The generalized Proudman–Johnson equation and its singular perturbation problems

We consider the generalized Proudman–Johnson equation with an external force. By varying the Reynolds number R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document} and another nondimensional parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, branching stationary solutions are computed numerically for the global picture of bifurcations of the equation. Asymptotic behavior of solutions as the Reynolds number tends to zero or infinity is also studied by a combination of heuristic analysis and the asymptotic expansion. In doing so, singular perturbation problems of new type are derived and analyzed. As a consequence, through the asymptotic analysis argument, the peculiarity of two dimensional Navier–Stokes flows related to the unimodality is re-confirmed.