On the existence of singular solutions of the stationary Navier–Stokes problem

Abstract

We consider the steady Navier–Stokes equations in the punctured regions (ı) Ω = Ω ₀ \ {o} (with {o} ∈ Ω ₀) and (ıı) $\varOmega ={{\mathbb{R}}^2}\backslash \left( {{{\overline{\varOmega}}_0}\cup \left\{ o \right\}} \right)$ (with $\left\{ o \right\}\notin {{\overline{\varOmega}}_0}$), where Ω ₀ is a simple connected Lipschitz bounded domain of ${{\mathbb{R}}^2}$. We regard o as a sink or a source in the fluid. Accordingly, we assign the flux $\mathcal{F}$ through a small circumference surrounding o and a boundary datum a on Γ = ∂Ω ₀ such that the total flux $\mathcal{F}+\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}}$ is zero in case (ı). We prove that if $\left| \mathcal{F} \right|<2\pi \nu$ and $\left| \mathcal{F} \right|+\left| {\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} } \right|<2\pi \nu$ in (ı) and (ıı), respectively, where ν is the kinematical viscosity, then the problem has a C ^∞ solution in Ω, which behaves at o like the gradient of the fundamental solution of the Laplace equation.