A Priori Estimates for Navier-Stokes Equations in a Strip: A Case With Nonzero Flux Condition in Uniformly Local Spaces: Page 1-11

Abstract

This paper computes energy estimates for Navier-Stokes equations in a strip in uniformly local spaces; focus is on a case with nonzero flux condition. In view of the instability properties of the poiseuille flow at certain critical lengths, a special auxiliary function ) of poiseuille solution with more stable properties is constructed and then subtracted from the solution  of the Navier-Stokes equations. The resulting system, which also has an extra term to be controlled, is then treated as the case of zero flux condition. This article specifies criteria for the choice of the poiseuille-like solution and provide bounds for it. Next, apriori estimates and bounds for the solutions of the entire system is computed in uniformly local spaces.