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Large weak solutions to Navier--Stokes--Maxwell systems are not known toexist in their corresponding energy space in full generality. Here, we mainlyfocus on the three-dimensional setting of a classical incompressibleNavier--Stokes--Maxwell system and --- in an effort to build solutions in thelargest possible functional spaces --- prove that global solutions exist underthe assumption that the initial velocity and electromagnetic fields have finiteenergy, and that the initial electromagnetic field is small in $\dotH^s\left({\mathbb R}^3\right)$ with $s\in \left[\frac 12,\frac 32\right)$. Wealso apply our method to improve known results in two dimensions by providinguniform estimates as the speed of light tends to infinity.  The method of proof relies on refined energy estimates and a Gr\"onwall-likeargument, 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-calledChemin--Lerner spaces altogether, which is crucial and could be of independentinterest.

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