Global well-posedness for the two-dimensional Maxwell-Navier-Stokes equations

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}.