Local regularity near boundary for the Stokes and Navier-Stokes equations

TL;DR

This paper investigates the local regularity of solutions to the Stokes and Navier-Stokes equations near boundaries, revealing singularities in some solutions and criteria for boundedness and regularity, with examples showing optimal regularity.

Contribution

It constructs solutions with boundary singularities, establishes criteria for bounded gradients, and provides examples demonstrating optimal regularity near boundaries.

Findings

Constructed solutions with singular normal derivatives near boundary.

Established criteria for boundedness and Hölder continuity of solutions.

Provided examples illustrating optimal boundary regularity.

Abstract

We are concerned with local regularity of the solutions for the Stokes and Navier-Stokes equations near boundary. Firstly, we construct a bounded solution but its normal derivatives are singular in any $L^p$ with $1<p$ locally near boundary. On the other hand, we present criteria of solutions of the Stokes equations near boundary to imply that the gradients of solutions are bounded (in fact, even further H\"{o}lder continuous). Finally, we provide examples of solutions whose local regularity near boundary is optimal.