Stationary stochastic Navier–Stokes on the plane at and above criticality
Stationary stochastic Navier–Stokes on the plane at and above criticality
复制标题
DOI：
10.1007/s40072-022-00283-5
复制
发表时间：
2023-01
期刊：
影响因子：
--
通讯作者：
G. Cannizzaro;Jacek Kiedrowski
中科院分区：
文献类型：
--
作者：
G. Cannizzaro;Jacek Kiedrowski
In the present paper, we study the fractional incompressible Stochastic Navier–Stokes equation on, formally defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t v = -\tfrac{1}{2} (-\Delta )^\theta v - \lambda v \cdot \nabla v + \nabla p + \nabla ^{\perp }(-\Delta )^{\frac{\theta -1}{2}} \xi , \qquad \nabla \cdot v = 0 \, , \end{aligned}$$\end{document}where,is the space-time white noise onandis the coupling constant. For any value ofthe previous equation is ill-posed due to the singularity of the noise, and is critical forand supercritical for. For, we prove that the weak coupling regime for the equation, i.e. regularisation at scaleNand coupling constant, is meaningful in that the sequenceof regularised solutions is tight and the nonlinearity does not vanish as. Instead, forwe show that the large scale behaviour ofvis trivial, as the nonlinearity vanishes andvis simply converges to the solution of (0.1) with.
In the present paper, we study the fractional incompressible Stochastic Navier–Stokes equation on, formally defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t v = -\tfrac{1}{2} (-\Delta )^\theta v - \lambda v \cdot \nabla v + \nabla p + \nabla ^{\perp }(-\Delta )^{\frac{\theta -1}{2}} \xi , \qquad \nabla \cdot v = 0 \, , \end{aligned}$$\end{document}where,is the space-time white noise onandis the coupling constant. For any value ofthe previous equation is ill-posed due to the singularity of the noise, and is critical forand supercritical for. For, we prove that the weak coupling regime for the equation, i.e. regularisation at scaleNand coupling constant, is meaningful in that the sequenceof regularised solutions is tight and the nonlinearity does not vanish as. Instead, forwe show that the large scale behaviour ofvis trivial, as the nonlinearity vanishes andvis simply converges to the solution of (0.1) with.