The Hydrostatic Stokes Operator on Non‐Smooth Layers

ABSTRACT In this paper, we establish the resolvent estimates in () spaces for the resolvent problem associated with the linearized primitive equations for large‐scaled geophysical flows with mixed boundary conditions in bounded layers with non‐smooth boundaries, such as ‐class. Since one cannot expect that the resolvent problem has a solution in the Sobolev spaces (), we deal with the weak problem. The key ingredient of the proof is to estimate the surface pressure, which is governed by two‐dimensional flow. Based on the idea of Shen ( Archive for Rational Mechanics and Analysis 205, no. 2 (2012): 395–424) and Geng–Shen ( Mathematische Annalen 391, no. 1 (2025): 1467–1503), we also define the hydrostatic Stokes operator on non‐smooth layers, which is a generator of an exponentially stable analytic semigroup.