Note on the analysis of Orr-Sommerfeld equations and application to boundary layer stability (Mathematical Analysis of Viscous Incompressible Fluid)

This note is about the stability of Prandtl boundary layer expansions, in the context of the two-dimensional Navier-Stokes equation. We explain the main arguments behind the recent analysis of [5], which shows an optimal Gevrey stability result for expansions of shear flow type. The core of the reasoning is a resolvent estimate for the fourth order Orr-Sommerfeld equation, which is derived by an iterative process reminiscent of splitting methods in numerical analysis. This iteration is explained here in an abstract setting, that might be helpful to other applications