Strong Well-Posedness for a Class of Dynamic Outflow Boundary Conditions for Incompressible Newtonian Flows

TL;DR

This paper establishes strong well-posedness and maximal regularity results for a class of dynamic outflow boundary conditions in incompressible Navier-Stokes and Stokes equations, including complex boundary geometries.

Contribution

It introduces a new class of dynamic outflow boundary conditions derived from energy considerations and proves their well-posedness and maximal regularity in various geometric settings.

Findings

Existence of strong solutions for the Stokes equations with dynamic outflow boundary conditions.

Maximal regularity results under natural compatibility conditions.

Application to wedge domains with mixed boundary conditions.

Abstract

Based on energy considerations, we derive a class of dynamic outflow boundary conditions for the incompressible Navier-Stokes equations, containing the well-known convective boundary condition but incorporating also the stress at the outlet. As a key building block for the analysis of such problems, we consider the Stokes equations with such dynamic outflow boundary conditions in a halfspace and prove the existence of a strong solution in the appropriate Sobolev-Slobodeckij-setting with $L_p$ (in time and space) as the base space for the momentum balance. For non-vanishing stress contribution in the boundary condition, the problem is actually shown to have $L_p$-maximal regularity under the natural compatibility conditions. Aiming at an existence theory for problems in weakly singular domains, where different boundary conditions apply on different parts of the boundary such that these surfaces meet orthogonally, we also consider the prototype domain of a wedge with opening angle $\frac{\pi}{2}$ and different combinations of boundary conditions: Navier-Slip with Dirichlet and Navier-Slip with the dynamic outflow boundary condition. Again, maximal regularity of the problem is obtained in the appropriate functional analytic setting and with the natural compatibility conditions.