Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations

We construct nonzero compactly supported stationary distributional solutions $u\in L^2({\mathbb{R}}^3;{\mathbb{R}}^3)$ of the unforced incompressible Navier–Stokes equations, together with compactly supported pressures in $L^1$, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized $h$-principle: nonzero stationary solutions are strongly dense in the localized $L^p$ classes for every $1\le p<2$ and in $H^{-1}$, and weakly dense at $p=2$, while strong $L^2$ density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive $L^2$ norms, periodize the construction to ${\mathbb{T}}^3$, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.