Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier-Stokes equations

TL;DR

This paper presents a new method for constructing solenoidal extensions in domains with obstacles, providing explicit bounds useful for analyzing Navier-Stokes equations and fluid flow stability.

Contribution

The paper introduces an explicit construction method for solenoidal extensions in domains with obstacles, with bounds based on geometric parameters, aiding fluid dynamics analysis.

Findings

Explicit bounds for solenoidal extensions in obstacle domains.

Application to Navier-Stokes equations for flow analysis.

Improved estimates for inflow velocities and stability.

Abstract

We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem with the resulting divergence to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds in terms of the geometric parameters of the obstacle. The natural applications of our results lie in the analysis of inflow-outflow problems, in which an explicit bound on the inflow velocity is needed to estimate the threshold for uniqueness in the stationary Navier-Stokes equations and, in case of symmetry, the stability of the obstacle immersed in the fluid flow.