We study two-dimensional characteristic free-boundary problems in ideal compressible magnetohydrodynamics. For current-vortex sheets, surface-wave effects yield derivative loss and only weak (neutral) stability; under a sufficient stability condition on the background state we obtain anisotropic weighted Sobolev energy estimates and prove local-in-time existence and nonlinear stability via a Nash-Moser scheme, confirming stabilization by strong magnetic fields against Kelvin-Helmholtz instability. For the plasma-vacuum interface, coupling hyperbolic MHD with elliptic pre-Maxwell dynamics, we establish local existence and uniqueness provided at least one magnetic field is nonzero along the initial interface.

In recent years, a more top-down approach to renormalisation for singular SPDEs has emerged within the theory of regularity structures, based on regularity structures of multi-indices. This approach adopts a geometric viewpoint, aiming to stably parametrise the solution manifold rather than the larger space of renormalised objects that typically arise in fixed-point formulations of the equation. While several works have established the construction of the renormalised data (the model) in this setting, less has been shown with regards to the corresponding solution theory since the intrinsic nature of the model leads to renormalised data that is too lean to apply Hairer’s fixed-point approach.

In this talk, I will discuss past and ongoing work with L. Broux and F. Otto addressing this issue for the Phi^4 equation in its full subcritical regime. We establish local and global well-posedness within the framework of regularity structures of multi-indices; first in a space-time periodic setting and subsequently in domains with Dirichlet boundary conditions.