Vanishing viscosity and the accumulation of vorticity on the boundary

Abstract

We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.2000 Mathematics Subject Classification. Primary 76D05, 76B99, 76D99.

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

“…There is a large literature concerned with related or similar conditional strong L 2 convergence results (a few examples are [1,2,4,5,9,10,19,20]). Some strong L 2 unconditional convergence results for short time do exist.…”

Heat Semigroup Bounds on BV(Ω) and Related Estimates

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On the zero viscosity limit in incompressible fluids

“…This phenomenon is perhaps the most interesting from the point of view of unsteady turbulence and turbulent mixing. Although symmetry prevents boundary layer separation, our analysis nevertheless rigorously shows that the boundary alone generates vorticity and vorticity production is affected by how irregular in time the motion of the boundary is (see also the recent preprint [28]). We recall that vorticity is the curl of the velocity, and in 2D curl = div ⊥ , so that the vorticity can be identified with a scalar.…”