Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6

Abstract

AbstractBy using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force $$f \in L_{\text {loc}}^q \cap L^{3/2}, q>3$$ f ∈ L loc q ∩ L 3 / 2 , q > 3 in general open subdomains of $${\mathbb {R}}^6$$ R 6 . These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in $${\mathbb {R}}^4 \times [0,\infty [$$ R 4 × [ 0 , ∞ [ .