On the asymptotic behaviour of the 2D Navier‐Stokes equations with Navier friction conditions towards Euler equations

Abstract

In this paper, the two dimensional incompressible Navier-Stokes problem in a bounded domain subject to nonhomogeneous Navier friction boundary conditions is considered. We establish the existence, uniqueness, and regularity of solutions to such problem and we demonstrate convergence towards the incompressible 2D Euler equations in the inviscid limit. We also show the stability and convergence of a time discrete implicit Euler's scheme as both parameters, viscosity and time step, go to zero.We consider in this paper the incompressible Navier-Stokes equations in a bounded domain Ω ⊂ R 2 subject to nonhomogeneous Navier friction boundary conditions. The problem is to find u : Ω × (0, T ) → R 2 the fluid velocity and π : Ω × (0, T ) → R the pressure, such thatwhere ν > 0 is the viscosity coefficient; n and τ are the unit outwards normal and counterclockwise tangent vectors to the boundary ∂Ω, respectively; D(u) = 1 2 (∂ i u j + ∂ j u i ) 1≤i,j≤2 is the deformation tensor; α = α(x) is a scalar friction function; b = b(x, t) and d = d(x, t) are boundary data, d describes the flow across ∂Ω and b represents some extra friction forces at the boundary. We also assume that Ω is simply connected with smooth boundary, then the incompressibility condition for the velocity leads to the following compatibility condition on the boundary data ∂Ω d(x, t) dσ = 0 for t ∈ (0, T ).The inviscid limit, also called vanishing viscosity limit, of the incompressible Navier-Stokes equations is a classical problem in Fluid Mechanics. In the whole space case, the periodic case and more generally in domains without boundaries, the inviscid limit was performed by several authors, we can refer for instance to [3,7,14]. In the physically interesting case where there exist physical boundaries the problem of convergence is essentially open. There is a discrepancy between the Dirichlet boundary conditions imposed to the Navier-Stokes equations and the tangential boundary conditions for the Euler equations, which leads to the formation of a boundary layer.Recently, the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions with homogeneous boundary data, that is, with b = d = 0 in (1), was studied in [2], [8]

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