Global small solutions to 2-D incompressible MHD system

TL;DR

This paper proves the global existence and uniqueness of small smooth solutions to the 2-D incompressible MHD system by reformulating the problem in Lagrangian coordinates and employing anisotropic Littlewood-Paley analysis.

Contribution

It introduces a novel approach using Lagrangian coordinates and anisotropic analysis to establish global wellposedness for the 2-D incompressible MHD system with small initial data.

Findings

Global wellposedness for small smooth initial data established.

Key a priori estimates derived using anisotropic Littlewood-Paley analysis.

Reformulation in Lagrangian coordinates is crucial for the proof.

Abstract

In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with small and smooth initial data. It is a coupled system between the Navier-Stokes equations and a free transport equation with an universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation in the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system \eqref{1.1} in the Lagrangian coordinates \eqref{a14}. Then we employ anisotropic Littlewood-Paley analysis to establish the key {\it a priori} $L^1(\R^+; Lip(\R^2))$ estimate to the Lagrangian velocity field $Y_t$. With this estimate, we prove the global wellposedness of \eqref{a14} with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of \eqref{a14} is crucial for the proofs to work. The global wellposedness of the original system \eqref{1.1} then follows by a suitable change of variables.