. Toolbox
  • Bibliography
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