On the reduction of PDE's problems in the half-space, under the slip boundary condition, to the corresponding problems in the whole space

摘要

The resolution of a very large class of linear and non-linear, stationary andevolutive partial differential problems in the half-space (or similar) underthe slip boundary condition is reduced here to that of the correspondingresults for the same problem in the whole space. The approach is particularlysuitable for proving new results in strong norms. To determine whether thisextension is available, turns out to be a simple exercise. The verificationdepends on a few general features of the functional space X related to thespace variables. Hence, we present an approach as much as possible independentof the particular space X. We appeal to a reflection technique. Hence a crucialassumption is to be in the presence of flat boundaries (see below). Instead ofstating "general theorems" we rather prefer to illustrate how to apply ourresults by considering a couple of interesting problems. As a main example, weshow that the resolution of a class of problems for the evolution Navier-Stokesequations under a slip boundary condition can be reduced to that of thecorresponding results for the Cauchy problem. In particular, we show that sharpvanishing viscosity limit results that hold for the evolution Navier-Stokesequations in the whole space can be extended to the boundary value problem inthe half-space. We also show some applications to non-Newtonian fluid problems.