Navier-stokes equations in thin 3D domains with navier boundary conditions

Dragoş Iftimie, Geneviève Raugel, George R. Sell

Research output: Contribution to journalArticlepeer-review

41 Scopus citations

Abstract

We consider the Navier-Stokes equations on a thin domain of the form Ωε = {x ∈ ℝ3 | x1, x 2 ∈ (0, 1), 0 < x3 < εg {x1, x2)} supplemented with the following mixed boundary conditions: periodic boundary conditions on the lateral boundary and Navier boundary conditions on the top and the bottom. Under the assumption that ∥u 0H1(Ωε ≤ Cε 1/2, par;Mu0iL2(Omega;ε) C for i ∈ {1,2} and similar assumptions on the forcing term, we show global existence of strong solutions; here u0i denotes the i-th component of the initial data u0 and M is the average in the vertical direction, that is, Mu0i(x1, x2 = (εg)-10εg u0 i(x1, x2, x3) dx3 Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as ε → 0, we prove convergence to a solution of a limiting system which is a Navier-Stokes-like equation where the function g plays an important role. Finally, we compare the attractor of the Navier-Stokes equations with the one of the limiting equation. Indiana University Mathematics Journal

Original languageEnglish (US)
Pages (from-to)1083-1155
Number of pages73
JournalIndiana University Mathematics Journal
Volume56
Issue number3
DOIs
StatePublished - 2007

Keywords

  • Global attractor
  • Global existence
  • Global regularity
  • Navier-stokes equations
  • Thin domain

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