2020/06/19 by Ilaria Fragalà, Filippo Gazzola, Fragalà, Ilaria +3 · 1 citation
Computer Science · Engineering · Mathematics · #35C05 #35Q35 #46E35 #49K20 #76D05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2006.11018
openalex publication_date 2020/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new method for constructing solenoidal extensions of fairly\ngeneral boundary data in (2d or 3d) cubes that contain an obstacle. This method\nallows us to provide explicit bounds for the Dirichlet norm of the extensions.\nIt runs as follows: by inverting the trace operator, we first determine\nsuitable extensions, not necessarily solenoidal, of the data; then we analyze\nthe Bogovskii problem with the resulting divergence to obtain a solenoidal\nextension; finally, by solving a variational problem involving the\ninfinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds\nin terms of the geometric parameters of the obstacle. The natural applications\nof our results lie in the analysis of inflow-outflow problems, in which an\nexplicit bound on the inflow velocity is needed to estimate the threshold for\nuniqueness in the stationary Navier-Stokes equations and, in case of symmetry,\nthe stability of the obstacle immersed in the fluid flow.\n