An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system

TL;DR

This paper presents a simplified proof of the global existence and uniqueness of smooth solutions for the 2D incompressible non-resistive MHD system, leveraging energy methods, inequalities, and algebraic structures.

Contribution

It offers a more straightforward proof of a known result, emphasizing the use of classical energy methods and algebraic structures in the analysis.

Findings

Established global existence and uniqueness of solutions.

Demonstrated the effectiveness of simplified energy and $L^ extinfty$ estimates.

Provided a clearer proof framework for the 2D incompressible non-resistive MHD system.

Abstract

In this paper, we provide a much simplified proof of the main result in [Lin, Xu, Zhang, arXiv:1302.5877] concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a 2D incompressible viscous and non-resistive MHD system under the assumption that the initial data are close to some equilibrium states. Beside the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the $L^\infty$ estimates for time slices to deduce the key $L^1$ in time estimates. The latter is responsible for the global in time existence.