On the zero viscosity limit in incompressible fluids

Abstract

Abstract. We discuss the convergence in the limit of vanishing viscosity of solutions of the Navier-Stokes equations for incompressible fluid flow to solutions of the Euler equations in the presence of boundaries. We present explicit examples in 2 and 3 dimensions for which convergence holds in the energy norm, even when the flow is forced through moving boundaries. We obtain convergence rates in viscosity and discuss concentration of vorticity at the boundary in the limit.

On the Inviscid Limit of the 3D Navier–Stokes Equations with Generalized Navier-Slip Boundary Conditions

“…Corresponding to the slip boundary condition (1.5) for the Euler equations, there are different choices of boundary conditions for the Navier-Stokes equations, one is the mostly-used no-slip boundary condition, i.e., u ε = 0, on Ω (1.6) This Dirichlet type boundary condition was proposed by Stokes ([33]) assuming that fluid particles are adherent to the boundary due to the positive viscosity. Although the well-posedness of the smooth solution to the initial boundary value problem of the Navier-Stokes equations with the no-slip boundary condition can be established quiet easily (at least local in time), the asymptotic convergence of the solution to the corresponding solution of the Euler equations (1.3) with the boundary condition (1.5) as the viscosity coefficient ε tends to zero is one of major open problems except special cases (see [15,29,31,32]) due to the possible appearance of the boundary layers. In general, only some sufficient conditions are obtained for the convergence in L 2 (Ω), see [21,23,38].…”

An elementary approach to the inviscid limits for the 3D Navier–Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations

“…u E (0, x) = u E 0 in Ω, see e.g. the review in Constantin [9] and Mazzucato [21]. In fact, even if both u E and u ν are very smooth and both exist in [0, T ] (for some positive T independent of the viscosity) certain extra-assumptions are needed in order to show, at least, that…”