TY - JOUR

T1 - Incompressible fluids in thin domains with navier friction boundary conditions (II)

AU - Hoang, Luan Thach

N1 - Funding Information:
The author acknowledges the support by NSF Grant DMS-0908177.

PY - 2013/6

Y1 - 2013/6

N2 - We study the Navier-Stokes equations for incompressible fluids in a three-dimensional thin two-layer domain whose top, bottom and interface boundaries are not flat. In addition to the Navier friction boundary conditions on the top and bottom boundaries of the domain, and the periodicity condition on the sides, the fluid velocities are subject to an interface boundary condition which relates the normal stress of each fluid to the relative velocity between them on the common boundary. We prove that the strong solutions exist for all time if the initial data and body force, measured in relevant norms, are appropriately large as the domain becomes very thin. In our analysis, the interface boundary condition is interpreted as a variation of the Navier boundary conditions containing an interaction part. The effect of that interaction on the Stokes operator and the nonlinear term of the Navier-Stokes equations is expressed and carefully estimated in different ways in order to obtain suitable inequalities.

AB - We study the Navier-Stokes equations for incompressible fluids in a three-dimensional thin two-layer domain whose top, bottom and interface boundaries are not flat. In addition to the Navier friction boundary conditions on the top and bottom boundaries of the domain, and the periodicity condition on the sides, the fluid velocities are subject to an interface boundary condition which relates the normal stress of each fluid to the relative velocity between them on the common boundary. We prove that the strong solutions exist for all time if the initial data and body force, measured in relevant norms, are appropriately large as the domain becomes very thin. In our analysis, the interface boundary condition is interpreted as a variation of the Navier boundary conditions containing an interaction part. The effect of that interaction on the Stokes operator and the nonlinear term of the Navier-Stokes equations is expressed and carefully estimated in different ways in order to obtain suitable inequalities.

KW - Interface boundary value problem

KW - Navier boundary conditions

KW - Navier-Stokes equations

KW - Strong solution

KW - Thin domain

UR - http://www.scopus.com/inward/record.url?scp=84879938180&partnerID=8YFLogxK

U2 - 10.1007/s00021-012-0123-0

DO - 10.1007/s00021-012-0123-0

M3 - Article

AN - SCOPUS:84879938180

VL - 15

SP - 361

EP - 395

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 2

ER -