Some bounds for the eigenfunctions of the Stokes problem under Navier boundary conditions in a cube
Gianni Arioli, Alessio Falocchi, Filippo Gazzola
TL;DR
This work analyzes eigenfunctions of the Stokes operator under Navier boundary conditions in the cube $\Omega=(0,\pi)^3$. It derives explicit eigenfunctions $X_{0,n,p}$, $Y_{m,0,p}$, $Z_{m,n,0}$, $V_{m,n,p}$, $W_{m,n,p}$ forming a basis and establishes sharp $L^{\infty}$ bounds and directional projections, along with integral bounds for sums of squared projections. A key contribution is the introduction of the auxiliary functions $\Upsilon(b,c)$ and $\Gamma(a,b,c)$, with a numerical maximum for $\Gamma$ around $10.91$, enabling compact, explicit constants in the spectral bounds. The proofs combine harmonic-maximum-principle arguments, boundary/interior critical-point analyses, and optimization via Lagrange multipliers, providing tools for subsequent spectral analysis in Navier-Stokes contexts. These results underpin sharper spectral estimates needed in related work and applications to stability and approximation questions.
Abstract
We prove some bounds for the eigenfunctions of the Stokes problem under Navier boundary conditions in a cube.