Stochastic Navier-Stokes equation and renormalization group theory
The authors deal with the stochastic Navier-Stokes equation for the velocity field \(u\) and the pressure \(p\) with periodic boundary conditions in a 2D domain \(\Omega=(0,2\pi a_1)\times (0,2\pi a_2)\): \[ \frac{\partial u}{\partial t}-\nu\Delta u+(u\cdot\nabla)u+\nabla p=f+\sqrt\varepsilon \eta,\quad \text{div} u=0\text{ in }\Omega, \quad \int_\Omega u dx=0,\quad u|_{t=0}=u_0, \] where \(\nu>0\), \(a_1,a_2>0\); \(a_1^{-2}\) and \(a_2^{-2}\) are rationally independent in order to avoid certain resonant terms. The forcing \(f+\sqrt\varepsilon \eta\) consists of a time-independent mean forcing \(f=f(x)\) and some small stochastic fluctuation \(\sqrt\varepsilon \eta\), \(\varepsilon\) is a small positive parameter. The authors present an application of a certain renormalization group theory (developed by Chen, Coldenfeld, and Oono). Since the classical renormalization group theory utilizes significantly on adding noise terms in order to similate high oscillating fluctuations, the authors' approach can be considered as another step towards a rigorous understanding of previous models.
scientific article; zbMATH DE number 1828079