The energy equality and regularity results for the stationary Navier–Stokes equations on the exterior of a rotating obstacle

Abstract

Consider the stationary Navier–Stokes equations on the exterior of a rotating body, which is also moving in the direction of the axis of rotation with constant velocity −ke1. For every external force f=divF, F∈L2(Ω), the existence of a weak solution u satisfying finite Dirichlet integral, i.e., ∇u∈L2(Ω), can be obtained by means of the classical Galerkin method. We first prove that the weak solution u satisfies the additional regularity property u−ke1∈L4(Ω) without any condition on F except for F∈L2(Ω). This regularity result is enough to ensure the validity of the generalized energy equality. Using these, we therefore obtain the uniqueness for weak solutions under some smallness condition on the data.