论文标题
3D Navier的全球适应性 - stokes方程式由确定性矢量字段扰动
Global well-posedness of the 3D Navier--Stokes equations perturbed by a deterministic vector field
论文作者
论文摘要
我们关注3D Navier的全球范围良好的问题 - 粘度单位粘度在圆环上的stokes方程。虽然对这个问题的完整答案似乎是当前技术无法实现的，但我们通过确定性矢量领域建立正规化。更确切地说，我们考虑了系统的额外运输类型术语扰动系统的涡度形式。这样的扰动可以保守肠胃，因此先验并不意味着任何平滑。我们的主要结果是构造确定性矢量字段$ v = v（t，x）$，该$提供了系统的正则化，并在任意小型集外的大型初始数据中产生了全球范围的良好性。证据依赖于弗兰多利和卢（Flandoli and Luo）开发的概率论点，霍夫曼诺瓦（Hofmanová），利希（Leahy）和尼尔斯森（Nilssen）的粗糙路径理论工具以及一个新的黄（Zakai） - Zakai近似结果，本身结合了概率和粗糙的路径技术。
We are concerned with the problem of global well-posedness of the 3D Navier--Stokes equations on the torus with unitary viscosity. While a full answer to this question seems to be out of reach of the current techniques, we establish a regularization by a deterministic vector field. More precisely, we consider the vorticity form of the system perturbed by an additional transport type term. Such a perturbation conserves the enstrophy and therefore a priori it does not imply any smoothing. Our main result is a construction of a deterministic vector field $v=v(t,x)$ which provides the desired regularization of the system and yields global well-posedness for large initial data outside arbitrary small sets. The proof relies on probabilistic arguments developed by Flandoli and Luo, tools from rough path theory by Hofmanová, Leahy and Nilssen and a new Wong--Zakai approximation result, which itself combines probabilistic and rough path techniques.
