Chanhong Min | Searcharxiv

arXiv2026

Flow reversal of the Stokes system with localized boundary data in the half space

We consider the unsteady Stokes system in the half-space with zero initial data and nonzero, space-time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign in the half-space. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half-space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.

math.AP↗

arXiv2023

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

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.

math.AP↗