The Well-posedness Of Global Weak Solutions Of The Dullin-Gottwald-Holm Equation With A Forcing Term

The shallow water wave equations,such as the Camassa-Holm equa-tion(C-H equation),derived from modern mechanics and physics,have becoming one of the important topics in the studies of nonlinear partial differential equa-tions.In this paper,we mainly study the well-posedness of the Cauchy problem for the Dullin-Gottwald-Holm equation(D-G-H equation,for short)with a forc-ing term,which is a generalized C-H equation.In fact,the D-G-H equation is a quasilinear partial differential equations and describes the one-way propaga-tion of surface waves in shallow water under the action of gravitation.Due to the nonlinearity containing derivatives,the classical Kato' s semigroup method is no longer applicable in H1(R).In terms of Bressan and Constantin' s new characteristic method proposed in 2007,we obtain the global well-posedness of weak solutions in H1(R)and the existence of dissipative solutions for the D-G-H equation with a forcing term.This paper is organized as follows.In Chapter 1,we introduce some related backgrounds and preliminaries for the D-G-H equation.In Chapter 2,first,according to the new characteristic method proposed by Bressan and Constantin in 2007,we transfer the quasi-linear partial differential equation into a semi-linear ordinary differential equation(ODE).Then,we prove the well-posedness of global weak solutions for this semi-linear ODE.Finally,we can obtain the well-posedness of the original equation by the inverse transforma-tion.However,under the effect of the forcing term,the D-G-H equation with a forcing term does not have the conservation of energy.we need to introduce some new estimates to obtain the well-posedness of the global weak solutions for the D-G-H equation with a forcing term in H1(R)by the balance law.In Chaper 3,we utilize the new characteristic method to prove the existence of the global dissipative solution for the D-G-H equation with a forcing term in H1.The forcing term results in that the corresponding ODE system contains a discontinuous non-local term,and then the cross terms are not continuous.First,by the local boundness of the variations under a suitable modification,we derive a new ODE system in L00.Then,we prove the existence of global weak solution for the semi-linear system.Finally,with the help of inverse transformation and Kato's semigroup method,we obtain a local dissipative solution for the D-G-H equation with a forcing term.