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

Abstract

We consider the steady Navier–Stokes equations in the punctured regions (i) Ω = Ω0\ {o} (with {o} ∈ Ω0) and (ii) \( \varOmega ={{\mathbb{R}}^2}\backslash \left( {{{\overline{\varOmega}}_0}\cup \left\{ o \right\}} \right) \) (with \( \left\{ o \right\}\notin {{\overline{\varOmega}}_0} \)), where Ω0 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 = ∂Ω0 such that the total flux \( \mathcal{F}+\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} \) is zero in case (i). 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 (i) and (ii), 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.