The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary
The author investigates the convergence of weak solutions of the Navier-Stokes equations when horizontal and vertical viscosities go to zero. Let \((v,p)\) satisfy the boundary value problem in \(Q=(0,T)\times \Omega\), \(\Omega=\omega\times (0,h)\) or \(\Omega=\omega\times (0,\infty)\), and \(\omega=\mathbb{T}^2\) or \(\omega=\mathbb{R}^2\), \(\frac{\partial v}{\partial t}+(v\cdot\nabla)v-\nu \frac{\partial^2 v}{\partial z^2 }-\eta\Delta' v+\nabla p=F\), \(\text{div} v=0\) in \(Q\), \(v| _{\partial Q}=0\), \(v| _{t=0}=v_0\), where \(\Delta' = \frac{\partial^2 }{\partial x^2 }+ \frac{\partial^2 }{\partial y^2 }\). Let further the Euler system in \(Q\) be written in the form \(\frac{\partial w}{\partial t}+(w\cdot\nabla)w +\nabla p=F^0\), \(\text{div} w=0\) in \(Q\), \(w\cdot n| _{\partial Q}=0\), \(w| _{t=0}=w_0\). It is proved that if \(v_0\to w_0,\;F\to F^0\) and \(\frac \nu\eta \to 0\) as \(\nu \to 0,\eta \to 0\), then there exist global weak solutions \(v\) of the first problem such that \(v\to w\) in certain spaces. A similar result is obtained for system describing rotating fluids.
scientific article; zbMATH DE number 1218193