Localization of Singularities in Inviscid Limit | Numerical Examples

We consider a family of stationary Navier-Stokes ows parameterized by the Reynolds number. When the Reynolds number tends to in nity, what asymptotic behaviors are observed? This is the problem which we would like to address in the present paper. The most renowned phenomenon is the appearance of the boundary layer. This is wellknown. However, what happens in the ow region away from the boundary is much less understood. DiPerna-Majda's concentration ([2]) is one example of singular behavior. Another example, which is less singular than concentration, is found in Kolmogorov ows, see [7]. In the present paper we add two examples whose singularities are less singular than concentration but more singular than the internal layer ([7]) in Kolmogorov ows. The singularity means those which appear as asymptotic behaviors in the limit of R ! +1 ( or ! 0 ). Our examples are Je ery-Hamel ows([9]) and Oseen's ows ([8]). One of the features of these examples is that the family of steady-states has a de nite limit as R!1 but the limit functions have discontinuities although all the given data are C 1 smooth. The degree of discontinuities and localization of singularities depend on individual problems, showing variety of interpretations. We hope such interpretations are helpful in understanding the increase of complexity of uid ows for large Reynolds numbers.