The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

Abstract

We show that, in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable.

A missed persistence property for the Euler equations and its effect on inviscid limits

“…The arbitrarily strong convergence results proved in [4], some estimates proved for non-flat boundaries in [2] and [3], and the strong convergence results available in the two-dimensional case, led to the conviction that strong convergence results held in the general three-dimensional case, at least in "moderately strong" topologies. In spite of this guess, in reference [5] we have shown that the result is false in a sphere.…”

“…Recently the vanishing viscosity limit problem under the above or similar Navier type conditions has been studied in [2], [8], [12], [14], in the 2D case, and in [2], [3], [5], [7], [21], in the 3D case. See also [23] for the magnetohydrodynamic system and [6] for a different approach to the inviscid limit for the slip-type boundary value problem.…”

“…In spite of the arbitrarily strong convergence results proved in the flat boundary case, see [4], it was shown in reference [5] that the result is false in general, by constructing an explicit family of smooth initial data in the sphere, for which the result fails. Our aim here is to present a more general, simpler and incisive proof.…”

“…In particular, counterexamples can be displayed in arbitrary, smooth, domains. As in [5], the proof is reduced to the lack of a suitable persistence property for the Euler equations. This negative result is proved by a completely different approach.…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…For the case of slip without friction, it was shown in [12] that the boundary condition (3.4) is not necessarily preserved under the Euler evolution in three space dimensions if the boundary is not flat. The strong zero-viscosity limit does hold for free boundary conditions, hence for arbitrary smooth domains in two space dimensions, and for slip-without friction boundary conditions on domains with flat boundary [9,10,147,11]. In fact, only the initial data need to satisfy the stronger free condition at the boundary [14].…”