The Stokes and Navier-Stokes equations in an aperture domain

Takayuki KUBO
2007 Journal of the Mathematical Society of Japan  
We consider the nonstationary Navier-Stokes equations in an aperture domain $\Omega\subset \mathbb{R}^{n},$ $n\geq 2$ . Main purpose of this paper is to discuss the existence of a unique solution to the Navier-Stokes problem with a zero and a non-zero flux condition through the aperture. To this end, we prove $L^{p}-L^{q}$ type estimate of the Stokes semigroup in the aperture domain. Applying them to the Navier-Stokes initial value problem in the aperture domain, we can prove the global
more » ... the global existence of a unique solution to the Navier-Stokes problem with the zero-flux condition and some decay properties as $tarrow\infty$ , when the initial velocity is sufficiently small in the $L^{n}$ space. Moreover we can prove the time-local existence of a unique solution to the Navier-Stokes problem with the non-trivial flux condition.
doi:10.2969/jmsj/05930837 fatcat:3cctovsforgyfp6mwtqkmdwa5a