Navier-Stokes Equations with Navier Boundary Conditions for an Oceanic Model

Luan T. Hoang, George R. Sell

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

We consider the Navier-Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a "large" set in the Sobolev space H1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray-Hopf weak solutions of the Navier-Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier-Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.

Original languageEnglish
Pages (from-to)563-616
Number of pages54
JournalJournal of Dynamics and Differential Equations
Volume22
Issue number3
DOIs
StatePublished - 2010

Keywords

  • Geophysical fluid dynamics
  • Global attractor
  • Global existence
  • Navier-Stokes equations
  • Strong solution
  • Thin domain

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