Global well-posedness for small data in a 3D temperature-velocity model with Dirichlet boundary noise

We study a three-dimensional Boussinesq-type temperature-velocity system on a bounded smooth domain $\mathcal{D}\subset {\mathbb{R}}^3$, where the velocity $u^{\epsilon }$ solves the Navier-Stokes equations and the temperature ${\theta }^{\epsilon }$ is driven by Dirichlet boundary noise of intensity $\sqrt{\epsilon }$. The boundary forcing produces a stochastic convolution $Z^{\epsilon }$ which is, in general, only continuous in time with values in $H^{-\frac{1}{2}-{\delta }_{\theta }}(\mathcal{D})$. To handle this roughness together with initial data ${\theta }_0\in W^{s,6\mathrm{/}5}(\mathcal{D})$, we work in the ambient space $H^{-\frac{1}{2}-{\delta }_u}(\mathcal{D})$ with ${\delta }_u\ge \max \{{\delta }_{\theta },\frac{1}{2}-s\}$. Given a finite time $T>0$, for any $p>4$ and sufficiently small initial data, we prove existence and uniqueness of a mild solution $(u^{\epsilon },{\theta }^{\epsilon })$ up to a stopping time ${\tau }^{\epsilon }\le T$ such that $$u^{\epsilon }\in W^{1,p}(0,{\tau }^{\epsilon };H^{-\frac{1}{2}-{\delta }_u}(\mathcal{D}))\cap L^p(0,{\tau }^{\epsilon };H^{\frac{3}{2}-{\delta }_u}(\mathcal{D})),{\theta }^{\epsilon }\in C(0,{\tau }^{\epsilon };H^{-\frac{1}{2}-{\delta }_u}(\mathcal{D}))\mathrm{.}$$ Moreover, we obtain a high-probability global existence estimate of the form $\mathbb{P}({\tau }^{\epsilon }=T)\ge 1-C\epsilon$, with $C=C({\delta }_{\theta },T)>0.$