Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

TL;DR

This paper provides explicit global estimates for solutions to the Stokes system with localized boundary flux in the half-space, showing finite energy but unbounded derivatives, and extends these results to Navier-Stokes solutions.

Contribution

It introduces explicit pointwise estimates for solutions with boundary flux, demonstrating finite energy yet unbounded derivatives, and constructs Navier-Stokes solutions exhibiting similar behavior.

Findings

Solutions have finite global energy but unbounded derivatives near the boundary.

Explicit pointwise estimates of the Green tensor are used to analyze solution behavior.

Normal derivatives tend to infinity in regions depending on the flux configuration.

Abstract

For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a solution generated by a compactly supported, H\"older continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in \cite{Green} (arXiv:2011.00134). We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.