| Abstract: |
| We consider the stochastic Rosensweig system, a complex model for the dynamics of ferrofluids under the influence of a magnetic field and thermal fluctuations. The system couples the Navier--Stokes equations for the fluid velocity with equations for the fluid`s magnetization and the magnetic field, leading to a highly nonlinear SPDE with non-convex constraints.
This talk presents a compactness and convergence analysis for a bi-layer approximation of the stochastic Rosensweig system: a Bloch--Torrey regularization to handle the magnetization constraint, coupled with a Galerkin approximation for spatial discretization. This approach yields the existence of a global renormalized weak martingale solution to the system.
From a numerical analysis perspective, the core contribution is the derivation of a series of \textbf{uniform a priori estimates} that are independent of both the regularization parameter and the Galerkin dimension. These estimates---including bounds on energy, dissipation, and higher-order moments---are precisely the \textbf{stability estimates required to prove convergence of fully discrete numerical schemes}, such as finite element or spectral methods. By establishing these foundational analytical results, this work lays the necessary groundwork for the future development and rigorous error analysis of computational methods for stochastic ferrohydrodynamic flows. |
|