Singular velocity of the Stokes and Navier–Stokes equations near boundary in the half-space

Local behavior near the boundary is analyzed for solutions of the Stokes and Navier–Stokes equations in the half space with localized non-smooth boundary data. We construct solutions to the Stokes equations whose velocity fields are unbounded near the boundary away from the support of boundary data, although the velocity and its gradient of solutions are locally square integrable. This is an improvement compared to known results in the sense that the velocity field itself is unbounded, since previously constructed solutions were bounded near the boundary, although their normal derivatives are singular. We also establish singular solutions and their derivatives that do not belong to $L_{\mathrm{loc}}^q$ near the boundary for $q>1$. For such examples, the corresponding pressures turn out not to be locally integrable. A similar construction, via a perturbation argument, is available to the Navier–Stokes equations near the boundary as well.