Existence and regularity for global weak solutions to the $\lambda$-family water wave equations

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as $\lambda$-family equations, where $\lambda$ is the power of nonlinear wave speed. The $\lambda$-family equations include Camassa-Holm equation ($\lambda =1$) and Novikov equation ($\lambda =2$) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent $1- \frac {1}{2\lambda }$. The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.