On symmetric Leray solutions of the stationary Navier-Stokes equations (Q547275)

The stationary Navier-Stokes equations are considered in an exterior plane Lipschitz domain \(\Omega\), symmetric with respect to the \(x_1\)-axis, \[ \begin{aligned} \Delta u-\lambda (u\cdot\nabla)u-\nabla p=0,\quad \text{div}\,u=0, \\ u=0\;\text{on}\;\partial\Omega,\quad u=O(1) \text{ as } |x|\rightarrow \infty , \end{aligned} \] where \(\lambda>0\) is the Reynolds number. It is proved that the problem has a nontrivial symmetric Leray solution such that \[ \lim_{r\rightarrow +\infty}u(r,\theta)=\alpha \overarrow{e}_1 \] for a certain \(\alpha\in [0,1]\). This result has two interesting applications presented in the paper. First, the Stokes paradox for the problem holds if and only if \(\lambda=0\). Second, the author proves the existence of a nontrivial solution to the problem with mixed boundary conditions.