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

• Published in: Calculus of variations and partial differential equations 2020-12, Vol.59 (6), Article 196
• Main Authors: Fragalà, Ilaria, Gazzola, Filippo, Sperone, Gianmarco
• Format: Article
• Language: English
• Subjects: Analysis; Barriers; Calculus of Variations and Optimal Control; Optimization; Control; Cubes; Dirichlet problem; Divergence; Flow stability; Fluid dynamics; Fluid flow; Inflow; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Operators (mathematics); Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-020-01844-z

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.