Invertibility of the Poiseuille Linearization for Stationary Two-Dimensional Channel Flows: Nonsymmetric Case
It was shown in the first part that in Sobolev spaces that incorporate symmetry requirements, the linearization at the Poiseuille solution of the Navier-Stokes system is an isomorphism regardless of the flux of the solution. This paper is devoted to the same question when the symmetry assumptions are removed. While the linearization remains one-to-one with dense range for all the fluxes, we are only able to prove its invertibility up to an upper bound for the flux. Although results of this type are known, the upper bound found here is more than 50 times larger than the currently known one. Some of the tools developed to derive this new bound have independent interest. This should be especially true of a simple criterion for the surjectivity of some linear operators in Hilbert space derived from Bessel‘s inequality. The applications to the Navier-Stokes system are similar to those in the symmetric case.