Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $\lambda$-family equations

In this paper, we prove the global existence of H\"older 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 another word, show wave breaking phenomenon. The global energy conservative solution we construct is H\"older 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.