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

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.

On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

“…The purpose of this section is to analyze the solvability of (1.5) and to obtain explicit bounds of its solution in terms of the known data (that is, in terms of h and of the geometric parameters of Ω). We follow closely the procedure described in [14]: we fix real numbers a, b, c such that…”

“…Problem (1.5) has been extensively studied in the past, not only because of its applicability in fluid mechanics (see the works by Ladyzhenskaya & Solonnikov [30,31], Bogovskii [4] and the book by Galdi [17,Section III.3]), but also due to its purely mathematical interest and connection with the Calderón-Zygmund theory of singular integrals, see the book by Acosta & Durán [2]. Our construction invokes the method described in [14]: by inverting the trace operator, an extension of h (not necessarily solenoidal) is determined; then the Bogovskii problem with the resulting divergence is studied to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cut-off functions, an explicit estimate of v 0 is given. As an application of our results, in Section 4 we study the fluid forces exerted over K. The lift force, understood as the force component that is perpendicular to the oncoming stream, plays a fundamental role in aerodynamics (where it must be maximized in order counter the force of gravity acting over the aircraft [1,Chapter 3]) and in civil engineering (where it must be minimized in order to avoid instabilities of structures such as suspension bridges or skyscrapers [19]).…”