Jim Kelliher, Brown University
Vanishing viscosity in the presence of a boundary
The behavior of an incompressible fluid as we let its viscosity approach zero is poorly understood when a boundary is present. Understanding this behavior is of both theoretical and practical interest. The most natural question to ask is whether in this vanishing viscosity limit solutions to the Navier-Stokes equations, which describe an incompressible viscous fluid, converge in the energy norm to a solution to the Euler equations, which describe an incompressible inviscid fluid. Under the assumption that the viscous fluid remains stationary on the boundary (no-slip boundary conditions) this is one of the oldest and most important questions in mathematical fluid mechanics, and one whose full solution does not appear likely to arrive anytime soon. I will describe some of the existing partial results concerning the vanishing viscosity limit and related problems, and will mention some potentially tractable approaches to extend some of these partial results.
TUESDAY, March 25, 2008