Weak and mild solutions to the MHD equations and the viscoelastic Navier–Stokes equations with damping in Wiener amalgam spaces

We study the three-dimensional incompressible magnetohydrodynamic (MHD) equations and the incompressible viscoelastic Navier–Stokes equations with damping. Building on techniques developed by Bradshaw, et al. (2024) [1], we prove the existence of mild solutions in Wiener amalgam spaces that satisfy the corresponding spacetime integral bounds. In addition, we construct global-in-time local energy weak solutions in these amalgam spaces using the framework introduced by Bradshaw and Tsai (2021) [4]. As part of this construction, we also establish several properties of local energy solutions with $L_{\mathrm{uloc}}^2$ initial data, including initial and eventual regularity as well as small-large uniqueness, extending analogous results obtained for the Navier–Stokes equations by Bradshaw and Tsai (2020) [3].