The 2d Nonlinear Fully Hyperbolic Inviscid Shallow Water Equations in a Rectangle

We continue our study of the inviscid shallow water equations (SWE) in a rectangle. In an earlier work (Huang and Temam in Arch Ration Mech Anal 211(3):1027–1063, 2014) we studied the well-posedness for all time of the linearized inviscid SWE in a non-smooth domain. We defined and classified the different sets of boundary conditions which make these equations well-posed for all time and showed the existence and uniqueness of solutions. As we show below totally different boundary conditions are needed in the full nonlinear cases. The case of supercritical flows was investigated in Huang et al. (Asymptot Anal 93:187–218, 2015), and the case of subcritical flows in a channel was studied in Huang and Temam (Commun Pure Appl Anal 13(5):2005–2038, 2014). We continue here and study subcritical flows in a rectangle which raises the additional issue of the compatibility of the boundary and initial conditions at t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document} and of the boundary conditions between them at the corners of the rectangle.