Global well-posedness of 2D incompressible MHD equations without magnetic diffusion
Shijin Ding, Ronghua Pan, Yi Zhu
TL;DR
This work addresses the global well-posedness of the 2D incompressible MHD equations without magnetic diffusion in the absence of negative-Sobolev data requirements. The authors develop a novel energy framework that leverages the dispersive effects of Alfvén waves in the direction where dissipation is inactive and recasts the most challenging nonlinear term as an artificial linear term using the variables $\Omega_{\pm}$ and $G_{\pm}$. They introduce and control the energy quantities $E_0(t)$, $E_1(t)$ and $A_{\pm}(t)$ to close a priori estimates, culminating in a bootstrapping argument that yields a global unique classical solution for small initial data in $H^2(\mathbb{R}^2)$. The result advances our understanding of non-resistive MHD in two dimensions and provides a robust method for handling nonlinear couplings without magnetic diffusion, with potential applicability to other dissipative systems.
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 Alfvén 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.