Methods and Applications of Analysis

Volume 23 (2016)

Number 4

The steady Navier–Stokes and Stokes systems with mixed boundary conditions including one-sided leaks and pressure

Pages: 329 – 364



Tujin Kim (Institute of Mathematics, Academy of Sciences, Pyongyang, DPR Korea)

Daomin Cao (Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China)


In this paper we are concerned with the steady Navier–Stokes and Stokes problems with mixed boundary conditions involving Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, rotation, stress and normal derivative of velocity together. Relying on the relations among strain, rotation, normal derivative of velocity and shape of boundary surface, we have variational formulations for the problems, which consist of five formulae with five unknowns. We get the variational inequalities equivalent to the formulated variational problems, which have one unknown. Then, we study the corresponding variational inequalities and relying the results for variational inequalities, we get existence, uniqueness and estimates of solutions to the Navier–Stokes and Stokes problems with the boundary conditions. Our estimates for solutions do not depend on the thresholds for slip and leaks.


Navier–Stokes equations, variational inequality, mixed boundary condition, Tresca slip, leak boundary conditions, one-sided leak, pressure boundary condition, existence, uniqueness

2010 Mathematics Subject Classification

35J87, 35Q30, 49J40, 76D03, 76D05

Published 3 April 2017