Global well-posedness of 2D incompressible MHD equations without magnetic diffusion

TL;DR

This paper proves the global existence and uniqueness of solutions to 2D incompressible MHD equations without magnetic diffusion for small initial data in H^2, using dispersive effects of Alfvén waves to control nonlinearities.

Contribution

It establishes the first global well-posedness result for 2D incompressible MHD without magnetic diffusion without relying on negative Sobolev space assumptions.

Findings

Global unique solutions exist for small initial perturbations in H^2.

Dispersive effects of Alfvén waves are used to control nonlinear terms.

The approach extends previous results by removing the need for negative Sobolev space assumptions.

Abstract

In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close to equilibrium states with nontrivial magnetic field, and the perturbation is small in some suitable spaces, say for instance, the Sobolev spaces with negative exponents. It leads to an interesting open question: Can one establish the global existence of classical solutions without the extra help from Sobolev spaces with negative exponents like its counterparts of ideal MHD ( i.e. without viscosity and magnetic diffusion), and fully dissipative MHD (i.e. with both viscosity and magnetic diffusion)? This paper offers an affirmative answer to this question. In fact, we will establish the existence of a global unique solution for initial perturbations being small in $H^2(\mathbb{R}^2)$. The key idea is further exploring the structure of system, using dispersive effects of Alfv\'{e}n waves in the direction which is transversal to the dissipation favorable direction. This motivates our key strategy to treat the wildest nonlinear terms as an artificial linear term. These observations help us to construct some interesting quantities which improves the nonlinearity order for the wildest terms, and to control them by terms with better properties.