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

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.