Weak Problem For A Class Of Shallow Water Wave Equations

In this paper,we consider the weak well-posedness for the Cauchy problem of the two-component Degasperis-Procesi system and the generalized Camassa-Holm equation arising from modern mechanics and physics.We mainly study the following contents in this paper:1)We consider the weak well-posedness for the Cauchy problem of the two-component Degasperis-Procesi sysytem with the initial value(u₀,?₀)in space(H¹(R)(40)W,^1?(R))?(L²(R)(40)L^?(R)).Firstly,by introducing the characteristics,the Cauchy problem of the two-component Degasperis-Procesi system is transformed into the ordinary differential equations(ODE system)in the Banach space by coordinate transformation.Secondly,using the classical theory of the ordinary differential,the existence and uniqueness of the solution to the ODE system is proved.Finally,the existence and unique of the local solution is investigated for the system by the relationship between the two-component Degasperis-Procesi system and the new system.Furthermore,we obtain that the unique weak solution of the two-component Degasperis-Procesi system is weak dependence on initial data.2)We study the Cauchy problem of the generalized Camassa-Holm equation in the condition ofm₀ changing sign,we prove the existence and uniqueness of global weak solutions for the equation if the initial datau₀?H¹(R).We first prove the existence and uniqueness of the global strong solutions,and show that theH(¹R)norm of the solution is conservative.Follwing,existence of the global weak solutions has been obtained by mollifiers initial data method and Helly's theorem.At last,we demonstrate the uniqueness of the global weak solutions.