Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains
The authors consider the Navier-Stokes equations in a smooth domain \(\Omega \subset \mathbb R^3\). The initial condition is posed on the velocity \(u\) and \(u=0\) on \(\partial \Omega\). If the smooth solution \(u \in C([0, T)\) breaks down at a finite time \(t = T\), then \(\int_0^t ||rot \; u (\tau)||_{L^{\infty}} d\tau \nearrow \infty\) as \(t \nearrow T\). The authors discuss similar break down criteria in various spaces. The largest normed space where the \(\mathrm{rot }u\) can break down is found under some additional assumptions.
scientific article; zbMATH DE number 6873210
scientific article