Solutions of Navier--Stokes--Maxwell systems in large energy spaces

Abstract: Large weak solutions to Navier--Stokes--Maxwell systems are not known to exist in their corresponding energy space in full generality. Here, we mainly focus on the three-dimensional setting of a classical incompressible Navier--Stokes--Maxwell system and --- in an effort to build solutions in the largest possible functional spaces --- prove that global solutions exist under the assumption that the initial velocity and electromagnetic fields have finite energy, and that the initial electromagnetic field is small in $\dot H^s\left({\mathbb R}^3\right)$ with $s\in \left[\frac 12,\frac 32\right)$. We also apply our method to improve known results in two dimensions by providing uniform estimates as the speed of light tends to infinity. The method of proof relies on refined energy estimates and a Gr\"onwall-like argument, along with a new maximal estimate on the heat flow in Besov spaces. The latter parabolic estimate allows us to bypass the use of the so-called Chemin--Lerner spaces altogether, which is crucial and could be of independent interest.