Qualitative properties of some equations related to fluid mechanics (Q707038)

The paper is based on the Lemma 1 (unpublished yet) obtained by J.M. Rakotoson, where a system of energy inequalities is considered and a condition involving initial data gives a bounded solution for all positive times. The authors use Lemma 1 to study the Navier-Stokes equation. It is proved that a maximal strong solution (Definition 1), for ``small'' initial data, is a global solution for all positive times. Using this method, some previous examples in 2D can be extended to the 3D case, also for small initial data end external forces. Estimations of the blow-up time are given, first for zero external forces. The last part contains two very interesting examples, with external forces in \(L^2\) space. The main tools are energetic inequalities for abstract Navier-Stokes equations.