Some Studies Of The Shallow Water Equations

In this thesis, we investigate to three kinds of the shallow water equations.First, we consider the b-family equation with weakly dissipative. We get some new blow-up results、global existence and infinite propagation speed. We also consider the DGH equation. We regard it as CH equation with the weakly dissi-pative.Then, we study the modified two component Camassa-Holm system which arises in shallow water theory.Some new blow-up criteria are presented which improve previous results.Finally,in this section we study the large time behavior for the support of momentum density of the Camassa-Holm equation. More precisely, we deduce the limit of the support of momentum density as t goes to+∞in some sense.