Global solutions to 3D incompressible MHD system with dissipation in only one direction

TL;DR

This paper proves the global well-posedness and stability of a 3D incompressible MHD system with dissipation in only one direction, inspired by experimental magnetic stabilization effects and overcoming significant mathematical challenges.

Contribution

It introduces novel techniques to handle derivative loss and demonstrates global solutions for a 3D MHD system with minimal dissipation, extending understanding beyond classical Navier-Stokes results.

Findings

Established global well-posedness for the 3D MHD system with one-directional dissipation.

Identified stabilizing effects of background magnetic fields in the mathematical model.

Developed innovative methods to control nonlinearities without Poincaré inequality.

Abstract

The small data global well-posedness of the 3D incompressible Navier-Stokes equations in $\mathbb R^3$ with only one-directional dissipation remains an outstanding open problem. The dissipation in just one direction, say $\partial_1^2 u$ is simply insufficient in controlling the nonlinearity in the whole space $\mathbb R^3$. The beautiful work of Paicu and Zhang \cite{ZHANG1} solved the case when the spatial domain is bounded in the $x_1$-direction by observing a crucial Poincar\'{e} type inequality. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, this paper intends to understand the global well-posedness and stability of a special 3D magnetohydrodynamic (MHD) system near a background magnetic field. The spatial domain is $\mathbb R^3$ and the velocity in this MHD system obeys the 3D Navier-Stokes with only one-directional dissipation. With no Poincar\'{e} type inequality, this problem appears to be impossible. By discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several innovative techniques to deal with the derivative loss difficulties, we are able to bound the Navier-Stokes nonlinearity and solve the desired global well-posedness and stability problem.