Studies On The Shallow Water Equations

The aim of this thesis is to study the model of the shallow water equations. Four chapters comprise this thesis.Research background and current status of the model of the shallow water equa-tions are displayed in detail in the first chapter.A class of the weakly dissipative one-dimensional shallow water equations is stud-ied in the second chapter, focused on establishing its blow-up theorem, proving the global existence of the solution and discussing its infinite propagation speed.The local well-posedness and blow-up scenarios for the cross-coupled Camassa-Holm system are presented in the third chapter.Some unsolved problems of the shallow water equations are collected in the fourth chapter.