Existence of periodic solutions of the Navier-Stokes equations (Q1359608)

The author considers the nonstationary Navier-Stokes system \[ \frac{\partial u}{\partial t}-\Delta u+u\cdot\nabla u+\nabla p=f ,\quad \nabla\cdot u=0\quad x\in\Omega,\;\;t\in \mathbb{R} \] with boundary condition \(u=0\) on \(\partial\Omega\) and periodicity condition \[ u(x,t+\omega)=u(x,t)\quad x\in\Omega,\;t\in \mathbb{R} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\;(n=3,4)\) with smooth boundary \(\partial\Omega\). It is proved that if \(f\) is a sufficiently small \(\omega\)-periodic function then the problem has a unique \(\omega\)-periodic strong solution.