J. Math. Fluid Mech. 20 (2018), 603–629
2017 Springer International Publishing AG
Journal of Mathematical
Stability Analysis for the Incompressible Navier–Stokes Equations with Navier
Shijin Ding, Quanrong Li
and Zhouping Xin
Communicated by G. P. Galdi
Abstract. This paper is concerned with the instability and stability of the trivial steady states of the incompressible Navier–
Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the
stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the
strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then
nonlinear asymptotic stability holds true. Otherwise, there is a sharp critical viscosity, which distinguishes the linear and
nonlinear stability from instability.
Mathematics Subject Classiﬁcation. 76N10, 35Q30, 35R35.
Keywords. Stability and instability, Navier–Stokes equations, Navier boundary conditions, critical viscosity.
1. Formulation of the Problem
In this paper, we are interested in the following incompressible Navier–Stokes equations
v + v ·∇v + ∇p − μΔv =0,
in Ω,t>0 where Ω is a domain in R
(N ≥ 2), v is the velocity vector ﬁeld, p is the pressure.
System (1.1) is mostly studied with no-slip boundary condition, i.e., Dirichlet boundary condition
which means that the ﬂuid does not slip along the boundary. However, this condition is not always
realistic and leads to induce a strong boundary layer in general. For example, hurricanes and tornadoes,
do slip along the ground, lose energy as they slip and do not penetrate the ground (see ). Other
examples about the slip of the ﬂuid on the boundary occur when moderate pressure is involved such
as in high altitude aerodynamics (see ), or in immiscible two phase ﬂows, the moving contact line
is not compatible with no-slip boundary condition, see [3,15]. As early as 1827, Navier  had taken
such factors into account and proposed a boundary condition as follows which is now called the Navier
boundary condition in which there is a stagnant layer of ﬂuid close to boundary allowing a ﬂuid to slip
v · n =0 and [(−pI + μ(∇v + ∇
v)) · n] · τ = αv · τ, on ∂Ω (1.2)
where n is the outward normal vector ﬁeld to ∂Ωandτ is the tangential vector. In Navier slip boundary
condition (1.2), α stands for a physical meaning parameter which is either a constant or a function in
(∂Ω) , even a smooth matrix . Here we restrict ourselves to the case that α is a constant
For such Navier bounadry value problems, the situation α ≤ 0 which reﬂects, in general, the friction
between the ﬂuid and the boundary, is the classical case and has got extensive attentions by physicists and
mathematicians in studying the existence, uniqueness, regularity and vanishing viscosity to system (1.1).
The ﬁrst pioneer paper on the mathematical rigorous analysis of the Navier–Stokes equation with Navier
boundary conditions should be due to Solonnikov and
Sˇcadilov  for the linearized stationary equations,