On regularity of stationary solutions to the Navier-Stokes equation in 3D torus (Q1780387)

As is well known, the Navier-Stokes equations appears in several different forms. The author considers the ``stationary'' version: \[ \partial_j(v^k, v^j)= -\partial_k p+\nu\Delta v^k+ f.\tag{1} \] Only real functions are considered in author's proof. The external force \(f\) is assumed to be smooth, divergence free and with zero mean value. The last property is important because later the author applies a specific inversion formula for the Laplace operator applied to \(f\), that works only if \(f\) has this zero mean value property. To prove his theorem that every weak solution is actually smooth, the author applies the inverse of the Laplace operator to all terms of equation (1), then uses the classical Sobolev imbedding theorem, plus a few other facts of Sobolev theory, and finishes with the conclusion that the weak solution \(v\) satisfies the relation \(v\in\bigcap_{p\geq 2} H^{k+1,p}\).