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

• Main Authors: Del Sarto, Gianmarco, Lenzi, Marta
• Format: Journal Article
• Language: English
• Published: 16.05.2025
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2505.11447

We analyse a Boussinesq coupled temperature-velocity system on a bounded, open, smooth domain$\mathcal{D} \subset \mathbb{R}^3$ . The fluid velocity$u^\varepsilon$evolves according to the three-dimensional Navier-Stokes equations, while the temperature$\theta^\varepsilon$is subject to Dirichlet boundary noise of intensity$\sqrt{\varepsilon}$ . Given a finite time$T>0$ , under natural assumptions on the stochastic forcing and for sufficiently small initial data, we show that there exists a unique mild solution$(u^\varepsilon, \theta^\varepsilon)$up to a random stopping time$\tau^\varepsilon \leq T$ . Moreover,$\mathbb{P}(\tau^\varepsilon = T) \geq 1 - C \varepsilon$ , and the solution trajectories lie in the optimal regularity class$ (u^\varepsilon, \theta^\varepsilon) \in \left( H^{1,p}_t(H_x^{-\tfrac{1}{2} - \delta}) \cap L^p_t(H_x^{\tfrac{3}{2} - \delta}) \right) \times C_t(H_x^{-\tfrac{1}{2} - \delta}) $for all sufficiently small$\delta > 0$and all$p > \tfrac{2}{1 - \delta}$ .