The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model

A nonlinear model, which characterizes motions of shallow water waves and includes the famous Degasperis-Procesi equation, is considered. The essential step is the derivation of the $ L^2(\mathbb{R}) $ uniform bound of solutions for the nonlinear model if its initial value belongs to space $ L^2(\mathbb{R}) $. Utilizing the bounded property leads to several estimates about its solutions. The viscous approximation technique is employed to establish the well-posedness of entropy weak solutions.