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Yasunori Maekawa

2014-03-06
09:30:00 - 12:00:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

In fluid dynamics it is well known that the vorticity field plays an important role. However, when the fluid domain has a nontrivial boundary, the study of the vorticity equations for viscous incompressible flows (under the no-slip boundary condition on the velocity field) is still less developed. The main obstacle is the boundary condition on the vorticity field, which is nonlocal and nonlinear.

In this lecture we firstly consider the vorticity formulation of the Navier-Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition on the velocity field. In particular, the boundary condition on the vorticity field is explicitly presented.

This vorticity formulation is useful to reveal the anisotropic nature of the vorticity field near the boundary. In the latter part of this lecture, we study the inviscid limit problem for viscous incompressible flows in the half plane.

By using the vorticity formulation we prove the local-in-time convergence of the Navier-Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary.