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

TL;DR

This paper proves the global existence and uniqueness of solutions for the 2D Maxwell-Navier-Stokes system in a near-energy space, solving an open problem in borderline functional spaces.

Contribution

It establishes the first global well-posedness result for the 2D Maxwell-Navier-Stokes equations in a borderline space close to $L^2$, using new energy estimates.

Findings

Proved global-in-time existence and uniqueness of solutions.

Developed new estimates in borderline functional spaces.

Solved an open problem posed by Masmoudi.

Abstract

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}.