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

TL;DR

This work analyzes a 3D temperature-velocity system of Boussinesq type on a bounded domain subjected to stochastic Dirichlet boundary forcing with intensity $\sqrt{\varepsilon}$. The authors develop a weak-regularity framework by solving the temperature with a prescribed velocity via a decomposition $\theta^\varepsilon = Z^\varepsilon + \zeta^\varepsilon$, where $Z^\varepsilon$ solves a linear boundary-noise heat problem and $\zeta^\varepsilon$ handles the nonlinear remainder, and by establishing maximal-regularity-based well-posedness for the velocity problem using a weak Stokes operator $A_w$. Under small data and $p>4$, they prove existence and uniqueness of a mild solution up to a stopping time with $u^\varepsilon \in W^{1,p}(0,\tau^\varepsilon;H^{-{1/2}-\delta_u}) \cap L^p(0,\tau^\varepsilon;H^{3/2-\delta_u})$ and $\theta^\varepsilon \in C(0,\tau^\varepsilon;H^{-{1/2}-\delta_u})$, and show a high-probability global bound $\mathbb P(\tau^\varepsilon=T) \ge 1- C\varepsilon$ that scales with the boundary-noise covariance. The approach integrates Dirichlet boundary-noise analysis, stochastic convolution factorisation, and interpolation-extrapolation scales for the Stokes operator to achieve global well-posedness in a low-regularity regime with rough boundary forcing, highlighting a robust framework for stochastic boundary perturbations in 3D coupled PDEs.

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 $θ^\varepsilon$ 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 $H^{-\frac12-δ_θ}(\mathcal D)$. To handle this roughness together with initial data $θ_0\in W^{s,6/5}(\mathcal D)$, we work in the ambient space $H^{-\frac12-δ_u}(\mathcal D)$ with $δ_u\ge \max\{δ_θ,\frac12-s\}$. Given a finite time $T>0$, for any $p>4$ and sufficiently small initial data, we prove existence and uniqueness of a mild solution $(u^\varepsilon,θ^\varepsilon)$ up to a stopping time $τ^\varepsilon\le T$ 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(τ^\varepsilon=T)\geq 1- C\varepsilon $, with $C= C( δ_θ, T)>0.$

Theorems & Definitions (28)

• Proposition 2.1
• Theorem 2.2
• Corollary 2.3
• Definition 2.4
• Theorem 2.5: Global well-posedness for small data
• Remark 2.6
• Definition 2.7
• Theorem 2.8
• Remark 2.9: Consistency of parameters
• proof : Proof of Proposition \ref{['prop: regularity for Z_t']} (i)-(ii)