Global existence for the stochastic Boussinesq equations with transport noise and small rough data

In this paper, we consider the stochastic Boussinesq equations on ${\mathbb{T}}^3$ with transport noise and rough initial data. We first prove the existence and uniqueness of the local pathwise solution with initial data in $L^p(\mathrm{\Omega };L^p)$ for $p>5$. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.