Gradient estimates for Stokes equations near curved boundary

We show that the gradient of velocity is bounded in $L^q$ up to the boundary for weak solutions of Stokes equations with Navier BC and curved boundary, whose correctness has already been shown near the flat boundary by us in the previous work [CPAA 2024]. We also prove the corresponding high order derivative estimates near curved boundary. For the infinite slip length case ($\alpha=0$ in the Navier BC), we find the regularity behavior under curved boundary is different from that under flat boundary.

Su Liang is a PhD student in Mathematics at UBC.