Nonlinear boundary layers for rotating fluids

TL;DR

This paper studies the complex behavior of rotating incompressible flows near uneven bottoms, establishing well-posedness and asymptotic properties of the nonlinear PDE system governing the flow, extending previous linear and periodic analyses.

Contribution

It proves well-posedness and asymptotic behavior for a nonlinear PDE system modeling rotating flows over variable bottoms, extending prior linear and periodic case results.

Findings

Established well-posedness of the nonlinear PDE system.

Analyzed the asymptotic behavior of solutions away from the boundary.

Extended previous results to non-periodic bottom variations.

Abstract

We investigate the behavior of rotating incompressible flows near a non-flat horizontal bottom. In the flat case, the velocity profile is given explicitly by a simple linear ODE. When bottom variations are taken into account, it is governed by a nonlinear PDE system, with far less obvious mathematical properties. We establish the well-posedness of this system and the asymptotic behavior of the solution away from the boundary. In the course of the proof, we investigate in particular the action of pseudo-differential operators in non-localized Sobolev spaces. Our results extend the older paper [18], restricted to periodic variations of the bottom. It ponders on the recent linear analysis carried in [14].