The Camassa--Holm (CH) equation is a nonlocal equation that manifests supercritical behaviour in ``wave-breaking" and non-uniqueness. In this talk, I will discuss the existence of global (dissipative weak martingale) solutions to the CH equation with multiplicative, gradient type noise, derived as an inviscid limit. The goal of the talk is twofold. The stochastic CH equation will be used to illustrate aspects of a stochastic compactness and renormalisation method which is popularly used to derive well-posedness and continuous dependence results in SPDEs. I shall also discuss how a lack of temporal compactness introduces fundamental difficulties in the case of the stochastic CH equation.
This talk is based on joint works with L. Galimbert and H. Holden, both at NTNU, and with K.H. Karlsen at the University of Oslo.