Existence proof of unimodal solutions of the Proudman–Johnson equation via interval analysis

Abstract

The Proudman–Johnson equation is considered as a representative of the two-dimensional fluid flow. Its steady-states are computed and their existence and unimodality are proved rigorously via the interval analysis and the interval Newton method. By using both the multiple shooting method and multiple-precision arithmetic our verification succeeds up to the Reynolds number $$\le 5000$$ .