Local and global regularity for the Stokes and Navier-Stokes equations with the localized boundary data in the half-space

TL;DR

This paper investigates the local regularity and singular behaviors of solutions to the Stokes and Navier-Stokes equations with localized boundary data in a half-space, revealing conditions under which solutions are smooth or exhibit boundary singularities.

Contribution

It provides new insights into how boundary data's smoothness and components influence solution regularity and singularities for the Stokes and Navier-Stokes systems in half-space.

Findings

Solutions are smooth near the boundary if boundary data are temporally smooth.

Normal boundary data absence can lead to second normal derivatives becoming singular.

Localized boundary data can produce solutions with unbounded second normal derivatives near the boundary.

Abstract

We study the Stokes system with the localized boundary data in the half-space. We are concerned with the local regularity of its solution near the boundary away from the support of the given boundary data which are product forms of each spatial variable and the temporal variable. We first show that if the boundary data are smooth in time, the corresponding solutions are also smooth in space and time near the boundary, even if the boundary data are only spatially integrable. Secondly, if the normal component of the boundary data is absent, we are able to construct a solution such that its second normal derivatives of the tangential components become singular near the boundary. Perturbation argument enables us to construct solutions of the Navier-Stokes equations with similar singular behaviors near the boundary in the half-space as the case of Stokes system. Lastly, we provide specific types of the localized boundary data to obtain the pointwise bounds of the solutions up to second derivatives. It turns out that such solutions are globally strong, and the second normal derivatives are, however, unbounded near the boundary. These results can be compared to the previous works in which only the normal component is present. In fact, the temporally non-smooth tangential boundary data can also cause spatially singular behaviors near the boundary, although such behaviors are milder than those caused by the normal boundary data of the same type.