  In this paper, we investigate Cauchy problem of the two-dimensional full
Maxwell-Navier-Stokes system, and prove the global-in-time existence and
uniqueness of solution in the borderline space which is very close to
$L^2$-energy space by developing the new estimate of $\sup_{j\in\mathbb Z}
2^{2j} \int_0^t \sum_{k\in\mathbb{Z}^2} \big\| \sqrt{\phi_{i,k}} u(\tau)
\big\|^2_{L^2(\mathbb{R}^2)} \text{d}\tau < \infty$. This solves the open
problem in the framework of borderline space purposed by Masmoudi in
\cite{Masmoudi-10}.
