Global small solutions to three-dimensional incompressible MHD system

TL;DR

This paper proves the global existence of small smooth solutions to the 3D incompressible MHD system by employing Lagrangian formulation and anisotropic Littlewood-Paley analysis to overcome degeneracy and anisotropic spectral challenges.

Contribution

It introduces a novel approach using Lagrangian coordinates and anisotropic analysis to establish global well-posedness for the 3D incompressible MHD system with small initial data.

Findings

Established global well-posedness for small initial data

Developed anisotropic Littlewood-Paley estimates for the velocity and pressure gradient

Handled degeneracy and anisotropic spectral properties effectively

Abstract

In this paper, we consider the global wellposedness of 3-D incompressible magneto-hydrodynamical system with small and smooth initial data. The main difficulty of the proof lies in establishing the global in time $L^1$ estimate for the velocity field due to the strong degeneracy and anisotropic spectral properties of the linearized system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the transport equation, we first write our system \eqref{B1} in the Lagrangian formulation \eqref{B11}. Then we employ anisotropic Littlewood-Paley analysis to establish the key $L^1$ in time estimates to the velocity and the gradient of the pressure in the Lagrangian coordinate. With those estimates, we prove the global wellposedness of \eqref{B11} with smooth and small initial data by using the energy method. Toward this, we will have to use the algebraic structure of \eqref{B11} in a rather crucial way. The global wellposedness of the original system \eqref{B1} then follows by a suitable change of variables together with a continuous argument. We should point out that compared with the linearized systems of 2-D MHD equations in \cite{XLZMHD1} and that of the 3-D modified MHD equations in \cite{LZ}, our linearized system \eqref{B19} here is much more degenerate, moreover, the formulation of the initial data for \eqref{B11} is more subtle than that in \cite{XLZMHD1}.