The pressure stabilization method for steady viscous flows in a system of pipes

This paper deals with a singular perturbation of the stationary Stokes and Navier–Stokes system. Thereby the term $\varepsilon^2 \Delta p$ is added to the continuity equation, where $\varepsilon$ is small parameter. For a domain with cylindrical outlets to infinity and exponentially decaying data, existence and uniqueness of solutions under flux conditions at infinity are shown for the linear problem, and for the nonlinear problem in the case of small data. Asymptotically precise estimates are proved, as $\varepsilon$ tends to zero. For sufficiently regular data, they lead to convergence in $H^{5/2-\delta}_\mathrm{loc}$ for the velocity parts and in $H^{3/2-\delta}_\mathrm{loc}$ for the pressure parts, respectively.