Global well-posedness for rougher stochastic forcing

Extend the global well-posedness theory for the viscous Camassa–Holm equation driven by the additive Gaussian noise $(1-\partial_x^2)^{-\eta}\xi$ from the established regime $\eta>1/4$ to the regime $\eta\leq 1/4$, by controlling the nonlinear interactions generated by the stronger stochastic irregularity.

Background

The paper proves local well-posedness for the stochastic viscous Camassa–Holm equation for every $\eta>-1/12$, but establishes global well-posedness only when $\eta>1/4$. In the latter regime, the stochastic convolution has sufficient spatial regularity for the authors’ $H^1$ energy estimates to yield uniform a priori bounds.

For $\eta\leq 1/4$, the stochastic forcing is rougher and the classical energy estimates used in Proposition 5.1 may not control the nonlinear interactions. The authors identify extending global well-posedness into this regime as a challenging unresolved problem and suggest combining rough-path techniques with suitable energy methods.