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

We prove that the steady–state Navier–Stokes problem in a plane Lipschitz domain $\mathrm{\Omega }$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $\text{a}\in L^2(\mathrm{\partial }\mathrm{\Omega })$ satisfies $\frac{1}{2\pi }\mathrm{\mid }{\int }_{\mathrm{\partial }\mathrm{\Omega }}\text{a}\cdot \text{n}\mathrm{\mid }<1$. If $\mathrm{\Omega }$ is of class $C^{1,1}$, we can assume $\text{a}\in W^{-1\mathrm{/}4,4}(\mathrm{\partial }\mathrm{\Omega })$. Moreover, we show that for every $D$–solution $(\breve{,}p)$ of the Navier–Stokes equations it holds $\nabla p = o(r^{-1}), \nabla_k p = O(r^{\epsilon-3/2}), \nabla_k\u = O(r^{\epsilon-3/4})$,forall$k\in{\Bbb N}\setminus{1}$andforallpositive$\epsilon$,andifthefluxof$\u$throughacircumferencesurrounding$\complement\Omega$iszero,thenthereisaconstantvector$\u_0$suchthat$\u=\u_0+o(1)$.