Global well-posedness for small data in a 3D temperature-velocity model with Dirichlet boundary noise
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| Format: | Preprint |
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2025
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| _version_ | 1866908878664368128 |
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| author | Del Sarto, Gianmarco Lenzi, Marta |
| author_facet | Del Sarto, Gianmarco Lenzi, Marta |
| contents | 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 $C= C( δ_θ, T)>0.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_11447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness for small data in a 3D temperature-velocity model with Dirichlet boundary noise Del Sarto, Gianmarco Lenzi, Marta Probability 60H15, 60H30, 76D03 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 $C= C( δ_θ, T)>0.$ |
| title | Global well-posedness for small data in a 3D temperature-velocity model with Dirichlet boundary noise |
| topic | Probability 60H15, 60H30, 76D03 |
| url | https://arxiv.org/abs/2505.11447 |