From a physical point of view, as well as from a mathematical point of view, horizontal layers (Ekman layers) are now well understood. This is not the casefor vertical layers which are much more complicated, from a physical, analytical and mathematical point of view, and many open questions in all these directions remain open. Let us, in this section, consider a domain Ω with vertical boundaries. Namely, let Ωh be a domain of R2 and let Ω=Ωh × [0, 1]. This domain has two types of boundaries: • horizontal boundaries Ωh × {0} (bottom) and Ωh ×{1} (top) where Ekman layers are designed to enforce Dirichlet boundary conditions; • vertical boundaries ∂Ωh × [0, 1] where again a boundary layer is needed to ensure Dirichlet boundary conditions. These layers, however, are not of Ekman type, since r is now parallel to the boundary. Vertical layers are quite complicated. They in fact split into two sublayers: one of size E1/3 and another of size E1/4 where E =νε denotes the Ekman number. This was discovered and studied analytically by Stewartson and Proudman. Vertical layers can be easily observed in experiments (at least the E1/4 layer, the second one being too thin) but do not seem to be relevant in meteorology or oceanography, where near continents, effects of shores, density stratification, temperature, salinity, or simply topography are overwhelming and completely mistreated by rotating Navier–Stokes equations. In MHD, however, and in particular in the case of rotating concentric spheres, they are much more important. Numerically, they are easily observed, at large Ekman numbers E (small Ekman numbers being much more difficult to obtain). The aim of this section is to provide an introduction to the study of these layers, a study mainly open from a mathematical point of view. First we will derive the equation of the E1/3 layer. Second we will investigate the E1/4 layer and underline its similarity with Prandtl’s equations. In particular, we conjecture that E1/4 is always linearly and nonlinearly unstable. We will not prove this latter fact, which would require careful study of what happens at the corners of the domain, a widely open problem.
On a Lp-estimate for the linearized compressible Navier–Stokes equations with the Dirichlet boundary conditions
