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

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