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

Abstract

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 $\hat{I}{¸}^{\epsilon }$ 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 $H^{-\frac{1}{2}-\hat{I}{´}_{\hat{I}}¸}(\mathcal{D})$. To handle this roughness together with initial data $\hat{I}{¸}_0\in W^{s,6/5}(\mathcal{D})$, we work in the ambient space $H^{-\frac{1}{2}-\hat{I}{´}_u}(\mathcal{D})$ with $\hat{I}{´}_u\ge \max \{\hat{I}{´}_{\hat{I}}¸,\frac{1}{2}-s\}$. Given a finite time $T>0$, for any $p>4$ and sufficiently small initial data, we prove existence and uniqueness of a mild solution $(u^{\epsilon },\hat{I}{¸}^{\epsilon })$ up to a stopping time $\ddot{I}{„}^{\epsilon }\le T$ such that \[ u^\varepsilon \in W^{1,p}(0,τ^\varepsilon;H^{-\frac12-δ_u}(\mathcal D)) \cap L^p (0,τ^\varepsilon;H^{\frac32-δ_u}(\mathcal D)), \quad θ^\varepsilon \in C(0,τ^\varepsilon;H^{-\frac12-δ_u}(\mathcal D)). \] Moreover, we obtain a high-probability global existence estimate of the form $\mathbb{P}(\ddot{I}{„}^{\epsilon }=T)\ge 1-C\epsilon$, with $C=C(\hat{I}{´}_{\hat{I}}¸,T)>0.$