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 $u^\varepsilon$ solves the Navier-Stokes equations and the temperature $\theta^\varepsilon$ is driven by Dirichlet boundary noise of intensity $\sqrt{\varepsilon}$. The boundary forcing produces a stochastic convolution $Z^\varepsilon$ which is, in general, only continuous in time with values in $H^{-\frac12-\delta_\theta}(\mathcal D)$. To handle this roughness together with initial data $\theta_0\in W^{s,6/5}(\mathcal D)$, we work in the ambient space $H^{-\frac12-\delta_u}(\mathcal D)$ with $\delta_u\ge \max\{\delta_\theta,\frac12-s\}$. Given a finite time $T>0$, for any $p>4$ and sufficiently small initial data, we prove existence and uniqueness of a mild solution $(u^\varepsilon,\theta^\varepsilon)$ up to a stopping time $\tau^\varepsilon\le T$ such that \[ u^\varepsilon \in W^{1,p}(0,\tau^\varepsilon;H^{-\frac12-\delta_u}(\mathcal D)) \cap L^p (0,\tau^\varepsilon;H^{\frac32-\delta_u}(\mathcal D)), \quad \theta^\varepsilon \in C(0,\tau^\varepsilon;H^{-\frac12-\delta_u}(\mathcal D)). \] Moreover, we obtain a high-probability global existence estimate of the form $\mathbb P(\tau^\varepsilon=T)\geq 1- C\varepsilon $, with $C= C( \delta_\theta, T)>0.$