Stationary Navier-Stokes equations under inhomogeneous boundary conditions in 3D exterior domains
This interesting recent work investigates the existence of weak solutions to the stationary Navier-Stokes equations in a 3D exterior domain \(\Omega\). The boundary \(\partial \Omega\) is a disjoint union of finitely many smooth, compact surfaces \(\{\Gamma_i\}_{i=1}^L\), and the boundary condition for \(v\) is that \(v|_{\Gamma_i}=\beta_i \in C^\infty(\Gamma_i)\) for each \(i \in \{0,1,\ldots, L\}\). This question is related to a famous open problem of Leray. In the paper, it is proved that if the \(L^3\)-norm of the harmonic part of \(\sum_{i=0}^L\int_{\Gamma_i}\beta_i \cdot\nu\,\mathrm{d}S\) is smaller than a universal constant times the viscosity, then there exists a weak solution to the stationary Navier-Stokes equations in the \(W^{1,2}\)-closure of smooth, divergence-free vector fields on \(\Omega\). The proof is based on the authors' works [J. Funct. Anal. 281, No. 8, Article ID 109144 (2021; Zbl 1472.35152); ``A characterization of harmonic \(L^r\)-vector fields in three dimensional exterior domains'', Preprint] on Hodge decomposition of vector fields on exterior domains: they first find a background solution, i.e., a divergence-free \(W^{1,2}\)-extension of the boundary data, and then apply a Galerkin approximation argument to the difference between the true solution and this background solution.
scientific article; zbMATH DE number 7384561
scientific article