具有 Dirichlet 边界噪声的三维温度-速度模型中小数据的全局适定性
概率论
2026-03-12 v3
摘要
我们研究了一个在有界光滑区域 D ⊂ R 3 \mathcal D\subset\mathbb R^3 D ⊂ R 3 u ε u^\varepsilon u ε θ ε \theta^\varepsilon θ ε ε \sqrt{\varepsilon} ε Z ε Z^\varepsilon Z ε H − 1 2 − δ θ ( D ) H^{-\frac12-\delta_\theta}(\mathcal D) H − 2 1 − δ θ ( D ) θ 0 ∈ W s , 6 / 5 ( D ) \theta_0\in W^{s,6/5}(\mathcal D) θ 0 ∈ W s , 6/5 ( D ) H − 1 2 − δ u ( D ) H^{-\frac12-\delta_u}(\mathcal D) H − 2 1 − δ u ( D ) δ u ≥ max { δ θ , 1 2 − s } \delta_u\ge \max\{\delta_\theta,\frac12-s\} δ u ≥ max { δ θ , 2 1 − s } T > 0 T>0 T > 0 p > 4 p>4 p > 4 ( u ε , θ ε ) (u^\varepsilon,\theta^\varepsilon) ( u ε , θ ε ) τ ε ≤ T \tau^\varepsilon\le T τ ε ≤ T u ε ∈ W 1 , p ( 0 , τ ε ; H − 1 2 − δ u ( D ) ) ∩ L p ( 0 , τ ε ; H 3 2 − δ u ( D ) ) , θ ε ∈ C ( 0 , τ ε ; H − 1 2 − δ u ( D ) ) . 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)). u ε ∈ W 1 , p ( 0 , τ ε ; H − 2 1 − δ u ( D )) ∩ L p ( 0 , τ ε ; H 2 3 − δ u ( D )) , θ ε ∈ C ( 0 , τ ε ; H − 2 1 − δ u ( D )) . P ( τ ε = T ) ≥ 1 − C ε \mathbb P(\tau^\varepsilon=T)\geq 1- C\varepsilon P ( τ ε = T ) ≥ 1 − C ε C = C ( δ θ , T ) > 0. C= C( \delta_\theta, T)>0. C = C ( δ θ , T ) > 0.
引用
@article{arxiv.2505.11447,
title = {Global well-posedness for small data in a 3D temperature-velocity model with Dirichlet boundary noise},
author = {Gianmarco Del Sarto and Marta Lenzi},
journal= {arXiv preprint arXiv:2505.11447},
year = {2026}
}