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

TL;DR

This paper constructs solutions to the Stokes and Navier-Stokes equations in a half-space with boundary data that lead to unbounded velocity fields near the boundary, revealing new singular behaviors not previously documented.

Contribution

It introduces explicit solutions exhibiting unbounded velocities near the boundary, advancing understanding of boundary singularities in fluid equations.

Findings

Constructed solutions with unbounded velocity near boundary

Identified solutions with non-$L^q_{loc}$ derivatives and non-integrable pressures

Extended constructions to Navier-Stokes equations near boundary

Abstract

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