On the vanishing viscosity limit in a disk

Details On the vanishing viscosity limit in a disk On the vanishing viscosity limit in a disk

2006-12-10

arxiv.org/abs/math-ph/0612027v1

Mathematische Annalen, 343:701-726, 2009

Physics

Mathematical Physics

Scientific paper

10.1007/s00208-008-0287-3

We say that the solution u to the Navier-Stokes equations converges to a solution v to the Euler equations in the vanishing viscosity limit if u converges to v in the energy norm uniformly over a finite time interval. Working specifically in the unit disk, we show that a necessary and sufficient condition for the vanishing viscosity limit to hold is the vanishing with the viscosity of the time-space average of the energy of u in a boundary layer of width proportional to the viscosity due to modes (eigenfunctions of the Stokes operator) whose frequencies in the radial or the tangential direction lie between L and M. Here, L must be of order less than 1/(viscosity) and M must be of order greater than 1/(viscosity).

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