发表机构
Tata Institute of Fundamental Research（塔塔基础研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带Kraichnan输运噪声的二维Euler方程，提出全离散逼近收敛性证明，通过有限维强制不等式和紧致性策略，得到弱鞅解。
AI 中文摘要
我们研究了定义在环面 $\mathbb{T}^{2}$ 上、以涡量形式表示、由Kraichnan型输运噪声驱动且初始涡量取值于 $\mathbb{H}^{-1}(\mathbb{T}^2)$ 的二维不可压缩Euler方程的全离散逼近理论。主要的新工具是环面Kraichnan模型的有限维强制不等式，该不等式在离散层面捕捉了粗糙输运噪声引起的正则化机制。结合基于连续方程插值和局部离散估计的新紧致性策略，该强制不等式给出了逼近的紧致性，并证明了子序列极限是Euler方程的弱鞅解。
英文摘要
We study a fully discrete approximation theory for the two-dimensional incompressible Euler equations on $\mathbb{T}^{2}$, in vorticity form, driven by Kraichnan-type transport noise and with $\mathbb{H}^{-1}(\mathbb{T}^2)$-valued initial vorticity. The main new ingredient is a finite-dimensional coercivity estimate for the torus Kraichnan model, which captures at the discrete level the regularizing mechanism induced by the rough transport noise. Together with a new compactness strategy based on continuous equation-based interpolants and localized discrete estimates, this coercivity estimate gives tightness of the approximations and shows that subsequential limit is a weak martingale solution of the Euler equations.
Comments58 pages
