The magnetization ripple: a nonlocal stochastic PDE perspective

The magnetization ripple is a microstructure formed by the magnetization in a thin-film ferromagnet. It is triggered by the random orientation of the grains in the poly-crystalline material. In an approximation of the micromagnetic model, which is sketched in this paper, this leads to a nonlocal (and strongly anisotropic) elliptic equation in two dimensions with white noise as a right hand side. However, like in singular Stochastic PDE, this right hand side is too rough for the non-linearity in the equation. In order to develop a small-date well-posedness theory, we take inspiration from the recent rough-path approach to singular SPDE. To this aim, we develop a Schauder theory for the non-standard symbol $|k_1|^3+k_2^2$.