\(W^{2,p}\)-Estimates of the Stokes System with Traction Boundary Conditions

The article deals with the 3D stationary Stokes system under traction boundary conditions, in interior and exterior domains. In the interior domain case, we obtain solutions with \(W^{2,p}\)-regular velocity and \(W^{1,p}\)-regular pressure globally in the domain, under suitable assumptions on the data. In the exterior domain case we construct two solutions classes, both of them consisting of functions which are \(W^{2,p}\)–\(W^{1,p}\)-regular in any vicinity of the boundary, with \(p \in (1, \infty )\) determined by the assumptions on the data. In addition the velocity part of these solutions is \(L^s\)-integrable near infinity, for some \(s>3\), provided that the right-hand side of the Stokes system is \(L^p\)-integrable near infinity for some \(p<3/2\). Moreover, the velocity part of the solutions in one of the two classes satisfies a zero flux condition on the boundary, whereas the pressure part of the solutions in the other class is \(L^s\)-integrable near infinity, for some \(s > 3/2\). The two solution classes are also uniqueness classes, one related to a zero flux condition for the velocity, the other one to decay of the pressure at infinity. This result confirms a conjecture by T. Hishida (University of Nagoya).