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

2025/05/16 by Gianmarco Del Sarto, Del Sarto, Gianmarco, Lenzi, Marta
Engineering · Mathematics · #60H15 #60H30 #76D03 #Advanced Mathematical Physics Problems #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.11447

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a three-dimensional Boussinesq-type temperature-velocity system on a bounded smooth domain \mathcal D⊂\mathbb R³, where the velocity u^ε solves the Navier-Stokes equations and the temperature θ^ε is driven by Dirichlet boundary noise of intensity √(ε). The boundary forcing produces a stochastic convolution Z^ε 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 θ₀∈ W^s,6/5(\mathcal D), we work in the ambient space H^{-\frac12-δ_u}(\mathcal D) with δ_u≥ 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^ε,θ^ε) up to a stopping time τ^ε≤ T such that u^ε ∈ W^1,p(0,τ^ε;H^{-\frac12-δ_u}(\mathcal D)) ∩ L^p (0,τ^ε;H^\frac32-δ_u(\mathcal D)), θ^ε ∈ C(0,τ^ε;H^{-\frac12-δ_u}(\mathcal D)). Moreover, we obtain a high-probability global existence estimate of the form \mathbb P(τ^ε=T)≥ 1- Cε , with C= C( δ_θ, T)>0.

Citations

• Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noise
• Global well-Posedness and Interior Regularity of 2D Navier-Stokes Equations with Stochastic Boundary Conditions
• Linear parabolic equation with Dirichlet white noise boundary conditions
• Multiplication in Sobolev Spaces, Revisited
• Second Order PDEs with Dirichlet White Noise Boundary Condition
• On the existence and uniqueness of weak solutions for a vorticity seeding model
• One-parameter semigroups for linear evolution equations

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• Local and global existence for the stochastic Prandtl equation driven by multiplicative noises in two and three dimensions
• Global existence for the stochastic Boussinesq equations with transport noise and small rough data