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

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

2010-08-19

arxiv.org/abs/1008.3252v1

Mathematics

Analysis of PDEs

Scientific paper

The resolution of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to that of the corresponding results for the same problem in the whole space. The approach is particularly suitable for proving new results in strong norms. To determine whether this extension is available, turns out to be a simple exercise. The verification depends on a few general features of the functional space X related to the space variables. Hence, we present an approach as much as possible independent of the particular space X. We appeal to a reflection technique. Hence a crucial assumption is to be in the presence of flat boundaries (see below). Instead of stating "general theorems" we rather prefer to illustrate how to apply our results by considering a couple of interesting problems. As a main example, we show that the resolution of a class of problems for the evolution Navier-Stokes equations under a slip boundary condition can be reduced to that of the corresponding results for the Cauchy problem. In particular, we show that sharp vanishing viscosity limit results that hold for the evolution Navier-Stokes equations in the whole space can be extended to the boundary value problem in the half-space. We also show some applications to non-Newtonian fluid problems.

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