Well-posedness of the Stokes-Coriolis system in the half-space over a rough surface

TL;DR

This paper establishes the well-posedness of the 3D stationary Stokes-Coriolis system in a half-space with rough boundary, using a Dirichlet to Neumann operator and analyzing low-frequency singularities.

Contribution

It introduces a novel approach to handle rough boundaries and infinite energy solutions for the Stokes-Coriolis system, including defining a Dirichlet to Neumann operator in Kato spaces.

Findings

Proves well-posedness of the system with rough bottom surfaces.

Identifies strong singularities at low tangential frequencies.

Develops a Dirichlet to Neumann operator in Kato space.

Abstract

This paper is devoted to the well-posedness of the stationary $3$d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of G\'erard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space $H^{1/2}_{uloc}$.