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

Abstract

By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force $f\in L_{\text{loc}}^q\cap L^{3/2},q>3$ in general open subdomains of ${\mathbb{R}}^6$. 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 ${\mathbb{R}}^4\times [0,\infty [$.

References in corpus (3)

• Partial Regularity of solutions to the Four-dimensional Navier-Stokes equations at the first blow-up time
• On partial regularity of steady-state solutions to the 6D Navier-Stokes equations
• Partially regular weak solutions of the Navier-Stokes equations in ${\mathbb{R}}^4\times [0,\infty [$