Weak Well-posedness For A Class Of Shallow Water Wave Equations

In this paper,we consider the weak well-posedness for the Cauchy problem of the Degasperis-Procesi equation and the modifed two-component Camassa-Holm system.The main content and innovations include the following two aspects1)we consider the weak well-posedness for the Cauchy problem of the Degasperis-Procesi equation with the initial value in space H'(R)?W1,?(R).First,by introducing the characteristics,the Cauchy problem of the Degasperis-Procesi equation is transformed into an ordinary differential equations in the Banach space through coordinate transformation.Second,using the theory of ordinary differential equations,the existence and uniqueness of the solution to the ordinary differential equations is proved.Finally,the existence and uniqueness of the solution is investigated for the Degasperis-Procesi equation by the existence and uniqueness of the solution to the ordinary differential equations and the existence and uniqueness of the characteristics2)we investigate the weak well-posedness for a modifed two-component Camassa-Holm system with the initial data in space(H1(R)?W1,?(R))×(H1(R)?W1,?(R)).Similar to the proof of the Degasperis-Procesi equation,we transform the modifed two-component Camassa-Holm system into an ordinary differential equations.Then,we prove the existence and uniqueness of the solution to the ordinary differential equations Finally,we obtain the existence and uniqueness of the solution to the modifed two-component Camassa-Holm system.Further,we investigate the weak continuous dependence on initial conditions to the modifed two-component Camassa-Holm system Since the problem of the modifed two-component Camassa-Holm system is a two-component system with the mutual effect between two variables u(·,t)and p(·,t),there are more difficulties in analyzing it than a single component equation.In addition,we need to introduce more variables for the ordinary differential equations,so the structure of this new system is more complicated.