受确定性向量场扰动的3D Navier–Stokes方程的全局适定性
Global well-posedness of the 3D Navier--Stokes equations perturbed by a deterministic vector field
AI总结：
针对3D Navier–Stokes方程全局适定性难题，研究构造了确定性输运型扰动向量场，结合概率与粗糙路径方法，证明了任意小集合外大初始数据下的全局适定性。
摘要：
我们研究单位黏度环面上3D Navier–Stokes方程的全局适定性问题。尽管该问题的完整解答似乎超出了现有技术的范畴，但我们通过确定性向量场建立了一种正则化机制。更具体地说，我们考虑添加了输运型附加项的系统涡度形式。这类扰动守恒涡拟能，因此先验地不会带来任何平滑效应。我们的主要结果是构造了一个确定性向量场$v=v(t,x)$，它能为系统提供所需的正则化，并在任意小集合之外的大初始数据下得到全局适定性。证明依赖于Flandoli和Luo提出的概率论证、Hofmanová、Leahy和Nilssen提出的粗糙路径理论工具，以及一个新的Wong–Zakai逼近结果，该结果本身结合了概率技术与粗糙路径技术。
英文摘要：
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.
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