On the existence of D-solutions of the steady-state Navier-Stokes equations in plane exterior domains

Abstract

We prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $\Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $\a \in L^2(\partial\Omega)$ satisfies ${1\over 2\pi}|\int_{\partial\Omega}\a\cdot\n|<1$. If $\Omega$ is of class $C^{1,1}$, we can assume $\a\in W^{-1/4,4}(\partial\Omega)$. Moreover, we show that for every $D$--solution $(\u,p)$ of the Navier--Stokes equations it holds $\nabla p = o(r^{-1}), \nabla_k p = O(r^{\epsilon-3/2}), \nabla_k\u = O(r^{\epsilon-3/4})$, for all $k\in{\Bbb N}\setminus\{1\}$ and for all positive $\epsilon$, and if the flux of $\u$ through a circumference surrounding $\complement\Omega$ is zero, then there is a constant vector $\u_0$ such that $\u=\u_0+o(1)$.