The adjoint problem in the presence of a deformed surface: the example of the Rosensweig instability on magnetic fluids

TL;DR

This paper investigates the mathematical challenges of defining an adjoint operator for the nonlinearities caused by a deformed surface in the Rosensweig instability of magnetic fluids, with implications for pattern formation analysis.

Contribution

It presents a novel analysis of the adjoint problem in systems with a deformed surface, specifically applied to the Rosensweig instability, highlighting the unique nonlinearities involved.

Findings

Illustration of the difficulties in defining an adjoint operator for deformed surface nonlinearities.

Discussion of implications for amplitude equations in pattern-forming systems.

Insights into the mathematical structure of the Rosensweig instability.

Abstract

The Rosensweig instability is the phenomenon that above a certain threshold of a vertical magnetic field peaks appear on the free surface of a horizontal layer of magnetic fluid. In contrast to almost all classical hydrodynamical systems, the nonlinearities of the Rosensweig instability are entirely triggered by the properties of a deformed and a priori unknown surface. The resulting problems in defining an adjoint operator for such nonlinearities are illustrated. The implications concerning amplitude equations for pattern forming systems with a deformed surface are discussed.