Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.

We examine the linear stability analysis of the equations governing Rayleigh–Bénard convection flows when the basic temperature profile is unstably stratified solely over a thin horizontal slice of the fluid region. We conduct both asymptotic and numerical analyses on three distinct shapes of the basic temperature: (i) a hyperbolic tangent profile, (ii) a piecewise linear profile and (iii) a step-function profile. In the first two cases, the thin unstably stratified layer is centrally located. The presence of stably stratified regions below and above the central layer diminishes the effect of the velocity and thermal boundary layers that form at the plates. This in turn allows for the analysis of the convection process without the constraints of the horizontal boundaries to be simulated in a finite domain. We obtain expressions for the threshold parameters for convection onset as well as flow features as function of the thickness of the unstably stratified layer. In the limit of a vanishingly small thickness, the hyperbolic tangent profile tends to a step-function profile with a heavy top layer overlying a lighter bottom layer. These two layers are separated by an interface where a jump in density occurs. This situation resembles the Rayleigh–Taylor instability of a horizontal interface except that neither is the interface free nor is the buoyancy diffusion absent. The exploration of this case uncovers new instability threshold values and flow patterns. Finally, we discuss some relevant applications.

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