Wolfgang Borchers

L2 decay for the Navier-Stokes flow in halfspaces

On etudie le comportement asymptotique dans L 2 des solutions faibles des equations de Navier-Stokes dans des demi-espaces

Existence, Uniqueness and Asymptotics of Steady Jets

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r−2) as r=¦x¦→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ+3. To investigate the Stokes problem in ℝ+3, we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.